Political Anaylisis ( Rationality behaviour instition exam) I have an exam tomorrow There will be two parts and each one has 30 min time limit. ( please check the attachment) need someone who have kno

Political Anaylisis ( Rationality behaviour instition exam) I have an exam tomorrow There will be two parts and each one has 30 min time limit. ( please check the attachment) need someone who have kno
Exercises (1) • In November 2008, a couple of weeks after the election of Barack Obama, Hillary Clinton was offered the job of Secretary of State of the United States. I • She faced the following trade -off: a) joining the new administration, in perhaps the highest profile cabinet position. b) continuing in the Senate, an option that promised less power (she would still be only one of a hundred) but greater autonomy. Moreover taking an administration job would preclude a primary challenge against Barack Obama in 2012, Hillary Clinton faced three Possibilities: C ) Remain in the Congress and not win the Presidency in 2012 , P ) Remain in the Congress and win the Presidency in 2012 S) Join the administration as secretary of state . If the probability of winning the White House in 2012 if she had remained in the Senate is p , then use an expected utility argument to determine the smallest p that would have induced Clinton to remain in the Senate in order to run in 2012. Exercises (1) • As H.Clinton choosed S it is reasonable to assume the following preference ordering : P>S>C • As she left the Senate then S> p (P) +(1 -p) (C) • S>p(P)+C -p(C) • S -C>p(P -C); S -C/P -C>p Therefore the smallest p that could induce H.C. to stay in the Congress was S -C/P -C=p Exercises (2) • Consider the following two sets of individuals (“Societies”) and their group preference rankings, aggregated using the same voting rule 1) Individual preferences a: (x>y>z>w) b: (y>z>w>x) c: (z>w>x>y) Group preferences (x>y), (z>x), (w>x), (y>w), (y>z), (z>w) 2) Individual preferences a: (y>z>x>w) b: (y>w>x>z) c: (y>w>z>x) Group preferences (y>x), (y>w), (z>y), (x>w),(z>x),(z>w) Which of the Arrow’s Conditions (P, D, I or transitivity ) is violated by the group preferences ? Exercises (2) • Society 1 violates “ transitivity ” (x >y>w>x ) ; (x>y>z >x ); (x>y>z>w>x ) • The rule that aggregates Society 2’s group references violates the (P) Pareto principle (unanimity) because all three members of the society individually prefer y to z, yet z >y. • The voting rule violates condition I (independence of irrelevant alternatives) because in both societies the three individuals share the same ordinal rankings over x and w (x>w, w>x and w>x, respectively) yet w>x in society 1 and x>w in society 2. Exercises (3) • May’s theorem suggests that any deviation from majority rule must be justified by a reasonable departure from one of 4 conditions : U, A, N, M. • Condition U (universal domain). All complete and transitive preference orderings over alternatives are admittable . • Condition A (Anonymity ). Social preferences depend only on the collection of individual preferences , not on who has which preference . • Condition N (Neutrality ). Interchanging the ranks of alternatives j and k in each group member’s preference ordering has the effect of interchanging the ranks of j and k in the group preference ordering . • Condition M ( Monotonicity ). If an alternative j beats or ties another alternative k — that is , j R G k — and j rises some group member’s preferences from below k to the same or a higher rank than k, then j now strictly beats k — that is , jP G k . Exercises (3) • For each of the following cases explain which of these conditions is violated by the electoral rule and suggest a possible justification . 1) Amendment of U.S. Constitution requires 2/3 majority in each Chamber. 2) IMF uses a system of weighted voting where weights are determined by contributions to IMF operating funds. US holds a veto in some circumstances . 3) Guilty Verdict in a (U.S.) criminal case requires unanimity or almost unanimity . 4) French President is elected under a two -stage majority rules . ( run -off vote) Exercises (3) 1) Amendment of U.S. Constitution requires 2/3 majority in each Chamber. The requirement of 2/3 majorities in Congress to propose an amendment is a violation of neutrality , because the status quo (no proposal) is given preferential treatment in the voting procedure . For example , if exactly 60% of Congress people in each house prefer the status quo, then the status quo wins . However, if we reverse the preferences so that now 60% of Congress people in each house prefer an amendment, the status quo nonetheless still wins out. In general, any voting rule which privileges the status quo (or some other outcome) violates neutrality; however, in many instances this departure from majority rule is justified as an attempt to make extraordinary changes difficult . Exercises (3) 2) IMF uses a system of weighted voting where weights are determined by contributions to IMF operating funds. US holds a veto in some circumstances . The IMF’s voting procedure violates anonymity in two separate ways. a) weighted voting privileges some members over others, so redistributing the preference orderings among the individuals can change the outcome. b) granting the US a veto in certain circumstances means that an identical collection of preferences but permuted among the members differently might lead to different outcomes e.g . if in one permutation 60% of members preferred some outcome a including the US, and if in another 60% of members preferred some outcome a and the US opposed it. In the case of the IMF, the weighting of voting rights is based on a normative argument (those who contribute more to IMF activities deserve more say) and a political justification (integrating a superpower into an international organization sometimes requires granting special rights and exceptions to that superpower, as in the UN Security Council ). Exercises (3) • Guilty Verdict in a (U.S.) criminal case requires unanimity or almost unanimity . • This is a violation of neutrality, because one outcome (acquittal) is given a privileged status relative to the others. Consider a case where 9 jurors favor an acquittal and 3 a guilty verdict. Clearly, acquittal wins out, but in most systems, acquittal will still prevail if the opposite set of preferences are held. • The usual justifications for this rule is that there should be consensus or near consensus among jurors before meting out life -altering criminal convictions and punishments, and that innocence should be heavily presumed and guilt only determined by overwhelmingly persuasive evidence. Exercises (3) 4) French President is elected under a two -stage majority rule . ( run -off vote) This rule in fact violates the monotonicity condition. Suppose that the following percentages of voters hold these preferences over three candidates a; b; c: a > b > c (40%); b > c > a (31%); c > a > b ( 29%). In a two -round election, a and b would win round 1, and then a would beat b in round 2 (assuming voters are sincere). Now suppose, that 3% of the of b > c > a voters change their preferences to a > b > c. In the two -round election, a and c win round 1, and then c defeats a in round 2. In other words, an increase in support for a has lead to a’s defeat. Two -stage elections are often supported because they guarantee that a winning candidate secures a majority of voters (thus , arguably enhancing the legitimacy of the eventual winner), allow voters to support smaller parties in the first stage (up to a point), and promote moderation in the second stage by creating the centripetal tendencies of two -candidate competition . Exercises (4) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 1) Write down the majority preference relation for this profile of preferences (pairwise comparison ) 2) Does Black’s Median voter Theorem support a prediction about which policy will be chosen if the group uses simple majority rule ? Why or why not 3) Suppose that the group is going to use a voting agenda v= (y, x, z), namely first y versus x then etc. Which is the outcome ? What about if the agenda is v’=( z,x,y ) and v’’ = ( z,y,x ) ? x y z 1 2 3 u y x z 2 3 1 z x y 3 1 2 z y x 1 2 3 x z y 1 3 2 y x z 1 2 3 u Exercises (4) 1) The group preferences over each pair of outcomes using majority rule are: xPGy , yPGz , and zPGx . 2) Black’s Median Voter Theorem does not support a prediction about which policy the group will choose because the preferences do not satisfy single -peakedness . Demonstrating this lack of single -peakedness graphically requires drawing six graphs. A faster check is to note that none of the three outcomes are agreed upon by the group to be `not worst’. One interesting thing to note is that the preferences could violate single – peakedness and still yield a coherent outcome e.g. with the following preferences: 1: x>y>z, 2: z>y>x , 3: z>x>y . These prefeences violate single -peakedness but still yield transitive social preferences . The preferences over x, y and z still satisfy Sen’s value -restriction criterion because z is agreed by all to be ‘ not middling ‘. 3) Under agenda v, z is the winner (x beats y then z beats x). Under agenda v’, y is the winner (z beats x then y beats z). Under agenda v’’, x is the winner (y beats z then x beats y). Exercises (5) • Downs takes politicians to be interested only in winning office. Does a different result other than convergence arise when politicians have strong policy preferences of their own ? Under which circumstances ? Exercises (6) 1) Suppose that strategy c3 is unavailable to Mr III. Solve the Game Exercises (6) 2 ) Suppose that strategy c3 is available but Mr I can no longer play a1 Solve the game. Exercises (6) 1) Final outcome If strategy c3 is unavailable to Mr. III, then the final outcome will be (a1; c2; b2) with payoffs (8; 9; 3) for I, II and III respectively. Exercises (6) 1) Final outcome If strategy c3 is unavailable to Mr. III, then the final outcome will be (a1; c2; b2) with payoffs (8; 9; 3) for I, II and III respectively. Exercises (6) 1) Final outcome If strategy c3 is unavailable to Mr. III, then the final outcome will be (a1; c2; b2) with payoffs (8; 9; 3) for I, II and III respectively. Exercises (6) 2 ) If strategy a1 is unavailable to Mr. I, then the final outcome will be (a2; c2; b1) with payoffs (7; 7; 6) going to I, II and III respectively. Exercises (6) 2 ) If strategy a1 is unavailable to Mr. I, then the final outcome will be (a2; c2; b1) with payoffs (7; 7; 6) going to I, II and III respectively. Exercises (6) 2 ) If strategy a1 is unavailable to Mr. I, then the final outcome will be (a2; c2; b1) with payoffs (7; 7; 6) going to I, II and III respectively. Exercises (7)1) Imagine that in a committee th ere are three legislators ( or three groups of legislators equal in size) A, B, C. They have to approve by majority rule a policy. There are three alternatives : X, Y, Z . The utiles given by each policy is described in the table and the utile of the status quo is 0 for all legislators. Bill(voted against the Status Quo =0 ) Legislato rs X Y Z A 3 -1 -1 B -1 3 -1 C -1 -1 3 a) Wh ich is the overall utility level the legislators ( no difference between them) enjoy if the bills are voted separately and independently ? (1): …………………………………………………………………………….. b) Imagine that each legislators knows the utility he can get from each alternative before starting voting. He /She considers the advantages of forming a coalition of two legislators but he/she does not know if he/she will be member of this coalition. How m any winning coalitions of two legislators can be formed (1) ?………………… Which is each legislator’s expected utility ? (2) ………………………….. 1) 0 2) 3 : AB, AC, CB 3) Imagine actor A : payoff of AB= (3 -1) ; payoff of AC=(3 -1), payoff of CB =( -2) Expected utility = 1/3*2+1/3*2 -1/3*2 = 2/3 Exercises (7) Solutions Exercises (8) 1) Imagine that in a committee there are three legislators ( or three groups of legislators equal in size) A, B, C. They have to approve by majority rule a policy. There are three alternatives : X, Y, Z. The utiles given by each policy is described in the tab le and the utile of the status quo is 0 for all legislators. Bill(voted against the Status Quo=0) Legislators X Y Z A 3 -2 -1 B -1 3 -2 C -2 -1 3 a) Which is the overall utility level the legislators ( no difference between them) enjoy if the bills are voted separately and independently ? (1): …………………………………………………………………………….. b) Imagine that each legislators knows the utility he can get from each alternati ve before starting voting. He /She considers the advantages of forming a coalition of two legislators but he/she does not know if he/she will be member of this coalition. How many winning coalitions of two legislators can be formed ( 1) ?………………… Which is each legislator’s expected utility ? (2) ………………………….. c) Once a coalition is formed is it stable ? Why yes or no 1) 0 2) 3 : AB, AC, CB 3) actor A : payoff of AB= ( 3 -2) ; payoff of AC=(3 -1), payoff of CB =( -1) Expected utility = 1/3*1+1/3*2 -1/3*1 = 2/3 Actor B: payoff of AB= ( 3 -1) ; payoff of AC =( -1), payoff of CB =(3 -2) Expected utility = 1/3*2 -1/3*1 +1/3*1 = 2/3 Actor C: payoff of AB= (-1) ; payoff of AC=( 3 -2), payoff of CB =( 3 -1) Expected utility = -1/3*1+1/3*1 + 1/3*2 = 2/3. 4) No coalition is stable. For instance for A AC>AB but but for C CB > AC and for B AB> CB; Cycle!! Exercises (8) Solutions Exercises (9) • For each of the following societies: 1) State whether the preferences satisfy Sen’s value – restriction criterion 2) If not, identify the tuple(s) of preferences that violate value – restricted preferences 3) Assuming majority rule, are the societies’ preferences transitive ? Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • Society 1: y is agreed to be not worst among xyz ; x is agreed to be not best among xyw ; no option satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . Using the method of majority rule , the only preference cycle occurs between x, z and w. Society 1 1: y>x>z >w 2: w> y>x>z 3: z>y> w> x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • Society 1: y is agreed to be not worst among xyz; x is agreed to be not best among xyw ; no option satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . Using the method of majority rule , the only preference cycle occurs between x, z and w. Society 1 1: y>x> z> w 2: w>y>x >z 3: z> y>w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • Society 1: y is agreed to be not worst among xyz; x is agreed to be not best among xyw ; no option satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . Using the method of majority rule , the only preference cycle occurs between x, z and w. Society 1 1: y> x>z>w 2: w> y> x>z 3: z> y> w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • Society 1: y is agreed to be not worst among xyz; x is agreed to be not best among xyw ; no option satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . Using the method of majority rule , the only preference cycle occurs between x, z and w. Society 1 1: y> x> z>w 2: w>y> x> z 3: z>y>w >x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz ; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y> w >z>x 2: w >x>y>z 3: z> w >y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w> z >x 2: w>x>y> z 3: z >w>y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y >w>z>x 2: w>x> y >z 3: z>w> y >x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w>z> x 2: w> x >y>z 3: z>w>y> x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz ; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y> w >z>x 2: z>x>y> w 3: x>y> w >z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w> z >x 2: z >x>y>w 3: x>y>w> z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y >w>z>x 2: z>x> y >w 3: x> y >w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w>z> x 2: z> x >y>w 3: x >y>w>z Exercises (10) • A legislature is going to vote on a policy that is well represented by a single -issue dimension, on a scale of zero to one . • The initial policy proposal will be supplied by a committee ( whose position on the dimension is supposed to “vary” ) to a legislature with median voter’s ideal point M ( 0.45) . The Status quo is at point SQ (0.25). Exercises (10) • Draw a line showing the equilibrium outcomes for any committee ideal point when the proposal is considered under the closed rule and when it is considered under the open rule. • Do the same exercise for both rules assuming that the legislature operates on the principal of zero -based budgeting (no decision  ZB=0) Exercises (10) • What is the impact of zero – based budgeting under a closed rule? And under the open rule ? Exercises (10) • Under a closed rule using the status quo as the `reversion’ or `default’ option, the kinks in the line occur at SQ = 0.25 and M +|M – SQ| = 0.65. • Exercises (10) • In between those two kinks, the Committee can propose and secure its ideal point in equilibrium, because the median voter prefers that proposal to the status quo . • If the committee’s ideal point is to the left of the first kink, then the legislative median will resist any attempt to move the policy to the left; to the right of the kink, the committee proposes an outcome it prefers which leaves the legislative median as well off as it is under the status quo. Exercises (10) • Under zero -based budgeting, the only kink in the line occurs at M + |M -ZB| = 0.90. • From ZB to M + | M -ZB|the committee can achieve always its ideal point. Exercises (10) • Under an open rule using the status quo as the default option, the break in the line occurs at the Committee’s point of indifference between the SQ and M, which is halfway between these points at 0.35 . • The committee will keep the gates closed when its ideal point is closer to the status quo than to the median because when it opens the gates, the full legislature will adopt M in equilibrium. Exercises (10) • Under zero -based budgeting (and open rule) this point of indifference occurs at the point 0.45/2 = 0.225 , the point where the 0, the de facto status quo now, is equivalent to M from the committee’s perspective. Exercises (10) • Under a closed rule, the committee is able to secure outcomes closer to its ideal point when zero -based budgeting is employed. When the committee has extreme preferences relative to the median voter in the legislature , the equilibrium outcomes are therefore more extreme than they would have been under an ordinary status quo rule . Exercises (10) • Under an open rule, the effects of zero -based budgeting on committee power are ambiguous . For some ranges of the line the committee’s best outcome is less – preferred than the equilibrium under the status quo rule (e.g. when the committee ideal point is near the SQ point), but for some ideal points the equilibrium outcome is preferred to the equilibrium under the status quo rule (e.g. when the committee ideal point is near zero). Therefore, in some instances the outcomes are further from the median ideal point than they would be under status quo budgeting, but in others the outcomes are closer to the median ideal point than they would be under status -quo budgeting. Exercises ( 11) • Political Actors Xa , Xb , Xc are located in a two dimensional policy space. Each actor would like to change the status quo SQ. All proposals are pitted against SQ in a final voting. Write down on the picture the final outcome (or the winset ) when (1) • a) Decision rule is unanimity • b) Decision rule is majority • c) Decision rule is majority, one dimension at a time, in some pre -set order. Exercises ( 11) Unanimity Unanimity Core Majority rule Majority , one dimension at a time, in some pre – set order. Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 1) Suppose that the group is going to use a voting agenda v= (y, x, z) or v’=( z,x,y ) or v’’ = ( z,y,x ) It is the last round. Will any player wish to misrepresent her true preferences ? 2) With the agenda v= (y, x, z) a) Determine what happens in the final round depending on whether y or x wins the first round b)Can player 2 ever do better by supporting (against her true preferences ) x in the first round c) What about 3 voting against x ? d) Should 1 misrepresent his preferences by playing y in round 1? Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 1) Suppose that the group is going to use a voting agenda v= (y, x, z) or v’=( z,x,y ) or v’’ = ( z,y,x ) It is the last round. Will any player wish to misrepresent her true preferences ? It is never optimal to misrepresent one’s preferences in a single round of majority rule voting over two outcomes . Any vote against one’s most – preferred outcome can only increase the support for the less -preferred option , possibly leading to its victory. Therefore, the final round of our agenda procedure here will never feature strategic voting . Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 2) With the agenda v= (y, x, z) a) Determine what happens in the final round depending on whether y or x wins the first round If y wins in the first round then y will win in the last one If x wins in the first round then z will win in the last one Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 2) With the agenda v= (y, x, z) b)Can player 2 ever do better by supporting (against her true preferences ) x in the first round No, the final outcome (z) would not change c) What about 3 voting against x ? Player 3 would not wish to vote for y rather than x because this would lead to a victory for y in the first round and in the last round. d) Should 1 misrepresent his preferences by playing y in round 1? Player 1 will wish to vote strategically in round 1 by voting for y. This leads to y being the overall winner, and player 1 prefers y to z . No player has an incentive to deviate from their strategy Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 2) With the agenda v’=( z,x,y ) a) Determine what happens in the final round depending on whether z or x wins the first round b)Can player 2 ever do better by supporting (against her true preferences ) z in the first round c) What about 3 voting against z ? d) Should 1 misrepresent his preferences by playing z in round 1? Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 2) With the agenda v’=( z,x,y ) a) Determine what happens in the final round depending on whether z or x wins the first round If z wins in the first round then y wins in the last one If x wins in the first round then x wins in the last one b)Can player 2 ever do better by supporting (against her true preferences ) x in the first round No as she achieves the worst outcome (x) in the last round c) What about 3 voting against z ? Player 3 has an incentive to misrepresent her vote in the first round by voting for x rather than z. The outcome under honest voting is y, however if 3 misrepresents her vote in this way , the final outcome is x (better than y). (Stable equilibrium) d) Should 1 misrepresent his preferences by playing z in round 1? No. She will achieve y as final outcome that is worse than x Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 2) With the agenda v’’ = ( z,y,x ) a) Determine what happens in the final round depending on whether z or x wins the first round b)Can player 2 ever do better by supporting (against her true preferences ) z in the first round c) What about 3 voting against z ? d) Should 1 misrepresent his preferences by playing z in round 1? Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 2) With the agenda v’’=( z,y,x ) a) Determine what happens in the final round depending on whether z or x wins the first round If z wins in the first round then z wins in the last one If y wins in the first round then x wins in the last one b)Can player 2 ever do better by supporting (against her true preferences ) z in the first round ? player 2 has an incentive to misrepresent his vote in the first round by voting for z rather than y . The outcome under honest voting is x, however if 2 misrepresents his vote in this way, the final outcome is z. (stable equilibrium) c) What about 3 voting against z ? Player 3 does not have an incentive to misrepresent her vote. She would achieve x in the final round that is worse than z. d) Should 1 misrepresent his preferences by playing z in round 1? No. She will achieve z as final outcome that is worse than x Exercises ( 12) Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 2) With the agenda v= (y, x, z) If y wins in the first round then y will win in the last one If x wins in the fitst round then z will win in the last one Whether two players might be able to form a strategic voting coalition ? With 1 voting strategically (which is an equilibrium ) 2 has no desire to change his behavior. Would it be possible for 1 and 3 (who secure their 2nd and 3rd most -preferred outcomes respectively) to team up and secure a better outcome? It would be possible, but not plausible. 1 and 3 agree to both vote for x in round 1, and then both vote for x in round 2. This eliminates y which 3 hates and 1 dislikes compared to x , and gives each a better outcome than y, which is the proposed equilibrium under strategic voting . However player 3 will be voting against her interest in the final round by voting for x over z. Thus, we might think that she would be tempted to renege on the deal with 1 and get her most -preferred outcome z . In the absence of some way of preventing herself from voting for z in the final round, 1 may not find 3’s promises very credible, and may prefer to stick with the strategic voting as described above . Exercises ( 13) • In the following one -dimension policy space describing the policy positions in a legislative arena, the Committee median voter’s ideal point is C and the Floor median voter is F. The committee has a gatekeeping power (It is the only actor who can propose a policy change). Given the location of the status quo SQ: 1) Identify the policy outcome if the committee enjoys a closed rule 2) Identify the policy outcome if the committee enjoys an open rule 3) Which rule is more convenient for the Floor (the Parliament) ? 4) Where should the Status quo be located to have a policy change and a different answer to the previous question ? Exercises ( 13) • In the following one -dimension policy space describing the policy positions in a legislative arena, the Committee median voter’s ideal point is C and the Floor median voter is F. The committee has a gatekeeping power (It is the only actor who can propose a policy change). Given the location of the status quo SQ: 1) Identify the policy outcome if the committee enjoys a closed rule Exercises ( 13) • In the following one -dimension policy space describing the policy positions in a legislative arena, the Committee median voter’s ideal point is C and the Floor median voter is F. The committee has a gatekeeping power (It is the only actor who can propose a policy change). Given the location of the status quo SQ: 2) Identify the policy outcome if the committee enjoys an open rule Exercises ( 13) • In the following one -dimension policy space describing the policy positions in a legislative arena, the Committee median voter’s ideal point is C and the Floor median voter is F. The committee has a gatekeeping power (It is the only actor who can propose a policy change). Given the location of the status quo SQ: 3) Which rule is more convenient for the Floor (the Parliament) ? Closed Rule Exercises ( 13) • In the following one -dimension policy space describing the policy positions in a legislative arena, the Committee median voter’s ideal point is C and the Floor median voter is F. The committee has a gatekeeping power (It is the only actor who can propose a policy change). Given the location of the status quo SQ: 4) Where should the Status quo be located to have a policy change and a different answer to the previous question ? Exercises ( 14) • A seven – member governance committee of a university is charged with allocating funds for post -doc fellowships. Last years committee allocated 3000$ and this is status quo level of funding. • A special subcommittee is appointed to research and then propose to the full governance committee a total value of all fellowships, between 0 and 12000 $. • University rules state that the subcommittee may bring a proposal before the full committee under open or closed rule. • All subcommittee and full committee decisions are made by majority rule. • The preferences of each members of the seven -member committee are single peaked: B and E: 0 ; A: 1500 ; C: 6000 ; F: 7500 ; G:10000; D: 12000 The subcommittee is composed of F, G, D . a) What is the subcommittee’s most -preferred level of funding ? b) Assuming the closed rule ( open rule) what is the subcommittee’s proposal and what is the outcome? Why ? c) Assume that the governance committee follows “zero -based” budgeting” in which each year’s allocation is initially set to zero. What will the outcome of voting be under closed rule (open rule) ? Exercises ( 14) • A seven – member governance committee of a university is charged with allocating funds for post -doc fellowships. Last years committee allocated 3000$ and this is status quo level of funding. • A special subcommittee is appointed to research and then propose to the full governance committee a total value of all fellowships, between 0 and 12000 $. • University rules state that the subcommittee may bring a proposal before the full committee under open or closed rule. • All subcommittee and full committee decisions are made by majority rule. • The preferences of each members of the seven -member committee are single peaked: B and E: 0 ; A: 1500 ; C: 6000 ; F: 7500 ; G:10000; D: 12000 The subcommittee is composed of F, G, D . a) What is the subcommittee’s most -preferred level of funding ? G: 10000 Exercises ( 14) • A seven – member governance committee of a university is charged with allocating funds for post -doc fellowships. Last years committee allocated 3000$ and this is status quo level of funding. • A special subcommittee is appointed to research and then propose to the full governance committee a total value of all fellowships, between 0 and 12000 $. • University rules state that the subcommittee may bring a proposal before the full committee under open or closed rule. • All subcommittee and full committee decisions are made by majority rule. • The preferences of each members of the seven -member committee are single peaked: B and E: 0 ; A: 1500 ; C: 6000 ; F: 7500 ; G:10000; D: 12000 The subcommittee is composed of F, G, D . a) Assuming the closed rule ( open rule) what is the subcommittee’s proposal and what is the outcome? Why ? If the subcommittee were to propose its ideal point of $10000, this would be rejected by the entire governance committee because B, E, A and C all prefer the status quo of $3000 to $ 10000. However, the subcommittee could propose $9000 and just secure the vote of C (median voter’s committee) in order to secure a more -preferred yet achievable outcome . Exercises ( 14) • A seven – member governance committee of a university is charged with allocating funds for post -doc fellowships. Last years committee allocated 3000$ and this is status quo level of funding. • A special subcommittee is appointed to research and then propose to the full governance committee a total value of all fellowships, between 0 and 12000 $. • University rules state that the subcommittee may bring a proposal before the full committee under open or closed rule. • All subcommittee and full committee decisions are made by majority rule. • The preferences of each members of the seven -member committee are single peaked: B and E: 0 ; A: 1500 ; C: 6000 ; F: 7500 ; G:10000; D: 12000 The subcommittee is composed of F, G, D . a) Assuming the ( closed rule ) open rule what is the subcommittee’s proposal and what is the outcome? Why ? The subcommittee will open the gates”. An open rule is likely to lead to the outcome being the ideal point of the median voter, who on the entire committee is C. Because all of the members of the sub -committee prefer an allocation of $6000 to $3000, they will vote to open the gates. Their actual proposal is immaterial because if all actors act in their best interests, no matter what they propose it will be amended to $6000 . Question: if the subcommittee was composed of A, G and D what would be the outcome ? Exercises ( 14) • A seven – member governance committee of a university is charged with allocating funds for post -doc fellowships. Last years committee allocated 3000$ and this is status quo level of funding. • A special subcommittee is appointed to research and then propose to the full governance committee a total value of all fellowships, between 0 and 12000 $. • University rules state that the subcommittee may bring a proposal before the full committee under open or closed rule. • All subcommittee and full committee decisions are made by majority rule. • The preferences of each members of the seven -member committee are single peaked: B and E: 0 ; A: 1500 ; C: 6000 ; F: 7500 ; G:10000; D: 12000 The subcommittee is composed of F, G, D . a) Assume that the governance committee follows “zero -based” budgeting” in which each year’s allocation is initially set to zero. What will the outcome of voting be under closed rule (open rule) ? The subcommittee can secure its most -preferred outcome, $10000, because a majority in the entire committee ( median voter C) prefer $10000 to $0. Exercises ( 14) • A seven – member governance committee of a university is charged with allocating funds for post -doc fellowships. Last years committee allocated 3000$ and this is status quo level of funding. • A special subcommittee is appointed to research and then propose to the full governance committee a total value of all fellowships, between 0 and 12000 $. • University rules state that the subcommittee may bring a proposal before the full committee under open or closed rule. • All subcommittee and full committee decisions are made by majority rule. • The preferences of each members of the seven -member committee are single peaked: B and E: 0 ; A: 1500 ; C: 6000 ; F: 7500 ; G:10000; D: 12000 The subcommittee is composed of F, G, D . a) Assume that the governance committee follows “zero -based” budgeting” in which each year’s allocation is initially set to zero. What will the outcome of voting be under ( closed rule ) open rule ? Under an open rule the committee will open the gates, as before, and $6000 is the equilibrium outcome . Powell amendment story (1956) • Democratic leadership sponsored a bill that authorized the distribution of federal funds to the states for the purpose of building schools (alternative y) • Powell, black representative from Harlem, proposed as amendment that “grants could be given only to states with school open to all children without regard to race in conformity with the requirements of U.S. Supreme court decisions. (alternative x) • Status quo= z x y x z y z H H H History 1 2 3 4 I II III Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 Who voted YY (as in the final passage no strategic voting is possibile) can have the following preference ordering : xyz , xzy , yxz. However if they voted non strategically the could not have yxz. As they do not like z also if they vote strategically yxz does not make any sense . xzy is higly unlikely for the Democrats as it means that they would have wanted school aid only with Powell amendment. YY voters has xyz preference ordering … Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 Powellians (xyz) Political group (60% D. 40% R.) Northern urban, big cities from midwest and north Atlantic Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 Who voted NY can have as the Powellians the following preference ordering : xyz , xzy , yxz. However we can eliminate the preference ordering of Powellians ( xyz). xzy is higly unlikely for the Democrats as it means that they would have wanted school aid only with Powell amendment. NY voters has yxz preference ordering if they voted sincerly . Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 School aiders: (yxz) 19% Democrats who followed the party leadership Some Republicans (24) from states like Maine, Colorado etc.who preffered school aid to a gesture for blacks Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 Those who voted NN could have the following preference orderings: zxy, zyx, yzx ; zxy is not possible if they vote sincerely. Conceivably the could have zxy and vote strategically. However it does not make any sense as they would have increased the chances of y against z in the final passage; NN voters can have zyx or yzx ; however if they held zyx the most convenient behaviour should have been voting strategically YN Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 Southerners (yzx): All southerners repr. (105 democrats and 11 republicans) and some Northerners (2 Democrats and 12 Repubblicans) Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 YN voters may have been either of the remaining unassigned : zxy or zyx ; The could have voted sincerely (and having zxy) or strategically (and having zyx) Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 Two political groups: 1) YN (zxy) Repubblicans against aid but symbolically pro black (49) 2) YN (zyx) Repubblicans against aid and indifferent to black issues.(48) Voting Final passage Voting Powell Amendme nt yea nay Totals yea 132 Powellians 78D. 54 R. xyz 97 R.against aid 49 R. zxy 48 R. zxy 229 nay 67 S. Aiders 42D. 25R. yxz 130 Southerners 107D. 23R. yzx 197 totals 199 227 426 What would have happened if all players had voted non strategically? x y x z y z H H H History 1 2 3 4 I II III Node I (sincere voting) x y Powellians, xyz 132 School aiders, yxz 67 Southerners, yzx 130 Republican against aid, zxy 49 Republican against aid, zyx 48 totals 181 245 Node III (sincere voting) y z Powellians, xyz 132 School aiders, yxz 67 Southerners, yzx 130 Republican against aid, zxy Republican against aid, zyx 97 totals 329 97 History 3 (that did not take place) x y x z y z H H H History 1 2 3 4 I II III • If the Repubblicans with zyx preference had voted strategically in order to defeat the bill…at node = instead of voting y they could misrepresent their preferences and vote for x (just to increase the chances to defeat x in the following step) Node I (strategic voting of R. against aid) x y Powellians, xyz 132 School aiders, yxz 67 Southerners, yzx 130 Republican against aid, zxy Republican against aid, zyx 97 totals 229 197 Node II (strategic voting of R. against aid) x z Powellians, xyz 132 School aiders, yxz 67 Southerners, yzx Republican against aid, zxy 130 Republican against aid, zyx 97 totals 199 227 History 2 (the real one) x y x z y z H H H History 1 2 3 4 I II III Puzzle • Why did the Powellians not vote strategically as if they preferred yxz instead of xyz ? Node I (strategic voting of Powellians) R. Holding zyx Vote non strat. R. Holding zyx Vote strat. X y x y Powellians, xyz 132 132 School aiders, yxz 67 67 Southerners, yzx 130 130 Republican against aid, zxy 49 49 Republican against aid, zyx 48 48 totals 49 377 97 329 Node III (strategic voting of Powellians) y z Powellians, xyz 132 School aiders, yxz 67 Southerners, yzx 130 Republican against aid, zxy Republican against aid, zyx 97 totals 329 97 History 3 (in case Powellians had voted strategically ) x y x z y z H H H History 1 2 3 4 I II III Choosing x or y in the node I means that at the node II the strategic equivalent is z or at the node III the strategic equivalent is y. Therefore the choice is in fact between z and y since the very beginning x y x z y z H H H History 1 2 3 4 I II III z y Puzzle..again why did the Powellians not vote strategically as if they preferred yxz instead of xyz ? • Problem: to explain why one set of politicians rationally voted strategically and another set rationally voted non strategically • 2 ways to earn credit with and future votes from people in their constituencies: 1. By producing legislative outcomes 2. By taking positions supported by some constituents • Powellians decided to take position • Rep. against aid decided to produce the best (for them) legislative outcome Costs of the different way can be captured by the different preference ordering (in terms of outcome utility) of the 2 groups • R. with zyx in order to obtain their best (z) must vote their worst • Powellians with xyz in order to obtain at most y (the second best) had to vote against the best alternative x • Republicans were able to vote strategically at low price; Powellians would have to pay a high price. Powell in fact obtained what he wanted: to humiliate the Democratic leadership. He was an herestetician When we consider the utility outcome we always we should add the utility and the cost in terms of “image” of the behaviour that makes possible a certain outcome Exercises ( 15) • The presidential election of 1844 featured two major -party candidates ( Polk for Democrats and Clay for Whigs ); final electoral vote count 170 for Polk and 105 for Clay. Birney , for a third party ( Liberty party) secured 2.3% of the popular vote. • Main issue : new states and slavery a) Polk in favor of new slave states b) Clay against new slave states (status quo) c) Birney strong abolitionist The result for the State of New York (with 36 electoral votes ): Polk 48.8 % pop.v . ; Clay 47.85%; Birney 3.25%. Assume that any Birney voters strictly preferred Clay to Polk . Exercises ( 15) • Suppose that New York’s electoral votes were allocated according to sequential runoff .Who would then have won the election and why ? • Suppose that New York’s electoral votes were allocated according to approval voting and a scenario in which Clay wins the U.S. presidential election. How plausible do you think your scenario is ? • IEC ( independence of entry clones) as criterion of fairness for electoral rules. Is it respected by plurality rule ? What about approval voting ? Exercises ( 15) • Sequential runoff If we assume that all Birney voters prefer Clay to Polk then Birney would be eliminated in round 1, and Birney’s 3.25% would be transferred to Clay, giving him more than 50% of the vote. It seems reasonable to assume that Birney voters did indeed prefer Clay to Polk because Clay was closer to their position on slavery, which was clearly the major political concern of Birney voters . Exercises ( 15) • Approval voting and Clay’s victory scenario 1) At least 30% of the Birney voters also approve of Clay, and at the same time no Clay voters approve of Polk and vice versa. It is not entirely plausible because the abolitionists supporting Birney were highly committed to ending slavery, and likely disapproved of Clay’s acceptance of the status quo . Scenario 2) More Polk supporters approve of Clay than Clay supporters approve of Polk (such that Clay pulls into the lead on number of approval votes). Exercises ( 15) • IEC Plurality rule does not satisfy IEC. Imagine that two candidates, A and B, have converged to the position of the median voter (with A an infinitesimal step to the left of B) and that voters consider only a single issue dimension in their evaluations of the candidates. A and B would then each receive 50% of the vote. If C enters the race only slightly to the left of A, then he will capture most of A’s votes, leading to B’s victory. Approval voting does satisfy IEC, because one candidate’s approval votes are not subtracted from another’s . Therefore , in a situation like above, C’s entry would not change the approval votes for A and B and so would leave the outcome unchanged. Exercises ( 16) • Suppose that candidates a and b. For 58% of population b>a; for 42% a>b. • Candidate a is to left of b and she contemplates paying the conservative «spoiler» candidate c to enter the race to take votes of 17% electorate that c>b>a. Sincere voting is assumed . Would this be a sound investment under 1) Plurality rule ? 2) Sequential runoff ? Exercises ( 16) 1) plurality rule: the entrance of the spoiler c would leave b with only 41% of the vote, so a would win with 42% of the vote. Thus, a should persuade c to enter the race. 2) sequential runoff . In the first round, c will be eliminated with only 17% of the vote. The second round result will lead to b winning with 58% of the vote . Therefore paying c is useless. Exercises ( 17) • Suppose three -way race where for opinion polls 40% a>b>c ; 31% c>b>a ; 29% b>a>c 1) Who will be the winner if an election were held today with sequential runoff ? • Suppose that by campaigning hard , a is able to steal 3% of c’ voters . 2) Who would win the election now ? Which May’s Theorem properties has been violated ? Exercises ( 17) • Suppose three -way race where for opinion polls 40% a>b>c ; 31% c>b>a ; 29% b>a>c 1) Who will be the winner if an election were held today with sequential runoff ? • Suppose that by campaigning hard , a is able to steal 3% of c’ voters . 2) Who would win the election now ? Which May’s Theorem properties has been violated ? Exercises ( 17) • Suppose three -way race where for opinion polls 40% a>b>c ; 31% c>b>a ; 29% b>a>c 1) Who will be the winner if an election were held today with sequential runoff ? b will be eliminated in round 1 with only 29% of the vote (to a and c’s 40 and 31%, respectively). In round 2 between c and a, a will secure 69% of the vote, winning the election . Exercises ( 17) • Suppose that by campaigning hard , a is able to steal 3% of c’ voters . • 2) Who would win the election now ? Which May’s Theorem properties has been violated ? • If a is able to steal a slice of 3% of the voters who support c, then the round 1 result changes. Outcome c will be eliminated in the first round (with only 28% of the vote to b’s 29 % and a’s 43%). Interestingly, in the second round of the vote now b will win with 57% of the vote, so a’s accumulation of extra supporters has lead to his defeat. • This is a violation of the monotonicity property employed in May’s theorem Exercises ( 18) • A small “society” of 9 people with the following preferences 3 people : ( w|zxy ) ; 4 people : ( xzy|w ) ; 2 people : ( y|zwx ) All outcomes to the left of | are « approval worthy » a) Which outcome will win if the society employs simple plurality voting ? What about if it employs b) P lurality runoff c) Borda counting d) Approval voting ? e) Would the society select a clear winner if it is used the Condorcet procedure ? Exercises ( 18) a ) Under simple plurality rule, outcome x will win with a plurality of 4 votes. b ) Under plurality runoff, w and x will proceed to the second round and w will defeat x in the second round, 5 votes to 4. c ) Under approval voting, outcome y will win with 6 votes. d) Under a Borda count (each top choice gets 4 points etc.), z wins with 27 total points. w, x and y have 20, 24 and 19 points, respectively. e) Under the Condorcet procedure, outcome z would win because its defeats each of w, x and y in head -to – head competitions. Exercises ( 19) Cox (1990) claims that if the number of candidates (m) is less than 2 times the number of votes per voter, then a centrist tendency is predominant. Preferences are single peaked and voters are honest: 1) What is a stable equilibrium for a first -past -the -post system when m=2; What voting model does this result reiterate ? 2) Is that same value an equilibrium when m= 3 or 4 Suppose same set up except now each voter has 2 votes (v=2) which are not cumulable (c=no) 1) If m=3, what is a stable equilibrium ? Is that same value an equilibrium when m=4 or 5 ? when m=2 and v=1 1) the ideal point of the median voter is a stable equilibrium as in the downsian model 2) The ideal point of the median voter is not an equilibrium when m = 3 or 4,because candidates will have an incentive to deviate slightly from the median position to secure a plurality of votes. when m=3 and v=1 r t when m=3 ,v=2 and c=0 1) If m = 3, v = 2 and c = no, then all of the candidates will again converge to the ideal point of the median voter. 2) Consider the case where each of the three candidates are already at the median voter’s ideal point. If one candidate were to deviate slightly to the left, then the other two would secure all of the votes of those to the right of the median, and a small sliver of those to the left of the median (up to the halfway point between the median and the new position of our deviating candidate). In addition, they would divide the second votes of the voters who most prefer the deviating candidate. 3) The deviating candidate would thus secure less votes than either of the other two, and lose. Therefore, concentrating at the median is an equilibrium. when m=3 ,v=2 and c=0 r z when m=5 ,v=2 and c=0 1) Suppose that all 5 candidates have positioned themselves at the median voter’s ideal point. One of the candidates could move slightly to the left of the other 4, and secure one of the votes from just less than 50% of the people (which is just less than 25% of the total available votes). The other four candidates would split the 75% of the remaining votes among them evenly, guaranteeing that the candidate who deviated wins one of the seats. 2) Therefore , all 5 positioning themselves at the median cannot be an equilibrium Exercises (20) Two pairs of lotteries over the three outcomes: x=$ 2.5 million ; y=$ 0.5 million; z=$0 First pair P1 vs P2; P1= (p1(x), p1(y), p1(z))=(0,1,0) P2= (p2(x), p2(y), p2(z))=(0.10,0.89,0.01) Second pair P3 vs P4 P3=(p3(x), p3(y), p3(z))=(0, 0.11, 0.89) P4=(p4(x ), p4(y ), p4(z))=(0.10,0,0.90) Empirically for most individuals P1>P2 and P4>P3. Is this behaviour consistent with the theory of expected utility ? No knowledge of the actual utility function is necessary to solve this problem. Exercises (20) P1>P2 implies 1u(y ) > 0.10u(x ) + 0.89u(y ) + 0.01u(z) P4>P3 implies implies 0. 10u(x ) + 0. 90u(z ) > 0. 11u(y ) + 0. 89u(z) • add 0.89u(z ) to both sides of the first expression, and then subtract 0.89u(y ) from both sides of the first expression . 1u(y) + 0.89u(z )-0.89u(y)> 0.10u(x)+0.01u(z)+ 0.89u(z) 0.11(y)+0.89u(z)>0.10u(x)+0.90u(z) 0.11(y )+0.89u(z)>0.10u(x)+0.90u(z ); (P1>P2) or 0. 10u(x) + 0. 90u(z) > 0. 11u(y) + 0. 89u(z ); (P4>P3) Both are not compatible . Inconsistency !!! Exercises (21) Consider the following two sets of individuals (“Societies”) and their group preference rankings, aggregated using the same voting rule 1) Individual preferences a: (x>y>z>w) b: (y>z>w>x) c: (z>w>x>y) Group preferences (x>y), (z>x), (x>w), (y>w), (y>z), (z>w) 2) Individual preferences a: (y>z>x>w) b: (y>w>x>z) c: (y>w>z>x) Group preferences (y>x), (y>w), (y>z), (x>w),(z>x),(z>w) Which of the Arrow’s Conditions (P, D, I or transitivity) is violated by the group preferences and why ? Exercises (21) Consider the following two sets of individuals (“Societies”) and their group preference rankings, aggregated using the same voting rule 1) Individual preferences a: (x>y>z>w) b: (y>z>w>x) c: (z>w>x>y) Group preferences (x>y), (z>x), (x>w), (y>w), (y>z), (z>w) Society 1’s group preferences violate transitivity. For example, x >G y >G w >G x; x >G y >G z >G x; and, x >G y >G z >G w >G x. Exercises (21) Consider the following two sets of individuals (“Societies”) and their group preference rankings, aggregated using the same voting rule 2 ) Individual preferences a: (y>z>x>w) b: (y>w>x>z) c: (y>w>z>x) Group preferences (y>x), (y>w), (z>y), (x>w),(z>x),(z>w) The rule that aggregates Society 2’s group preferences violates the Pareto principle (also known as unanimity) because all three members of the society individually prefer y to z , yet for the group z > y. Exercises (21) Consider the following two sets of individuals (“Societies”) and their group preference rankings, aggregated using the same voting rule 1) Individual preferences a: (x>y>z>w) b: (y>z>w>x) c: (z>w>x>y) Group preferences (x>y), (z>x), (x>w), (y>w), (y>z), (z>w) 2) Individual preferences a: (y>z>x>w) b: (y>w>x>z) c: (y>w>z>x) Group preferences (y>x), (y>w), (y>z), (w>x),( z>x),(z>w ) The voting rule violates condition I (independence of irrelevant alternatives) because in both societies the three individuals share the same ordinal rankings over x and w (x > w, w > x and w > x, respectively) yet x > w in society 1 and w > x in society 2. Exercises ( 22) Three individuals : i,j,k ; P = > i: q>s>r>t j: r>q>t>s k: t>r>s>q 1) If they vote honestly using a round -robin tournament , are there any group preference cycles ? 2) If k changes his mind and switches his preferences to t>s>r>q, are there any group preference cycles ? Exercises ( 22) 1) If they vote honestly using a round -robin tournament , are there any group preference cycles ? i: q>s>r> t j: r>q> t>s k: t>r>s>q i: q>s> r>t j: r>q>t>s k: t> r>s>q i: q> s>r>t j: r>q>t> s k: t>r> s>q i: q >s>r>t j: r> q >t>s k: t>r>s> q Exercises ( 22) 1) Value Restriction Theorem’s conditions are always respected therefore there are not group preferences cycles . The winner is r. i: q>s>r> t j: r>q> t>s k: t>r>s>q i: q>s> r>t j: r>q>t>s k: t> r>s>q i: q> s>r>t j: r>q>t> s k: t>r> s>q i: q >s>r>t j: r> q >t>s k: t>r>s> q Exercises ( 22) 2) If k changes his mind and switches his preferences to t>s>r>q, are there any group preference cycles ? i: q>s>r> t j: r>q> t>s k: t>s>r>q i: q>s> r>t j: r>q>t>s k: t>s> r>q i: q> s>r>t j: r>q>t> s k: t> s>r>q i: q >s>r>t j: r> q >t>s k: t>s>r> q Exercises ( 22) 2) There are two subsets of three alternatives that do not respect value restriction theorem . There group preference cycles . i: q>s>r> t j: r>q> t>s k: t>s>r>q i: q >s>r>t j: r> q >t>s k: t>s>r> q Exercises ( 23) Three individuals : i,j,k ; P = > i: q>s>r>t j: r>q>t>s k: t>r>s>q 1) If they vote honestly using a round -robin tournament , can i fashion a sequential agenda such that her top choice wins ? Identify (if they exist ) agendas that j and k could design to secure their top choices ) . 2) What about if k changes his mind and switches his preferences to t>s>r>q ? Identify (if they exist ) agendas that j and k could design to secure their top choices ) Exercises ( 23) Three individuals : i,j,k ; P = > i: q>s>r>t j: r>q>t>s k: t>r>s>q 1) If they vote honestly using a round -robin tournament , can i fashion a sequential agenda such that her top choice wins ? Identify (if they exist ) agendas that j and k could design to secure their top choices ) . Nobody can (or has interest in) fashion( ing ) a « convenient » sequential agenda as there is a C ondorcet winner , r, that will win always . Exercises ( 23) Three individuals : i,j,k ; P = > i: q>s>r>t j: r>q>t>s k: t> s > r >q 2) What about if k changes his mind and switches his preferences to t>s>r>q ? Identify (if they exist ) agendas that q, j and k could design to secure their top choices ) Exercises ( 23) Three individuals : i,j,k ; P = > i: q>s>r>t ; j: r>q>t>s ; k: t> s > r >q 2) What about if k changes his mind and switches his preferences to t>s>r>q ? Identify (if they exist ) agendas that q, j and k could design to secure their top choices ) In this case there is no Condorcet winner , therefore different agendas can drive to different results . Exercises ( 23) Three individuals : i,j,k ; P = > i: q>s>r>t ; j: r>q>t>s ; k: t> s> r>q 2) First you have to identify the best alternative for each individual and which alternatives are defeated by this alternative. These alterative must located in the last round of the agenda. Then you have to select an alternative that is defeated by the alternative defeated by the top choice . i: q; (….s q); (….t q); (r t s q) (r t q s) (r s t q) (r s q t) j: r; (…..q r); (….t r); (s t q r) ( s t r q) (s q t r) (s q r t) k: t; (…..s t) ; (q r s t) (q r t s) Other two agendas drive to s as winner (q t r s ); (q t s r) Exercises ( 24) Three individuals : i,j,k ; P = > i: q>s>r>t j: r>q>t>s k: t>s>r>q 1) Suppose that player j proposes the agenda ( tsrq ). Has player i any incentive to strategically misrepresent her vote if she assumes the others vote honestly ? 2) Suppose k proposes the agenda ( rqst ). Can j do better by misrepresenting his preferences than by voting sincerly ? Exercises ( 24) Three individuals : i,j,k ; P = > i: q>s>r>t j: r>q>t>s k: t>s>r>q 1) Suppose that player j proposes the agenda ( tsrq ). Has player i any incentive to strategically misrepresent her vote if she assumes the others vote honestly ? If i vote sincerly the outcome is r . If during the second round she votes for t instead of r then q ( her top preference ) will win . Exercises ( 24) Three individuals : i,j,k ; P = > i: q>s>r>t j: r>q>t>s k: t>s>r>q 1) Suppose k proposes the agenda ( rqst ). Can j do better by misrepresenting his preferences than by voting sincerly ? If j vote sincerly the outcome is t. If during the first round he votes for q instead of r then q ( his second choice ) will win . Exercises ( 25)The following table illustrates the probability of cyclical majority as the number of voters and/or the number of alternatives increase. a) Why these probabilities can overemphasize the real occurence of the phenomenon ? ( 1) .……………………………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………………………………… …………………………… …………………………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………………………………………………… b) Why, on the contrary, can they underestimate it ? (2) ………………………………………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………………………………………… Hume’s Marsh – Draining game • No matter what Farmer B does, Farmer A always gets a higher payoff if he chooses not to drain. The reasoning is precisely the same • No matter what Farmer A does, Farmer B always gets a higher payoff if he chooses not to drain. Stag Hunt game (Rousseau) • An alternative vision of the problem of social cooperation is provided by the Stag Hunt Game Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • What is the most preferred outcome ? Is there another outcome in which neither player has an incentive to alter his strategy ? Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • The most -preferred outcome for both players is (Stag, Stag), for which each player receives a payoff of 3. This outcome is an equilibrium inasmuch as neither player wishes to alter his strategy when he believes the other player will be playing Stag. • (Hare, Hare) is also a stable outcome or equilibrium because if A believes that B is going to play Hare, than A’s best response is also to play Hare. Likewise, if B believes A is going to play Hare, than B will play Hare, too. Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • Does either player end up doing better playing either Stag or Hare no matter what his partner chooses to do ( as in the marsh -draining game)? Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • Unlike the marsh -draining game,neither Stag nor Hare is always the optimal strategy regardless of the strategy employed by the other player. If a player believes his partner will play Stag then his best option is to play Stag. But if a player believes his parter will play Hare, than his best response is to play Hare. Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • How certain must A be that B will playing Stag to do the same ? Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • Let’s define pB as the probability that B plays Stag . Then A’s expected utilities associated with the two strategies are: • EUA[Stag] = pB ⋅ 3 + (1 − pB ) ⋅ 0 = 3pB • EUA[Hare] = pB ⋅ 1 + (1 − pB ) ⋅ 1 = 1 Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • A will wish to play Stag when EUA[Stag] > EUA[Hare] , namely 3pB > 1 or pB > 1/3 Achieving the most -preferred outcome in this game then requires that both players believe that the other player will play Stag with at least probability 1/3. One interpretation of this is that the equilibrium depends on each player’s conjecture about the other’s behavior . Another interpretation is that the players must trust one another to play a certain outcome (at least up to a point) in order to secure the socially -optimal outcome. Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Chicken game ( Rebel Without a Cause, 1955) • Another famous coordination Game. Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • What is (are) the most preferred outcome (s) ? What are the «pure» equilibria outcomes ? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • Does either player end up doing better playing either Go Straight or Swerve no matter what his competitor chooses to do ( as in the marsh -draining game)? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • Does either player end up doing better playing either Go Straight or Swerve no matter what his competitor chooses to do ( as in the marsh -draining game)? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game (« Rebel Without a Cause», 1955) • How certain must A be that B will playing Swerve to play the Go Straight ? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game (« Rebel Without a Cause», 1955) Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 • Let’s define pB as the probability that B plays Go Straight . Then A’s expected utilities associated with the two strategies are: • EUA[Go straight] = pB ⋅ -5 + (1 − pB ) ⋅ 3 = 3 -8pB • EUA[Swerve] = pB ⋅ -1 + (1 − pB ) ⋅ 0 = -pB Chicken game (« Rebel Without a Cause», 1955) Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 • A will wish to play Go Straight when • EUA[Go Straight] > EUA[Swerve] , namely • 3 -8pB > -pB or 3 >7pB or pB <3/7 Gangs’rewarding cooperation • In order to prevent the prisoner’s dilemma outcome , criminal organizations can also reward the “cooperation” ( do not confess) for instance by looking after an individual’s family while the criminal is in prison. Gangs’rewarding cooperation • Suppose that a bonus of is given to a criminal who cooperates but whose partner defects, while a payoff of is given to a criminal who cooperates and whose partner also cooperates. a) Rewrite the payoff matrix b) For what values of and is cooperation an equilibrium ? c) For what values is it the only equilibrium ? Gangs’rewarding cooperation • Mutual cooperation is an equilibrium if ≥ 1. For what values is it the only equilibrium ? • Mutual cooperation is the only equilibrium if ≥1 Apartment cleaning • 4 friends (X, Y , Z ,W) live together in a college apartment and must work together to clean common areas . Outcome is dichotomous and has the feature of the following collective action problem . Assume that B (utility coming from cleaniless )>C ( cost of cleaning ) Apartment cleaning • What are the possible equilibrium outcomes when all 4 friends must contribute ? Which do you think is likely ? Apartment cleaning • What are the possible equilibrium outcomes when all 4 friends must contribute ? Which do you think is likely ? • There are two outcomes which are equilibria: 1) everyone contributes or 2) no one contributes. If everyone contributes, each individual secures a benefit B and pays cost C. Thus, their net payoff is B -C > 0. 1) With everyone contributing, if one person decides to not contribute, than the apartment is not cleaned. Those contributing then get net payoff -C, while the person who didn’t contribute gets a payoff of 0. Because B -C > 0, the now non -contributor is worse off than she had been when she contributed along with all of her apartment mates . 2) If no one is contributing, than each player earns a payoff of 0 . No player will wish to unilaterally start contributing because that will only lead to them paying the cost of contribution without securing any benefit, hence the net payoff goes from 0 to -C, and 0 is preferable. Apartment cleaning • What are the possible equilibrium outcomes when only 2 of the 4 friends must contribute to clean the apartment (k=2)? Can you predict which outcome will occur without further information ? Apartment cleaning • What are the possible equilibrium outcomes when only 2 of the 4 friends must contribute to clean the apartment (k=2)? Can you predict which outcome will occur without further information ? • If only two members are required to clean the apartment, than any combination of the two apartment mates contributing is an equilibrium. XY; XZ; XW; YZ; YW; WZ • Imagine for example, suppose X and Y contribute and Z and W don’t. Z and W certainly don’t want to start contributing because they are already getting B without having to pay C. Nevertheless X and Y still prefer B -C to 0, which is what will occur if either one of them decides to not contribute. There are 6 possible ‘cooperative ’ equilibria. • But there is also one ‘non -cooperative’ equilibrium as no one wants to be a sucker and start contributing on their own, because one person cleaning is insufficient to fully clean the apartment. Thus, there are 7 possible equilibria, and it is hard to predict beforehand which will occur . • However , compared to the previous condition, now each roommate has ½ probability to enjoy the cleaniless without any effort Apartment cleaning • How might the prediction change ( when k=2) if B increases or C increases ? What about if B is different for different members of the group ? Apartment cleaning 1)How might the prediction change ( when k=2) if B increases or C increases ? 2) What about if B is different for different members of the group ? 1) Higher B or lower C means that the incentives to coordinate on a cooperative equilibrium are greater . 2) we might expect to see high -B individuals exploited by low -B members, who free ride confident in the knowledge that the very high benefits secured by high -B individuals will motivate them to coordinate on cleaning the apartment. Typology of goods Excludability Yes No Non Rivalrous Yes No Cable or satellite TV «Premium» version on line Journal A Pizza The Global Positioning System (GPS) Public beaches Knowledge Street lighting National Parks One individual public good provision game with mixed strategies • Suppose that there are n individuals who desire a collective good that yields benefit B to all n individuals . Provision of the good requires only one individual (k=1) to expend C to provide it (B>C). Show that there are n possible sets of «pure» strategy equilibria ( each player i plays « contribute » or « Don’t contribute » with probability 1) One individual public good provision game with mixed strategies • The n possible pure strategy are all the same: one individual contributes and no other player does. • If one individual is contributing, no one else wishes to add on their own contribution because the group benefit B has already been secured by all, and adding a contribution would only waste C units of utility. • The net payoff for the sole contributor is B -C; however, we know this is a positive quantity which the contributor prefers to receiving no B and paying no C ( that would be equal to 0) , which is what will occur if he withdraws his contribution. • There is one ‘single -contribution’ equilibrium for each of our n individuals, and hence n pure strategy equilibria. One individual public good provision game with mixed strategies • Now suppose that all players are playing an indentical mixed strategy . In other terms they probabilistically choose whether to play C (oop .)or D( on’t ) . Call p the probability that any one player plays C. a) Show that for any player i, if he does not contribute , the probability that the good is supplied by some one else is 1 -(1 -p) n -1 One individual public good provision game with mixed strategies if A represents some event occurring and A’ represents that event not occuring , then Pr (A’) = 1 -Pr(A) 2. If A and B are two independent events, then Pr (A and B both occur) = Pr (A)* Pr (B). The probability that a player contributes is p, the probability of that player not contributing is (1 − p). Each player makes their decision independently , so the probability that every player but i does not contribute is (1 − p)(1 − p)…(1 − p) = (1 -p) n -1 If (1 -p) n -1 is the probability that everyone but i does not contribute, then 1 − (1 − p) n−1 is the probability that at least one person (excluding i for the moment) contributes. One individual public good provision game with mixed strategies • Equate the expected utility for i of contributing with the expected utility of not contributing . • Solve the expression you found for p • Show that p is decreasing in C, increasing in B and decreasing in n. One individual public good provision game with mixed strategies • Equate the expected utility for i of contributting with the expected utility of not contributing . • The utility for i of contributing is : EUi [Contribute] = B − C; • while the expected utility for i of not contributing is EUi [Don’t] = B[1 − (1 − p) n−1 ] One individual public good provision game with mixed strategies • When these two expected utilities are equal, i is ambivalent about whether to play Contribute or Don’t, which opens up the possibility of playing a probabilistic mixture of Contribute or Don’t, otherwise known as a mixed strategy. • A player is only willing to play this probabilistic mixture of strategies when the payoffs associated with each strategy are exactly equal. One individual public good provision game with mixed strategies • B − C = B[1 − (1 − p) n−1 ] • B(1 − p) n−1 = C • p = 1 − (C/B ) 1/n− 1 • B > C, therefore C/B < 1. As C increases, the second term gets larger so p gets smaller. • An intuitive interpretation of this is that as the costs of contribution increase, each person is less willing to contribute, holding all other factors constant. • Similarly, as the benefits of contribution (B) increase, individuals are more likely to contribute, reflecting the extra gains from contribution. Finally, as the number of individuals increases, each person is less likely to contribute. • This makes sense, because as more individuals contribute with some probability p the more likely it is the good will be provided (only one needs to contribute, after all so each person can relax a little bit. Externalities • A factory located in a small village produces a good with increasing marginal costs MC(q) = 12 + q; so the first unit costs 13 , the second 14 etc. This firm can produce at most 15 and no fractional amount can be produced . The market price for the good is p=20$ and firm’s level of production does not affect this price . Assume that the factory owner maximizes profit and her utility is measured in dollar . Profit is calculated by summing up differences between the price and the marginal cost of each unit produced . • The factory is noisy and interfere with the practice of a neighboring doctor . For every extra unit produced the doctor loses $2 worth of profits . • The doctor’s welfare depends only on his profits , which are 50 -2q Externalities 1) How many units of the good will the factory produce if it ignores the externality imposed on the doctor in its profit maximization ? What will be the aggregate social utility ( factory’s and doctor’s total profits ) ? Externalities (1) • If the factory ignores the external effects of its production, then it will produce up to the point where the marginal revenue of an extra unit equals the marginal cost of an extra unit. The marginal revenue for each unit is $20, and is invariant to the level of production. The marginal cost increases steadily, and will equal $20 when q = 8, which yields a profit of 7+6+5+4+3+2+1+0 = $28. • At this level of production, the doctor’s profits are 50 − 2q = $34. Therefore, aggregate social utility is 28 + 34 = $62 unit revenue cost profit 1 20 13 7 2 40 27 13 3 60 42 18 4 80 58 22 5 100 75 25 6 120 93 27 7 140 112 28 8 160 132 28 9 180 153 27 10 200 175 25 11 220 198 22 0 5 10 15 20 25 30 0 2 4 6 8 10 12 Titolo del grafico marginal revenue marginal cost profit Externalities 2) Identify the level of production that is socially most preferred , in other terms that maximizes aggregate social utility. Externalities (2) • Social utility, S, has been defined as the sum of the factory’s and doctor’s profits. • S(q) = 20 − σ = 1 12 + + 50 − 2 . • This is maximized where q = 6. • We need to consider the marginal profit for an extra unit of production for the factory owner against the marginal cost of that extra unit imposed on the doctor. For example, if the factory increases q from 0 to 1, this garners the factory $7 units of profit while imposing a cost of only $2 on the doctor. • When q = 6, the extra two dollars of profit for the factory are exactly cancelled out in the social utility function by the two dollars of loss in the doctor’s profits. • At this level of production, S = 120 -93+38 = $65 -10 0 10 20 30 40 50 60 70 0 2 4 6 8 10 12 Titolo del grafico Marginal profit Marginal d’s cost Social utilityunit revenue marginal revenue marginal cost cost profit Doctor’s tot utility Marginal profit Marginal d’s cost Social utility 1 20 20 13 13 7 48 7 2 55 2 40 20 14 27 13 46 6 2 59 3 60 20 15 42 18 44 5 2 62 4 80 20 16 58 22 42 4 2 64 5 100 20 17 75 25 40 3 2 65 6 120 20 18 93 27 38 2 2 65 7 140 20 19 112 28 36 1 2 64 8 160 20 20 132 28 34 0 2 62 9 180 20 21 153 27 32 -1 2 59 10 200 20 22 175 25 30 -2 2 55 11 220 20 23 198 22 28 -3 2 50 Externalities 2) Propose a government taxation scheme that will lead to socially preferred outcome … Obviously a tax of $2 per unit of production levied against the factory would lead to the socially optimal amount of production. The factory would only produce up to 6 units (after this point, the marginal profit of an extra unit turns negative) Donation of time • 5 civic -minded patrons of a public library contemplate donations of time to its annual fundraiser . Each individual i bases his or her decision of how much time to donate, x i, on the following utility function : Where namely the total amount of time given by all library patrons and is the cost of losing x i of one’s leisure time Donation of time • What is the socially optimal amount of time donated ? Determine this by summing the utility functions of five individuals , and finding the q that maximizes this function . Donation of time • What is the socially optimal amount of time donated ? Determine this by summing the utility functions of five individuals , and finding the q that maximizes this function . • First we need an expression for society’s utility, (S ). We can find this by adding up the utilities of the individuals who we will index by j 2 (1; 2; 3; 4; 5): Donation of time • We can maximize with respect to q to find the socially optimal net contribution (let’s call it q*). Because all of our individuals are identical, we can then divide this by 5 to find the socially optimal individual contribution x* i At the maximum of this function, this derivative will equal zero. We can solve this for q to find that The socially optimal individual contribution is thus Donation of time • Imagine that an individual assumes that everyone else will contribute no time. How much time will this individual donate ? Is it at the socially optimal level ? Donation of time • Imagine that an individual assumes that everyone else will contribute no time. How much time will this individual donate ? Is it at the socially optimal level ? • If some individual i believes that no one else will contribute, then he will behave as if ; his expected utility will be ;Maximizing this with respect to x i, ; x i=1 < 1.710 , the socially optimal individual contribution < 8.550, the “social” optimal level. Donation of time • Imagine that each individual assumes that the other 4 members will donate 0.8 units of time. Does this individual donate more or less time than before ? Why this ? Donation of time • Imagine that each individual assumes that the other 4 members will donate 0.8 units of time. Does this individual donate more or less time than before ? Why this ? If i assumes that the other individuals will collectively supply .8 units he will believe that ; Maximizing with respect to x i, we get this expression which equals zero where EUi is maximized ; x i=.2 Donation of time • Imagine that each individual assumes that the other 4 members will donate 0.8 units of time. Does this individual donate more or less time than before ? Why this ? The level of contribution .2 turns out to be a symmetric equilibrium , where each individual gives .2 towards the cause. This results in only 1 unit total of time donated, which is far less than the socially optimal level of 8.550 . Each individual gives only .2 , rather than the 1.710 units which would maximize the group’s welfare. The individual’s belief that the others will collectively provide .8 units is confirmed in equilibrium. Thus , this belief is “rational”. Dimension by dimension decision making process • 5= SQ • Committees of members with gatekeeping powers • Specific jurisidictions attached to committees • Rules of amendment once a committee has sent a bill to the full legislative body SQ Dimension by dimension decision making process • Which partioning of legislatures into commitees and which structure of jurisdictions for those committees can give rise to a stable , predictable equilibrium under: 1) Closed rule 2) Open rule with germaneness rule in effect (related to the substance of the original bill ) SQ Dimension by dimension decision making process Closed rule • Dimension X = comm. 1,2,3 • Dimension Y = comm. 4,6,7. • If SQ=5 then 2 on X dimension is approved by 1,2,3, 7 • On Y dimension SQ=5 is the median voter therefore Committee will keep the gate closed . SQ Dimension by dimension decision making process O pen rule (with germaneness ) • Dimension X = comm. 1,2,3 • Dimension Y = comm. 4,6,7. • If SQ=5 then on dimension X Floor would approve 1 Comm. will open the gate. • On Y dimension Floor would approve 4 (worse than 5 for the comm.) Committee will keep the gate closed . SQ Dimension by dimension decision making process Would a change in the committee membership invalidate the previous equilibria ? SQ Dimension by dimension decision making process Closed rule =Open rule • Dimension X = comm. 1,3,5 • Dimension Y = comm. 4,6,7. • If SQ=5 then 1 on X dimension is approved by 1,7,2,3 • On Y dimension SQ=5=6 is the median voter therefore Committee will keep the gate closed . SQ Dimension by dimension decision making process Under what circumstances would a stable equilibrium not exist? SQ Dimension by dimension decision making process 1) If the committees had multi – dimensional jurisdictions. Imagine committee 1,2,3 with gatekeeping power and closed rule SQ Dimension by dimension decision making process 1) If the committees had multi -dimensional jurisdictions. Imagine committee 1,2,3 with gatekeeping power and closed rule .. There are a wide range of (x,y ) pairs that each member of the committee (and player 7 or 6, needed to form a majority) would prefer to 5’s ideal point, but in the multi -dimensional spatial setting it is impossible to predict which of these will be the committee’s proposal. But at least there is a range of plausible proposals. Dimension by dimension decision making process 1) If instead, the committee had multidimensional juridiction and was composed of 1, 2, 3, 4 and 7, then it is impossible to even state a plausible range of proposals. Similarly, if committee jurisdiction is multi -dimensional and the full house operates under an open rule, then chaos is likely to prevail. Dimension by dimension decision making process 2) The second circumstance is when a germaneness rule is not in effect (but it is in effect the open rule) .Under this scenario, even if committee jurisdictions are limited to single issue dimensions, any proposal made by the committee will be subject to the chaos of the multidimensional spatial world once it reaches the full house. Multidimensional decision making process • Three equivalent blocs of voters (1,2,3) • 3 has gatekeeping power and legislature operates under open rule . • Status quo=q Multidimensional decision making process • If 3 open the gates and propose p to the whole house could it achieve final passage of that bill ? In general would the committee be guaranteed final passage of a bill that it prefers to q ? Multidimensional decision making process • P oint p is strictly preferred to q by both 2 and 3 (a majority) . However 3 cannot guarantee passage of a bill that it prefers to q under an open rule… Multidimensional decision making process • The committee proposes p. When p reaches the full legislature, it can be amended under the open rule. 1 might propose some alternative r which is strictly preferred by 2 to p and which 1 also prefers. • The proposal r is then a plausible alternative but in fact, this process of amendments could continue ad libitum and it is impossible to predict what will happen. • In an open -rule setting the committee can’t guarantee an outcome which is preferred to q Multidimensional decision making process • Suppose that there is a rule which grants the members of the committee , 3, an after -the -fact veto ( it can reinstate the status quo q). If the committee opens the gates proposing the p, would it be guaranteed final passage of a bill that it prefers to q ? Multidimensional decision making process • Any final legislation must be strictly preferred by the committee to q .For example, if 1 and 2 were to propose a point like r again, it might pass the whole house but would then be vetoed by the committee resulting in the policy remaining at the status quo, q. • In fact, any point that 1 prefers to q would be vetoed by the committee, therefore 1 is unlikely to be part of any coalition with 2 to amend the status quo. • A wide array of points that 2 and 3 can agree are preferred to q, and this range of points are one set of reasonable predictions for the outcome . Specifically, any point on the dashed line connecting 2 and 3’s ideal points which falls in the preferred -to -q sets is a plausible outcome. Multidimensional decision making process • Granting the committee an after -the -fact veto provides the committee a measure of control over the eventual outcome. • This feature grants committee’s real power, tempering the otherwise chaotic nature of the multi -dimensional spatial setting with an open rule. In this way, specialized committee’s become a vehicle for legislator’s with special interests to secure influence over outcomes in those areas. Krehbiel model • A persistent feature of American political life is the legislative gridlock (high policy stability ) • Khrebiel insist on the importance of the real rules of the U.S. law making 1) Congress can override a presidential veto if a 2/3 of the members vote to do so 2) Most bills can only escape ( for ending the filbustering ) the Senate with a vote of cloture (3/5 of senators ) Krehbiel model • Given president p c = median voter of the Congress v= the ideal point of the pivotal member of Congress need to override a presidential veto (2/3th) f= the ideal point of the pivotal member of the Senate necessary to close the debate (3/5th) Krehbiel model Can the Congress secure the implementation of any preferable law when the status quo is …? SQ SQ SQ SQ The possibile pivots are in fact 4 ( v, v’ and f and f’) but when the position of the president is known than f’ and v’ are not influent . So we have to consider only v and f and c. v ’ f’ 2/3 2/3 3/5 3/5 Krehbiel model • Given president p c = median voter of the Congress v= the ideal point of the pivotal member of Congress need to override a presidential veto (2/3th) f= the ideal point of the pivotal member of the Senate necessary to close the debate (3/5 ° ) Outcome =c Outcome =(cx* 01.2  1,5Q 2 + p[f(Q -1,2 )] + (1 -p)0 2) If p=0.2 and f=5 for Q>1.2  1,5Q 2 + 0.2[5(Q -1,2 )] + ( 1 -0.8)0 = 1,5Q 2 + Q – 1,2 Niskanian Bureaucracy 1) If p=0.2 and f=5 for Q>1.2  1,5Q 2 + 0.2[5(Q – 1,2)] + (0,8)0 = 1,5Q 2 + Q – 1,2 for Q≤1,2  EC(Q )= 1,5Q 2 The legislature’s demand constraint remains binding, and so Q** = 2. Niskanian Bureaucracy 1) If p=0.2 and f=5 for Q>1.2  1,5Q 2 + 0.2[5(Q – 1,2)] + (0,8)0 = 1,5Q 2 + Q – 1,2 for Q≤1,2  EC(Q )= 1,5Q 2 The legislature’s demand constraint remains binding, and so Q** = 2. 2,16 Niskanian Bureaucracy 1) If p=0.5 and f=10 for Q>1.2  1,5Q 2 + 0,5[10(Q – 1,2)] + (0,5)0 = 1,5Q 2 + 5Q – 6 for Q≤1,2  EC(Q)= 1,5Q 2 The legislature’s demand constraint is not binding anymore . Niskanian Bureaucracy We set the cost constraint B=TC; 1,5Q 2 + 5Q – 6= 8Q -2Q 2 3,5Q 2 – 3Q – 6= 0 Then we have to solve the equation for Q Q= 1,806 Niskanian Bureaucracy We set the cost constraint B=TC; 1,5Q 2+ 5Q – 6= 8Q – 2Q 2 3,5Q 2- 3Q – 6= 0 The we have to solve the equation for Q Q= 1,806 .Now the monitoring system makes the cost constraints more binding than the demand constraints Principal – agent game • A principal delegates some authority to an agent and can choose whether or not to audit that agent’s effort in any period • An audit is costly to the principal , but he does not have to pay the agent if he detects shirking • The principal earns 4 if his agent works ; she earns 0 if the agent shirks • The principal pays the agent 3 to work but if she audits and catches the agent shirking he does not have to pay the agent • It costs the agent 2 to do his work • The audit costs 1 to the principal Principal – agent game Principal Agent Audit Don’t Audit Work (3 -2), (4 -3-1) (3 -2), (4 -3) Shirk 0, -1 3, -3 Principal Agent Audit Don’t Audit Work 1, 0 1, 1 Shirk 0, -1 3, -3 • Does either individual have a strategy that is optimal no matter what the other individual plays ? • Are any of the four cells equilibria ? • What percentage of time must the principal inspect to make the agent indifferent between working and shirking ? • What percentage of time must the agent work to make the principal indifferent between auditing and not auditing ? Principal – agent game Principal Agent Audit Don’t Audit Work 1, 0 1, 1 Shirk 0, -1 3, -3 • Does either individual have a strategy that is optimal no matter what the other individual plays ? Are any of the four cells equilibria ? • Neither individual has a strategy which is optimal no matter what the other person plays (there is no dominant strategy .) • If the agent works, the principal prefers not to audit, but if the agent does not work, he of course prefers to audit. If the principal does not audit the agent prefers to shirk, but if the principal audits, the agent prefers to work. None of the four cells represents a pure strategy equilibrium . In other terms, given a certain strategy of a player it is not true that conditional on the other player’s choice of strategy, the player has no incentive to play a different strategy. Principal – agent game Principal Agent Audit Don’t Audit Work 1, 0 1, 1 Shirk 0, -1 3, -3 • What percentage of time must the principal inspect to make the agent indifferent between working and shirking ? • What percentage of time must the agent work to make the principal indifferent between auditing and not auditing ? pA = the probability of an audit or inspection. The expected utilities of the principal’s two strategies ( Audit or No Audit): EU[Work ] = pA ⋅ 1 + (1 − pA ) ⋅ 1 = 1 EU[Shirk ] = pA ⋅ 0 + (1 − pA ) ⋅ 3 = 3 − 3pA: Thus, the agent is indifferent between working and shirking when 1 = 3 − 3pA or pA = 2/3 . Principal – agent game Principal Agent Audit Don’t Audit Work 1, 0 1, 1 Shirk 0, -1 3, -3 • What percentage of time must the principal inspect to make the agent indifferent between working and shirking ? • What percentage of time must the agent work to make the principal indifferent between auditing and not auditing ? pW the probability that the agent works. The expected utilities of the agent’s two strategies are then: EU[Audit] = pW ⋅ 0 + (1 − pW ) ⋅ −1 = −1 + pW EU[ Don’tAudit ] = pW ⋅ 1 + (1 − pW ) ⋅ −3 = −3 + 4pW : The principal is therefore indifferent between auditing and not auditing when -1+3 =3pW ; pW = 2/3 Principal – agent game Principal Agent Audit Don’t Audit Work 1, 0 1, 1 Shirk 0, -1 3, -3 • These strategies (the agent randomly chooses to work with probability pW = 2/3 and the principal randomly chooses to audit with probability pA = 2/3 ) are a mixed strategy equilibrium. • They share the defining property of an equilibrium with pure (i.e. non – probabilistic) strategies: conditional on the other player’s strategy remaining the same, neither player wishes to alter his or her strategy. Each players’ strategy leaves the other player indifferent between his two strategies, and therefore content to play a probabilistic mixture himself. Principal – agent game Principal Agent Audit Don’t Audit Work 1*(2/3*2/3) , 0 *(2/3*2/3) 1*(2/3*1/3) , 1*(2/3*1/3) Shirk 0*(2/3*1/3) , -1*(2/3*1/3) 3*(2/3*1/3) , – 3 *(1/3*1/3) • The average payoff under mixed strategy equilibrium for agent is 1 and for the principal is -1/3 Principal Agent Audit Don’t Audit Work 4/9, 0 2/9, 2/9 Shirk 0, -2/9 3/9, -3/9 Ferejohn (1986) about accountability • Suppose the median voter’s V ideal point in 0 and an elected leader’s ideal point in 1 in on a single -dimensional issue space ( valence issue , corruption ) V L 0 1 • L’s utility for any outcomes is equal to p ( 0≤p≤1) and T for each term in office . Only two terms in office are possible . • In the first term the total payoff is p+T . • In the second term the total payoff is λ ( p+T ) where λ is a discount factor <1 Ferejohn (1986) about accountability • If L is reelected for a second term , what policy will be implemented ? • Assume voters use a « retrospective voting strategy » of the form : reelect if p ≤ r, and vote out otherwise . a) Come up with two expressions for L’s utilities, one if he is reelected and one if he is not , assuming for each case that L sets p as high as possible consistent with the desired electoral outcome . b) Show that for voters the optimal voting rule has r=1 – λ – λT or 0 depending on the values of λ and T c) How does voter utility in equilibrium change with λ and T ? Ferejohn (1986) about accountability • If L is elected for a second term then he will implement p = 1 in that term. This is because he can no longer be held accountable by the electorate and so freely selects his most -preferred policy without facing any negative consequences . • L has two electoral strategies to consider. 1) First , he can attempt to satisfy voters by choosing a p ≤ r in the first round. The overall payoff attached to this strategy is r + T + λ (1 + T) = r + λ + (1 + λ )T . 2)Alternatively , L can forget about reelection and attempt to milk everything possible out of a single term in office by choosing p = 1 in the first round. This results in an overall payoff of 1 + T. Ferejohn (1986) about accountability • Voters can use these possible payoffs to determine an optimal voting rule, i.e. a value for r that just guarantees `good behavior’ in the first term at minimal cost. The politician will choose his `seek re -election’ strategy only if r+ λ +( 1 + λ )T ≥ 1+T . In other terms if r ≥ 1 – λ – λ T. The lowest r at which voters can ensure that the politician behaves himself in the first term is r = 1 – λ – λ T • Depending on the values of λ and T, it may be that r = 0, i.e. any politician will prefer to behave himself in the first term to guarantee the payoffs in the second term . This is more likely as λ gets larger (meaning the politician doesn’t discount future payoffs heavily) and as T gets larger (meaning there is a big payoff associated with simply being in office). • Voters always get a payoff of 0 in the second round, therefore we only need to consider their payoff of p = r = 1 – λ – λ T in the first round . Ferejohn & Weingast model of Court’s behaviour • XH= median voter in House • XS= median voter in Senate Supreme Court ( XSc , median voter ) is hearing an argument about some statute ( at XQ) passed in a previous session of Congress . XH XS XSc XS/XH XH/XS New legislation XQ New legislation XQ XQ Ferejohn & Weingast model of Court’s behaviour • Assume that XSc =XQ Would the Supreme Court alter the bill when XQ is ? XH XS XSc XS/XH XH/XS New legislation XQ New legislation XQ XQ XQ Ferejohn & Weingast model of Court’s behaviour • Assume that XSc =XQ Because xQ is between xH and xS , there will be no way for one house to move the law closer to its ideal point which doesn’t make the other house worse off. Anticipating this, the Supreme Court will leave the law unchanged and secure its ideal point, xQ , in equilibrium XH XS XSc XS/XH XH/XS New legislation XQ New legislation XQ XQ XQ Ferejohn & Weingast model of Court’s behaviour • Assume that XSc =XQ Would the Supreme Court alter the bill when XQ is ? XH XS XSc XS/XH XH/XS New legislation XQ New legislation XQ XQ XQ Ferejohn & Weingast model of Court’s behaviour • Assume that XSc =XQ • The House and the Senate will be able to agree on a variety of proposals which both houses strictly prefer to xQ . Proposals between xH and either xH + jxH − xQj (that is, proposals which are up to equally far from xH as xQ is,but on the right side) or xS , whichever is smaller. These proposals constitute the bargaining range. XH XS XSc XS/XH XH/XS New legislation XQ New legislation XQ XQ XQ XH+|XH -XQ| Bargaining range Ferejohn & Weingast model of Court’s behaviour • Assume that XSc =XQ • if the Supreme Court proposes XSc • the outcome will be in the range [XH , xH + |XH − XSc |] or [ xH,xS ], whichever is shorter. • What is the Supreme Court’s optimal proposal? It will be XSc = xH . • Anything less than xH raises the possibility of a final bargained outcome between the Senate and House which is greater than or equal to xH . This strategy guarantees an outcome equal to xH , because the House will have no interest in compromising with the Senate to move the status quo closer to the Senate’s position XH XS XSc XS/XH XH/XS New legislation XQ New legislation XQ XQ XQ XH+|XH -XQ| Bargaining range Ferejohn & Weingast model of Court’s behaviour • Assume that XSc =XQ • These results suggest that Supreme Court justices (and judges on lower courts who may be asked to interpret or amend existing law) might have an incentive to change the law, if their main goal is as little change in the law as possible. XH XS XSc XS/XH XH/XS New legislation XQ New legislation XQ XQ XQ XH+|XH -XQ| Bargaining range Example government formation according to different hypotheses • Suppose some country with PR ends up with the following distribution of 100 Parliamentary seats among 5 parties from left to right A (15) B(28) C(11) D(33) E(14) 1) Find all minimum winning coalitions (MWC) 2) Find the smallest MWC 3) Find the MWC with the fewest members 4) Find all MWCs for which the parties are adjacent in the political space Example government formation according to different hypotheses • Suppose some country with PR ends up with the following distribution of 100 Parliamentary seats among 5 parties from left to right A (15) B(28) C(11) D(33) E(14) 1) Find all minimum winning coalitions (MWC) The set of minimum winning coalitions is: ABC (54 members), ABE (57), ACD (59), ADE (62 ), BCE (53), BD (61), and CDE (58). Example government formation according to different hypotheses • Suppose some country with PR ends up with the following distribution of 100 Parliamentary seats among 5 parties from left to right A (15) B(28) C(11) D(33) E(14) 2) Find the smallest MWC The smallest MWC in terms of number of parliamentarians is BCE with 53 . Example government formation according to different hypotheses • Suppose some country with PR ends up with the following distribution of 100 Parliamentary seats among 5 parties from left to right A (15) B(28) C(11) D(33) E(14) 3) Find the MWC with the fewest members 4)Find all MWCs for which the parties are adjacent in the political space The MWC with the fewest parties is BD. The only coalitions with adjacent parties are ABC and CDE. Cabinet formation • Two dimensions ranging from 0 to 10 • X dimension = Finance • Y dimension = Defense • Three parties A, B, C ; no party has a majority of seats , two parties are sufficient to have a majority of seats Cabinet formation • What is the set of possible minority and majority governments ? • Is there a stable majority government coalition ? • Is there a minority government that is a stable equilibrium ? Cabinet formation • What is the set of possible minority and majority governments ? • Is there a stable majority government coalition ? • Is there a minority government that is a stable equilibrium ? Cabinet formation • What is the set of possible minority and majority governments ? • Is there a stable majority government coalition ? • Is there a minority government that is a stable equilibrium ? Cabinet formation • What is the set of possible minority and majority governments ? • Is there a stable majority government coalition ? • Is there a minority government that is a stable equilibrium ? Chicken game ( Rebel Without a Cause, 1955) • Another famous coordination Game. Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • What is (are) the most preferred outcome (s) ? What are the «pure» equilibria outcomes ? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • Does either player end up doing better playing either Go Straight or Swerve no matter what his competitor chooses to do ( as in the marsh -draining game)? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • Does either player end up doing better playing either Go Straight or Swerve no matter what his competitor chooses to do ( as in the marsh -draining game)? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game (« Rebel Without a Cause», 1955) • How certain must A be that B will playing Swerve to play the Go Straight ? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game (« Rebel Without a Cause», 1955) Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 • Let’s define pB as the probability that B plays Go Straight . Then A’s expected utilities associated with the two strategies are: • EUA[Go straight] = pB ⋅ -5 + (1 − pB ) ⋅ 3 = 3 -8pB • EUA[Swerve] = pB ⋅ -1 + (1 − pB ) ⋅ 0 = -1 Chicken game (« Rebel Without a Cause», 1955) Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 • A will wish to play Go Straight when • EUA[Go Straight] > EUA[Swerve] , namely • 3 -8pB > -1 or 4 >8pB or pB <1/2 Fees and cooperation • A cultural association, composed of 100 members, has to pay as soon as possible 1000$ of annual rent for the room where its weekly meetings take place. Each member enjoys on from these meetings a utility equal to 15$ per year. All this is common knowledge. Some members suggest the association chair to fix the fee 15> f > 10 just have some margin in case of few member’s defection; Others insist on keeping the fee as low as possible, namely f=10. Which advice should the chair follow and why ? • 2 0 members decide to voluntarily donate to the chair 280 $ to cope with such emergency and this donation is common knowledge. • Should the association chair change the previous fee ? Which is the final fee ? Fees and cooperation • A cultural association, composed of 100 members, has to pay as soon as possible 1000$ of annual rent for the room where its weekly meetings take place. Each member enjoys on from these meetings a utility equal to 15$ per year. All this is common knowledge. Some members suggest the association chair to fix the fee 15> f > 10 just have some margin in case of few member’s defection; Others insist on keeping the fee as low as possible, namely f=10. Which advice should the chair follow and why ? Chair should fix the fee as low as possible . In this case unless all members pay the rent won’t meet the required amount . In other words as the unanimity is required the cooperation will be very likely . Fees and cooperation • 20 members decide to voluntarily donate to the chair 280 $ to cope with such emergency and this donation is common knowledge. Should the association chair change the previous fee ? Which is the final fee ? Now the members that do not donate and that have still to contribute are 80 and the required amount is 720. Therefore the chair has to fix the fee equal to 9. Committe membership and closed rule • The parliamentary floor has to decide the composition of two committees (with 3 members ) that control dimension Y and dimension X and have gatekeeping power . The Parliament members are 1,2,3,4,5. Only one Mp can ( and has to) be member of both committees and each committee enjoys the closed rule . • Given the following position of SQ and the positions of MPs , which committes ( and controlling which dimension ) will be formed ? Committe membership and closed rule SQ 1 2 3 4 5 Committe membership and closed rule SQ 1 2 3 4 5 Committe membership and closed rule SQ 1 2 3 4 5 Committe membership and open rule . • What happens if the rule is open with germaneness ( amendments are possible only on the dimension that is considered by the committee ) ? Vote trading Each legislator has one vote on any issue coming before the body. He cannot aggregate the votes in his possession and cast them all, or some large fraction of them, for a motion on a subject near and dear to his heart (or those of his constituents). What prevent the vote trading to be an efficient solution to this type of problem ? Filling the matrix ..2) Write down payoffs to create the “ Stag Hunt game ” ( assurance game ) ( 1) Player B Cooperate Do not cooperate -1 Player A Cooperate Do not cooperate Constitutional and Statutory Interpretation Supreme Court ( XSc , Supreme Court median voter) is hearing an argument about some statute ( at XQ) passed in a previous session of Congress formed by House and Senate ( XH and XS, median voters of the two Chambers) . Assume that XSc =XQ and that in order to change a law a majority in both chambers is necessary, while it is necessary a majority of 2/3 in order to change the constitution. Assume also that 1/3 of the House members is on the left of XQ. Draw where the status quo will be located after the Supreme Court ’s statutory interpretation (1) and after a constitutional interpretation (1) Constitutional and Statutory Interpretation Supreme Court ( XSc , Supreme Court median voter) is hearing an argument about some statute ( at XQ) passed in a previous session of Congress formed by House and Senate ( XH and XS, median voters of the two Chambers) . Assume that XSc =XQ and that in order to change a law a majority in both chambers is necessary, while it is necessary a majority of 2/3 in order to change the constitution. Assume also that 1/3 of the House members is on the left of XQ. Draw where the status quo will be located after the Supreme Court ’s statutory interpretation (1) and after a constitutional interpretation (1) Multidimensional decision making process A is an agenda setter that enjoys a closed rule . Where is the policy outcome if a) the decision rule is the majority rule b) The decision rule is the unanimity rule Multidimensional decision making process A is an agenda setter that enjoys a closed rule . Where is the policy outcome if a) the decision rule is the majority rule b) The decision rule is the unanimity rule Accountability In the country Referendumland citizens have to vote about the MPs salary . They have already decided that each Mp can be in office only for two terms . The current salary is 6000 Euro per month The political group that sponsorized the referendum proposes to reduce such a salary . New salary would be 3000 Euro. • According to Ferejohn’s model how should citizens vote in order to increase accountability ? Coalitional drift • What is ? • Which is its relationship with the Bureaucratic drift ? p q r . . . 1 2 3 Multidimensional decision making process ( again ) p q r . . . 1 2 3 Three equivalent blocs of voters (1,2,3) 1 ( the committee )has gatekeeping power and legislature operates under open rule . Status quo=q p q r . . . 1 2 3 1 proposes to 3 p that is for both better than q. However such a alliance is broken by 2 that proposes r that is better for 2 than p p q r . . . 1 2 3 Suppose that there is a rule which grants the members of the committee , 1, an after – the -fact veto ( it can reinstate the status quo q). If the committee opens the gates proposing the p, would it be guaranteed final passage of a bill that it prefers to q ? Which coalition will be formed ? p q r . . . 1 2 3 If 1 can threat to reinstate q then 3 could bargain an alternative between p and the border of the indifference curve of 1 as all these alternatives are better than q. However the coalition between 1 and 3 is not the most plausible .. p q r . . . 1 2 3 In fact 2 is available to concede to 1 slightly morethan 2, at most w that is sligltly closer to 1 than p. . w Paradox of voting • Illustrate the calculus of voting according to an « instrumental » approach and how and why it should be changed according to Riker & Ordershook
Political Anaylisis ( Rationality behaviour instition exam) I have an exam tomorrow There will be two parts and each one has 30 min time limit. ( please check the attachment) need someone who have kno
Exercises (1) • In November 2008, a couple of weeks after the election of Barack Obama, Hillary Clinton was offered the job of Secretary of State of the United States. I • She faced the following trade -off: a) joining the new administration, in perhaps the highest profile cabinet position. b) continuing in the Senate, an option that promised less power (she would still be only one of a hundred) but greater autonomy. Moreover taking an administration job would preclude a primary challenge against Barack Obama in 2012, Hillary Clinton faced three Possibilities: C ) Remain in the Congress and not win the Presidency in 2012 , P ) Remain in the Congress and win the Presidency in 2012 S) Join the administration as secretary of state . If the probability of winning the White House in 2012 if she had remained in the Senate is p , then use an expected utility argument to determine the smallest p that would have induced Clinton to remain in the Senate in order to run in 2012. Exercises (1) • As H.Clinton choosed S it is reasonable to assume the following preference ordering : P>S>C • As she left the Senate then S> p (P) +(1 -p) (C) • S>p(P)+C -p(C) • S -C>p(P -C); S -C/P -C>p Therefore the smallest p that could induce H.C. to stay in the Congress was S -C/P -C=p Exercises (2) • Consider the following two sets of individuals (“Societies”) and their group preference rankings, aggregated using the same voting rule 1) Individual preferences a: (x>y>z>w) b: (y>z>w>x) c: (z>w>x>y) Group preferences (x>y), (z>x), (w>x), (y>w), (y>z), (z>w) 2) Individual preferences a: (y>z>x>w) b: (y>w>x>z) c: (y>w>z>x) Group preferences (y>x), (y>w), (z>y), (x>w),(z>x),(z>w) Which of the Arrow’s Conditions (P, D, I or transitivity ) is violated by the group preferences ? Exercises (2) • Society 1 violates “ transitivity ” (x >y>w>x ) ; (x>y>z >x ); (x>y>z>w>x ) • The rule that aggregates Society 2’s group references violates the (P) Pareto principle (unanimity) because all three members of the society individually prefer y to z, yet z >y. • The voting rule violates condition I (independence of irrelevant alternatives) because in both societies the three individuals share the same ordinal rankings over x and w (x>w, w>x and w>x, respectively) yet w>x in society 1 and x>w in society 2. Exercises (3) • May’s theorem suggests that any deviation from majority rule must be justified by a reasonable departure from one of 4 conditions : U, A, N, M. • Condition U (universal domain). All complete and transitive preference orderings over alternatives are admittable . • Condition A (Anonymity ). Social preferences depend only on the collection of individual preferences , not on who has which preference . • Condition N (Neutrality ). Interchanging the ranks of alternatives j and k in each group member’s preference ordering has the effect of interchanging the ranks of j and k in the group preference ordering . • Condition M ( Monotonicity ). If an alternative j beats or ties another alternative k — that is , j R G k — and j rises some group member’s preferences from below k to the same or a higher rank than k, then j now strictly beats k — that is , jP G k . Exercises (3) • For each of the following cases explain which of these conditions is violated by the electoral rule and suggest a possible justification . 1) Amendment of U.S. Constitution requires 2/3 majority in each Chamber. 2) IMF uses a system of weighted voting where weights are determined by contributions to IMF operating funds. US holds a veto in some circumstances . 3) Guilty Verdict in a (U.S.) criminal case requires unanimity or almost unanimity . 4) French President is elected under a two -stage majority rules . ( run -off vote) Exercises (3) 1) Amendment of U.S. Constitution requires 2/3 majority in each Chamber. The requirement of 2/3 majorities in Congress to propose an amendment is a violation of neutrality , because the status quo (no proposal) is given preferential treatment in the voting procedure . For example , if exactly 60% of Congress people in each house prefer the status quo, then the status quo wins . However, if we reverse the preferences so that now 60% of Congress people in each house prefer an amendment, the status quo nonetheless still wins out. In general, any voting rule which privileges the status quo (or some other outcome) violates neutrality; however, in many instances this departure from majority rule is justified as an attempt to make extraordinary changes difficult . Exercises (3) 2) IMF uses a system of weighted voting where weights are determined by contributions to IMF operating funds. US holds a veto in some circumstances . The IMF’s voting procedure violates anonymity in two separate ways. a) weighted voting privileges some members over others, so redistributing the preference orderings among the individuals can change the outcome. b) granting the US a veto in certain circumstances means that an identical collection of preferences but permuted among the members differently might lead to different outcomes e.g . if in one permutation 60% of members preferred some outcome a including the US, and if in another 60% of members preferred some outcome a and the US opposed it. In the case of the IMF, the weighting of voting rights is based on a normative argument (those who contribute more to IMF activities deserve more say) and a political justification (integrating a superpower into an international organization sometimes requires granting special rights and exceptions to that superpower, as in the UN Security Council ). Exercises (3) • Guilty Verdict in a (U.S.) criminal case requires unanimity or almost unanimity . • This is a violation of neutrality, because one outcome (acquittal) is given a privileged status relative to the others. Consider a case where 9 jurors favor an acquittal and 3 a guilty verdict. Clearly, acquittal wins out, but in most systems, acquittal will still prevail if the opposite set of preferences are held. • The usual justifications for this rule is that there should be consensus or near consensus among jurors before meting out life -altering criminal convictions and punishments, and that innocence should be heavily presumed and guilt only determined by overwhelmingly persuasive evidence. Exercises (3) 4) French President is elected under a two -stage majority rule . ( run -off vote) This rule in fact violates the monotonicity condition. Suppose that the following percentages of voters hold these preferences over three candidates a; b; c: a > b > c (40%); b > c > a (31%); c > a > b ( 29%). In a two -round election, a and b would win round 1, and then a would beat b in round 2 (assuming voters are sincere). Now suppose, that 3% of the of b > c > a voters change their preferences to a > b > c. In the two -round election, a and c win round 1, and then c defeats a in round 2. In other words, an increase in support for a has lead to a’s defeat. Two -stage elections are often supported because they guarantee that a winning candidate secures a majority of voters (thus , arguably enhancing the legitimacy of the eventual winner), allow voters to support smaller parties in the first stage (up to a point), and promote moderation in the second stage by creating the centripetal tendencies of two -candidate competition . Exercises (4) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 1) Write down the majority preference relation for this profile of preferences (pairwise comparison ) 2) Does Black’s Median voter Theorem support a prediction about which policy will be chosen if the group uses simple majority rule ? Why or why not 3) Suppose that the group is going to use a voting agenda v= (y, x, z), namely first y versus x then etc. Which is the outcome ? What about if the agenda is v’=( z,x,y ) and v’’ = ( z,y,x ) ? x y z 1 2 3 u y x z 2 3 1 z x y 3 1 2 z y x 1 2 3 x z y 1 3 2 y x z 1 2 3 u Exercises (4) 1) The group preferences over each pair of outcomes using majority rule are: xPGy , yPGz , and zPGx . 2) Black’s Median Voter Theorem does not support a prediction about which policy the group will choose because the preferences do not satisfy single -peakedness . Demonstrating this lack of single -peakedness graphically requires drawing six graphs. A faster check is to note that none of the three outcomes are agreed upon by the group to be `not worst’. One interesting thing to note is that the preferences could violate single – peakedness and still yield a coherent outcome e.g. with the following preferences: 1: x>y>z, 2: z>y>x , 3: z>x>y . These prefeences violate single -peakedness but still yield transitive social preferences . The preferences over x, y and z still satisfy Sen’s value -restriction criterion because z is agreed by all to be ‘ not middling ‘. 3) Under agenda v, z is the winner (x beats y then z beats x). Under agenda v’, y is the winner (z beats x then y beats z). Under agenda v’’, x is the winner (y beats z then x beats y). Exercises (5) • Downs takes politicians to be interested only in winning office. Does a different result other than convergence arise when politicians have strong policy preferences of their own ? Under which circumstances ? Exercises (6) 1) Suppose that strategy c3 is unavailable to Mr III. Solve the Game Exercises (6) 2 ) Suppose that strategy c3 is available but Mr I can no longer play a1 Solve the game. Exercises (6) 1) Final outcome If strategy c3 is unavailable to Mr. III, then the final outcome will be (a1; c2; b2) with payoffs (8; 9; 3) for I, II and III respectively. Exercises (6) 1) Final outcome If strategy c3 is unavailable to Mr. III, then the final outcome will be (a1; c2; b2) with payoffs (8; 9; 3) for I, II and III respectively. Exercises (6) 1) Final outcome If strategy c3 is unavailable to Mr. III, then the final outcome will be (a1; c2; b2) with payoffs (8; 9; 3) for I, II and III respectively. Exercises (6) 2 ) If strategy a1 is unavailable to Mr. I, then the final outcome will be (a2; c2; b1) with payoffs (7; 7; 6) going to I, II and III respectively. Exercises (6) 2 ) If strategy a1 is unavailable to Mr. I, then the final outcome will be (a2; c2; b1) with payoffs (7; 7; 6) going to I, II and III respectively. Exercises (6) 2 ) If strategy a1 is unavailable to Mr. I, then the final outcome will be (a2; c2; b1) with payoffs (7; 7; 6) going to I, II and III respectively. Exercises (7)1) Imagine that in a committee th ere are three legislators ( or three groups of legislators equal in size) A, B, C. They have to approve by majority rule a policy. There are three alternatives : X, Y, Z . The utiles given by each policy is described in the table and the utile of the status quo is 0 for all legislators. Bill(voted against the Status Quo =0 ) Legislato rs X Y Z A 3 -1 -1 B -1 3 -1 C -1 -1 3 a) Wh ich is the overall utility level the legislators ( no difference between them) enjoy if the bills are voted separately and independently ? (1): …………………………………………………………………………….. b) Imagine that each legislators knows the utility he can get from each alternative before starting voting. He /She considers the advantages of forming a coalition of two legislators but he/she does not know if he/she will be member of this coalition. How m any winning coalitions of two legislators can be formed (1) ?………………… Which is each legislator’s expected utility ? (2) ………………………….. 1) 0 2) 3 : AB, AC, CB 3) Imagine actor A : payoff of AB= (3 -1) ; payoff of AC=(3 -1), payoff of CB =( -2) Expected utility = 1/3*2+1/3*2 -1/3*2 = 2/3 Exercises (7) Solutions Exercises (8) 1) Imagine that in a committee there are three legislators ( or three groups of legislators equal in size) A, B, C. They have to approve by majority rule a policy. There are three alternatives : X, Y, Z. The utiles given by each policy is described in the tab le and the utile of the status quo is 0 for all legislators. Bill(voted against the Status Quo=0) Legislators X Y Z A 3 -2 -1 B -1 3 -2 C -2 -1 3 a) Which is the overall utility level the legislators ( no difference between them) enjoy if the bills are voted separately and independently ? (1): …………………………………………………………………………….. b) Imagine that each legislators knows the utility he can get from each alternati ve before starting voting. He /She considers the advantages of forming a coalition of two legislators but he/she does not know if he/she will be member of this coalition. How many winning coalitions of two legislators can be formed ( 1) ?………………… Which is each legislator’s expected utility ? (2) ………………………….. c) Once a coalition is formed is it stable ? Why yes or no 1) 0 2) 3 : AB, AC, CB 3) actor A : payoff of AB= ( 3 -2) ; payoff of AC=(3 -1), payoff of CB =( -1) Expected utility = 1/3*1+1/3*2 -1/3*1 = 2/3 Actor B: payoff of AB= ( 3 -1) ; payoff of AC =( -1), payoff of CB =(3 -2) Expected utility = 1/3*2 -1/3*1 +1/3*1 = 2/3 Actor C: payoff of AB= (-1) ; payoff of AC=( 3 -2), payoff of CB =( 3 -1) Expected utility = -1/3*1+1/3*1 + 1/3*2 = 2/3. 4) No coalition is stable. For instance for A AC>AB but but for C CB > AC and for B AB> CB; Cycle!! Exercises (8) Solutions Exercises (9) • For each of the following societies: 1) State whether the preferences satisfy Sen’s value – restriction criterion 2) If not, identify the tuple(s) of preferences that violate value – restricted preferences 3) Assuming majority rule, are the societies’ preferences transitive ? Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • Society 1: y is agreed to be not worst among xyz ; x is agreed to be not best among xyw ; no option satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . Using the method of majority rule , the only preference cycle occurs between x, z and w. Society 1 1: y>x>z >w 2: w> y>x>z 3: z>y> w> x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • Society 1: y is agreed to be not worst among xyz; x is agreed to be not best among xyw ; no option satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . Using the method of majority rule , the only preference cycle occurs between x, z and w. Society 1 1: y>x> z> w 2: w>y>x >z 3: z> y>w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • Society 1: y is agreed to be not worst among xyz; x is agreed to be not best among xyw ; no option satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . Using the method of majority rule , the only preference cycle occurs between x, z and w. Society 1 1: y> x>z>w 2: w> y> x>z 3: z> y> w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • Society 1: y is agreed to be not worst among xyz; x is agreed to be not best among xyw ; no option satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . Using the method of majority rule , the only preference cycle occurs between x, z and w. Society 1 1: y> x> z>w 2: w>y> x> z 3: z>y>w >x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz ; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y> w >z>x 2: w >x>y>z 3: z> w >y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w> z >x 2: w>x>y> z 3: z >w>y>x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y >w>z>x 2: w>x> y >z 3: z>w> y >x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w>z> x 2: w> x >y>z 3: z>w>y> x Society 3 1: y>w>z>x 2: z>x>y>w 3: x>y>w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz ; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y> w >z>x 2: z>x>y> w 3: x>y> w >z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w> z >x 2: z >x>y>w 3: x>y>w> z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y >w>z>x 2: z>x> y >w 3: x> y >w>z Exercises (9) • For each society we must check four “tuples”, (xyz; xyw ; xzw ; yzw ), to determine if at least one outcome satisfies value restriction, • For Society 2, y is agreed to be not worst among xyz; w is agreed to be not worst among xyw ; w is agreed to be not worst among xzw ; and z is agreed to be not middling among yzw . Using the method of majority rule , there are no preference cycles. • For society 3 , no outcome satisfies value restriction among xyz; x is agreed to be not middling among xyw ; no outcome satisfies value restriction among xzw ; and y is agreed to be not worst among yzw . There are preference cycles among the social outcomes when majority rule is used, including among xyz, xzw and wxyz . Society 1 1: y>x>z>w 2: w>y>x>z 3: z>y>w>x Society 2 1: y>w>z>x 2: w>x>y>z 3: z>w>y>x Society 3 1: y>w>z> x 2: z> x >y>w 3: x >y>w>z Exercises (10) • A legislature is going to vote on a policy that is well represented by a single -issue dimension, on a scale of zero to one . • The initial policy proposal will be supplied by a committee ( whose position on the dimension is supposed to “vary” ) to a legislature with median voter’s ideal point M ( 0.45) . The Status quo is at point SQ (0.25). Exercises (10) • Draw a line showing the equilibrium outcomes for any committee ideal point when the proposal is considered under the closed rule and when it is considered under the open rule. • Do the same exercise for both rules assuming that the legislature operates on the principal of zero -based budgeting (no decision  ZB=0) Exercises (10) • What is the impact of zero – based budgeting under a closed rule? And under the open rule ? Exercises (10) • Under a closed rule using the status quo as the `reversion’ or `default’ option, the kinks in the line occur at SQ = 0.25 and M +|M – SQ| = 0.65. • Exercises (10) • In between those two kinks, the Committee can propose and secure its ideal point in equilibrium, because the median voter prefers that proposal to the status quo . • If the committee’s ideal point is to the left of the first kink, then the legislative median will resist any attempt to move the policy to the left; to the right of the kink, the committee proposes an outcome it prefers which leaves the legislative median as well off as it is under the status quo. Exercises (10) • Under zero -based budgeting, the only kink in the line occurs at M + |M -ZB| = 0.90. • From ZB to M + | M -ZB|the committee can achieve always its ideal point. Exercises (10) • Under an open rule using the status quo as the default option, the break in the line occurs at the Committee’s point of indifference between the SQ and M, which is halfway between these points at 0.35 . • The committee will keep the gates closed when its ideal point is closer to the status quo than to the median because when it opens the gates, the full legislature will adopt M in equilibrium. Exercises (10) • Under zero -based budgeting (and open rule) this point of indifference occurs at the point 0.45/2 = 0.225 , the point where the 0, the de facto status quo now, is equivalent to M from the committee’s perspective. Exercises (10) • Under a closed rule, the committee is able to secure outcomes closer to its ideal point when zero -based budgeting is employed. When the committee has extreme preferences relative to the median voter in the legislature , the equilibrium outcomes are therefore more extreme than they would have been under an ordinary status quo rule . Exercises (10) • Under an open rule, the effects of zero -based budgeting on committee power are ambiguous . For some ranges of the line the committee’s best outcome is less – preferred than the equilibrium under the status quo rule (e.g. when the committee ideal point is near the SQ point), but for some ideal points the equilibrium outcome is preferred to the equilibrium under the status quo rule (e.g. when the committee ideal point is near zero). Therefore, in some instances the outcomes are further from the median ideal point than they would be under status quo budgeting, but in others the outcomes are closer to the median ideal point than they would be under status -quo budgeting. Exercises ( 11) • Political Actors Xa , Xb , Xc are located in a two dimensional policy space. Each actor would like to change the status quo SQ. All proposals are pitted against SQ in a final voting. Write down on the picture the final outcome (or the winset ) when (1) • a) Decision rule is unanimity • b) Decision rule is majority • c) Decision rule is majority, one dimension at a time, in some pre -set order. Exercises ( 11) Unanimity Unanimity Core Majority rule Majority , one dimension at a time, in some pre – set order. Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 1) Suppose that the group is going to use a voting agenda v= (y, x, z) or v’=( z,x,y ) or v’’ = ( z,y,x ) It is the last round. Will any player wish to misrepresent her true preferences ? 2) With the agenda v= (y, x, z) a) Determine what happens in the final round depending on whether y or x wins the first round b)Can player 2 ever do better by supporting (against her true preferences ) x in the first round c) What about 3 voting against x ? d) Should 1 misrepresent his preferences by playing y in round 1? Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 1) Suppose that the group is going to use a voting agenda v= (y, x, z) or v’=( z,x,y ) or v’’ = ( z,y,x ) It is the last round. Will any player wish to misrepresent her true preferences ? It is never optimal to misrepresent one’s preferences in a single round of majority rule voting over two outcomes . Any vote against one’s most – preferred outcome can only increase the support for the less -preferred option , possibly leading to its victory. Therefore, the final round of our agenda procedure here will never feature strategic voting . Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 2) With the agenda v= (y, x, z) a) Determine what happens in the final round depending on whether y or x wins the first round If y wins in the first round then y will win in the last one If x wins in the first round then z will win in the last one Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 2) With the agenda v= (y, x, z) b)Can player 2 ever do better by supporting (against her true preferences ) x in the first round No, the final outcome (z) would not change c) What about 3 voting against x ? Player 3 would not wish to vote for y rather than x because this would lead to a victory for y in the first round and in the last round. d) Should 1 misrepresent his preferences by playing y in round 1? Player 1 will wish to vote strategically in round 1 by voting for y. This leads to y being the overall winner, and player 1 prefers y to z . No player has an incentive to deviate from their strategy Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 2) With the agenda v’=( z,x,y ) a) Determine what happens in the final round depending on whether z or x wins the first round b)Can player 2 ever do better by supporting (against her true preferences ) z in the first round c) What about 3 voting against z ? d) Should 1 misrepresent his preferences by playing z in round 1? Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 2) With the agenda v’=( z,x,y ) a) Determine what happens in the final round depending on whether z or x wins the first round If z wins in the first round then y wins in the last one If x wins in the first round then x wins in the last one b)Can player 2 ever do better by supporting (against her true preferences ) x in the first round No as she achieves the worst outcome (x) in the last round c) What about 3 voting against z ? Player 3 has an incentive to misrepresent her vote in the first round by voting for x rather than z. The outcome under honest voting is y, however if 3 misrepresents her vote in this way , the final outcome is x (better than y). (Stable equilibrium) d) Should 1 misrepresent his preferences by playing z in round 1? No. She will achieve y as final outcome that is worse than x Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 2) With the agenda v’’ = ( z,y,x ) a) Determine what happens in the final round depending on whether z or x wins the first round b)Can player 2 ever do better by supporting (against her true preferences ) z in the first round c) What about 3 voting against z ? d) Should 1 misrepresent his preferences by playing z in round 1? Exercises ( 12) • Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 2) With the agenda v’’=( z,y,x ) a) Determine what happens in the final round depending on whether z or x wins the first round If z wins in the first round then z wins in the last one If y wins in the first round then x wins in the last one b)Can player 2 ever do better by supporting (against her true preferences ) z in the first round ? player 2 has an incentive to misrepresent his vote in the first round by voting for z rather than y . The outcome under honest voting is x, however if 2 misrepresents his vote in this way, the final outcome is z. (stable equilibrium) c) What about 3 voting against z ? Player 3 does not have an incentive to misrepresent her vote. She would achieve x in the final round that is worse than z. d) Should 1 misrepresent his preferences by playing z in round 1? No. She will achieve z as final outcome that is worse than x Exercises ( 12) Suppose that a society consists of three individuals 1,2,3 who must choose one among three proposed budgets : x, y, z. Their preferences over them are: 1: x>y>z 2: y>z>x 3: z>x>y 2) With the agenda v= (y, x, z) If y wins in the first round then y will win in the last one If x wins in the fitst round then z will win in the last one Whether two players might be able to form a strategic voting coalition ? With 1 voting strategically (which is an equilibrium ) 2 has no desire to change his behavior. Would it be possible for 1 and 3 (who secure their 2nd and 3rd most -preferred outcomes respectively) to team up and secure a better outcome? It would be possible, but not plausible. 1 and 3 agree to both vote for x in round 1, and then both vote for x in round 2. This eliminates y which 3 hates and 1 dislikes compared to x , and gives each a better outcome than y, which is the proposed equilibrium under strategic voting . However player 3 will be voting against her interest in the final round by voting for x over z. Thus, we might think that she would be tempted to renege on the deal with 1 and get her most -preferred outcome z . In the absence of some way of preventing herself from voting for z in the final round, 1 may not find 3’s promises very credible, and may prefer to stick with the strategic voting as described above . Exercises ( 13) • In the following one -dimension policy space describing the policy positions in a legislative arena, the Committee median voter’s ideal point is C and the Floor median voter is F. The committee has a gatekeeping power (It is the only actor who can propose a policy change). Given the location of the status quo SQ: 1) Identify the policy outcome if the committee enjoys a closed rule 2) Identify the policy outcome if the committee enjoys an open rule 3) Which rule is more convenient for the Floor (the Parliament) ? 4) Where should the Status quo be located to have a policy change and a different answer to the previous question ? Exercises ( 13) • In the following one -dimension policy space describing the policy positions in a legislative arena, the Committee median voter’s ideal point is C and the Floor median voter is F. The committee has a gatekeeping power (It is the only actor who can propose a policy change). Given the location of the status quo SQ: 1) Identify the policy outcome if the committee enjoys a closed rule Exercises ( 13) • In the following one -dimension policy space describing the policy positions in a legislative arena, the Committee median voter’s ideal point is C and the Floor median voter is F. The committee has a gatekeeping power (It is the only actor who can propose a policy change). Given the location of the status quo SQ: 2) Identify the policy outcome if the committee enjoys an open rule Exercises ( 13) • In the following one -dimension policy space describing the policy positions in a legislative arena, the Committee median voter’s ideal point is C and the Floor median voter is F. The committee has a gatekeeping power (It is the only actor who can propose a policy change). Given the location of the status quo SQ: 3) Which rule is more convenient for the Floor (the Parliament) ? Closed Rule Exercises ( 13) • In the following one -dimension policy space describing the policy positions in a legislative arena, the Committee median voter’s ideal point is C and the Floor median voter is F. The committee has a gatekeeping power (It is the only actor who can propose a policy change). Given the location of the status quo SQ: 4) Where should the Status quo be located to have a policy change and a different answer to the previous question ? Exercises ( 14) • A seven – member governance committee of a university is charged with allocating funds for post -doc fellowships. Last years committee allocated 3000$ and this is status quo level of funding. • A special subcommittee is appointed to research and then propose to the full governance committee a total value of all fellowships, between 0 and 12000 $. • University rules state that the subcommittee may bring a proposal before the full committee under open or closed rule. • All subcommittee and full committee decisions are made by majority rule. • The preferences of each members of the seven -member committee are single peaked: B and E: 0 ; A: 1500 ; C: 6000 ; F: 7500 ; G:10000; D: 12000 The subcommittee is composed of F, G, D . a) What is the subcommittee’s most -preferred level of funding ? b) Assuming the closed rule ( open rule) what is the subcommittee’s proposal and what is the outcome? Why ? c) Assume that the governance committee follows “zero -based” budgeting” in which each year’s allocation is initially set to zero. What will the outcome of voting be under closed rule (open rule) ? Exercises ( 14) • A seven – member governance committee of a university is charged with allocating funds for post -doc fellowships. Last years committee allocated 3000$ and this is status quo level of funding. • A special subcommittee is appointed to research and then propose to the full governance committee a total value of all fellowships, between 0 and 12000 $. • University rules state that the subcommittee may bring a proposal before the full committee under open or closed rule. • All subcommittee and full committee decisions are made by majority rule. • The preferences of each members of the seven -member committee are single peaked: B and E: 0 ; A: 1500 ; C: 6000 ; F: 7500 ; G:10000; D: 12000 The subcommittee is composed of F, G, D . a) What is the subcommittee’s most -preferred level of funding ? G: 10000 Exercises ( 14) • A seven – member governance committee of a university is charged with allocating funds for post -doc fellowships. Last years committee allocated 3000$ and this is status quo level of funding. • A special subcommittee is appointed to research and then propose to the full governance committee a total value of all fellowships, between 0 and 12000 $. • University rules state that the subcommittee may bring a proposal before the full committee under open or closed rule. • All subcommittee and full committee decisions are made by majority rule. • The preferences of each members of the seven -member committee are single peaked: B and E: 0 ; A: 1500 ; C: 6000 ; F: 7500 ; G:10000; D: 12000 The subcommittee is composed of F, G, D . a) Assuming the closed rule ( open rule) what is the subcommittee’s proposal and what is the outcome? Why ? If the subcommittee were to propose its ideal point of $10000, this would be rejected by the entire governance committee because B, E, A and C all prefer the status quo of $3000 to $ 10000. However, the subcommittee could propose $9000 and just secure the vote of C (median voter’s committee) in order to secure a more -preferred yet achievable outcome . Exercises ( 14) • A seven – member governance committee of a university is charged with allocating funds for post -doc fellowships. Last years committee allocated 3000$ and this is status quo level of funding. • A special subcommittee is appointed to research and then propose to the full governance committee a total value of all fellowships, between 0 and 12000 $. • University rules state that the subcommittee may bring a proposal before the full committee under open or closed rule. • All subcommittee and full committee decisions are made by majority rule. • The preferences of each members of the seven -member committee are single peaked: B and E: 0 ; A: 1500 ; C: 6000 ; F: 7500 ; G:10000; D: 12000 The subcommittee is composed of F, G, D . a) Assuming the ( closed rule ) open rule what is the subcommittee’s proposal and what is the outcome? Why ? The subcommittee will open the gates”. An open rule is likely to lead to the outcome being the ideal point of the median voter, who on the entire committee is C. Because all of the members of the sub -committee prefer an allocation of $6000 to $3000, they will vote to open the gates. Their actual proposal is immaterial because if all actors act in their best interests, no matter what they propose it will be amended to $6000 . Question: if the subcommittee was composed of A, G and D what would be the outcome ? Exercises ( 14) • A seven – member governance committee of a university is charged with allocating funds for post -doc fellowships. Last years committee allocated 3000$ and this is status quo level of funding. • A special subcommittee is appointed to research and then propose to the full governance committee a total value of all fellowships, between 0 and 12000 $. • University rules state that the subcommittee may bring a proposal before the full committee under open or closed rule. • All subcommittee and full committee decisions are made by majority rule. • The preferences of each members of the seven -member committee are single peaked: B and E: 0 ; A: 1500 ; C: 6000 ; F: 7500 ; G:10000; D: 12000 The subcommittee is composed of F, G, D . a) Assume that the governance committee follows “zero -based” budgeting” in which each year’s allocation is initially set to zero. What will the outcome of voting be under closed rule (open rule) ? The subcommittee can secure its most -preferred outcome, $10000, because a majority in the entire committee ( median voter C) prefer $10000 to $0. Exercises ( 14) • A seven – member governance committee of a university is charged with allocating funds for post -doc fellowships. Last years committee allocated 3000$ and this is status quo level of funding. • A special subcommittee is appointed to research and then propose to the full governance committee a total value of all fellowships, between 0 and 12000 $. • University rules state that the subcommittee may bring a proposal before the full committee under open or closed rule. • All subcommittee and full committee decisions are made by majority rule. • The preferences of each members of the seven -member committee are single peaked: B and E: 0 ; A: 1500 ; C: 6000 ; F: 7500 ; G:10000; D: 12000 The subcommittee is composed of F, G, D . a) Assume that the governance committee follows “zero -based” budgeting” in which each year’s allocation is initially set to zero. What will the outcome of voting be under ( closed rule ) open rule ? Under an open rule the committee will open the gates, as before, and $6000 is the equilibrium outcome . Powell amendment story (1956) • Democratic leadership sponsored a bill that authorized the distribution of federal funds to the states for the purpose of building schools (alternative y) • Powell, black representative from Harlem, proposed as amendment that “grants could be given only to states with school open to all children without regard to race in conformity with the requirements of U.S. Supreme court decisions. (alternative x) • Status quo= z x y x z y z H H H History 1 2 3 4 I II III Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 Who voted YY (as in the final passage no strategic voting is possibile) can have the following preference ordering : xyz , xzy , yxz. However if they voted non strategically the could not have yxz. As they do not like z also if they vote strategically yxz does not make any sense . xzy is higly unlikely for the Democrats as it means that they would have wanted school aid only with Powell amendment. YY voters has xyz preference ordering … Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 Powellians (xyz) Political group (60% D. 40% R.) Northern urban, big cities from midwest and north Atlantic Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 Who voted NY can have as the Powellians the following preference ordering : xyz , xzy , yxz. However we can eliminate the preference ordering of Powellians ( xyz). xzy is higly unlikely for the Democrats as it means that they would have wanted school aid only with Powell amendment. NY voters has yxz preference ordering if they voted sincerly . Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 School aiders: (yxz) 19% Democrats who followed the party leadership Some Republicans (24) from states like Maine, Colorado etc.who preffered school aid to a gesture for blacks Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 Those who voted NN could have the following preference orderings: zxy, zyx, yzx ; zxy is not possible if they vote sincerely. Conceivably the could have zxy and vote strategically. However it does not make any sense as they would have increased the chances of y against z in the final passage; NN voters can have zyx or yzx ; however if they held zyx the most convenient behaviour should have been voting strategically YN Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 Southerners (yzx): All southerners repr. (105 democrats and 11 republicans) and some Northerners (2 Democrats and 12 Repubblicans) Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 YN voters may have been either of the remaining unassigned : zxy or zyx ; The could have voted sincerely (and having zxy) or strategically (and having zyx) Voting Final passage Voting Powell Amendm ent yea nay Totals yea 132 97 229 nay 67 130 197 totals 199 227 426 Two political groups: 1) YN (zxy) Repubblicans against aid but symbolically pro black (49) 2) YN (zyx) Repubblicans against aid and indifferent to black issues.(48) Voting Final passage Voting Powell Amendme nt yea nay Totals yea 132 Powellians 78D. 54 R. xyz 97 R.against aid 49 R. zxy 48 R. zxy 229 nay 67 S. Aiders 42D. 25R. yxz 130 Southerners 107D. 23R. yzx 197 totals 199 227 426 What would have happened if all players had voted non strategically? x y x z y z H H H History 1 2 3 4 I II III Node I (sincere voting) x y Powellians, xyz 132 School aiders, yxz 67 Southerners, yzx 130 Republican against aid, zxy 49 Republican against aid, zyx 48 totals 181 245 Node III (sincere voting) y z Powellians, xyz 132 School aiders, yxz 67 Southerners, yzx 130 Republican against aid, zxy Republican against aid, zyx 97 totals 329 97 History 3 (that did not take place) x y x z y z H H H History 1 2 3 4 I II III • If the Repubblicans with zyx preference had voted strategically in order to defeat the bill…at node = instead of voting y they could misrepresent their preferences and vote for x (just to increase the chances to defeat x in the following step) Node I (strategic voting of R. against aid) x y Powellians, xyz 132 School aiders, yxz 67 Southerners, yzx 130 Republican against aid, zxy Republican against aid, zyx 97 totals 229 197 Node II (strategic voting of R. against aid) x z Powellians, xyz 132 School aiders, yxz 67 Southerners, yzx Republican against aid, zxy 130 Republican against aid, zyx 97 totals 199 227 History 2 (the real one) x y x z y z H H H History 1 2 3 4 I II III Puzzle • Why did the Powellians not vote strategically as if they preferred yxz instead of xyz ? Node I (strategic voting of Powellians) R. Holding zyx Vote non strat. R. Holding zyx Vote strat. X y x y Powellians, xyz 132 132 School aiders, yxz 67 67 Southerners, yzx 130 130 Republican against aid, zxy 49 49 Republican against aid, zyx 48 48 totals 49 377 97 329 Node III (strategic voting of Powellians) y z Powellians, xyz 132 School aiders, yxz 67 Southerners, yzx 130 Republican against aid, zxy Republican against aid, zyx 97 totals 329 97 History 3 (in case Powellians had voted strategically ) x y x z y z H H H History 1 2 3 4 I II III Choosing x or y in the node I means that at the node II the strategic equivalent is z or at the node III the strategic equivalent is y. Therefore the choice is in fact between z and y since the very beginning x y x z y z H H H History 1 2 3 4 I II III z y Puzzle..again why did the Powellians not vote strategically as if they preferred yxz instead of xyz ? • Problem: to explain why one set of politicians rationally voted strategically and another set rationally voted non strategically • 2 ways to earn credit with and future votes from people in their constituencies: 1. By producing legislative outcomes 2. By taking positions supported by some constituents • Powellians decided to take position • Rep. against aid decided to produce the best (for them) legislative outcome Costs of the different way can be captured by the different preference ordering (in terms of outcome utility) of the 2 groups • R. with zyx in order to obtain their best (z) must vote their worst • Powellians with xyz in order to obtain at most y (the second best) had to vote against the best alternative x • Republicans were able to vote strategically at low price; Powellians would have to pay a high price. Powell in fact obtained what he wanted: to humiliate the Democratic leadership. He was an herestetician When we consider the utility outcome we always we should add the utility and the cost in terms of “image” of the behaviour that makes possible a certain outcome Exercises ( 15) • The presidential election of 1844 featured two major -party candidates ( Polk for Democrats and Clay for Whigs ); final electoral vote count 170 for Polk and 105 for Clay. Birney , for a third party ( Liberty party) secured 2.3% of the popular vote. • Main issue : new states and slavery a) Polk in favor of new slave states b) Clay against new slave states (status quo) c) Birney strong abolitionist The result for the State of New York (with 36 electoral votes ): Polk 48.8 % pop.v . ; Clay 47.85%; Birney 3.25%. Assume that any Birney voters strictly preferred Clay to Polk . Exercises ( 15) • Suppose that New York’s electoral votes were allocated according to sequential runoff .Who would then have won the election and why ? • Suppose that New York’s electoral votes were allocated according to approval voting and a scenario in which Clay wins the U.S. presidential election. How plausible do you think your scenario is ? • IEC ( independence of entry clones) as criterion of fairness for electoral rules. Is it respected by plurality rule ? What about approval voting ? Exercises ( 15) • Sequential runoff If we assume that all Birney voters prefer Clay to Polk then Birney would be eliminated in round 1, and Birney’s 3.25% would be transferred to Clay, giving him more than 50% of the vote. It seems reasonable to assume that Birney voters did indeed prefer Clay to Polk because Clay was closer to their position on slavery, which was clearly the major political concern of Birney voters . Exercises ( 15) • Approval voting and Clay’s victory scenario 1) At least 30% of the Birney voters also approve of Clay, and at the same time no Clay voters approve of Polk and vice versa. It is not entirely plausible because the abolitionists supporting Birney were highly committed to ending slavery, and likely disapproved of Clay’s acceptance of the status quo . Scenario 2) More Polk supporters approve of Clay than Clay supporters approve of Polk (such that Clay pulls into the lead on number of approval votes). Exercises ( 15) • IEC Plurality rule does not satisfy IEC. Imagine that two candidates, A and B, have converged to the position of the median voter (with A an infinitesimal step to the left of B) and that voters consider only a single issue dimension in their evaluations of the candidates. A and B would then each receive 50% of the vote. If C enters the race only slightly to the left of A, then he will capture most of A’s votes, leading to B’s victory. Approval voting does satisfy IEC, because one candidate’s approval votes are not subtracted from another’s . Therefore , in a situation like above, C’s entry would not change the approval votes for A and B and so would leave the outcome unchanged. Exercises ( 16) • Suppose that candidates a and b. For 58% of population b>a; for 42% a>b. • Candidate a is to left of b and she contemplates paying the conservative «spoiler» candidate c to enter the race to take votes of 17% electorate that c>b>a. Sincere voting is assumed . Would this be a sound investment under 1) Plurality rule ? 2) Sequential runoff ? Exercises ( 16) 1) plurality rule: the entrance of the spoiler c would leave b with only 41% of the vote, so a would win with 42% of the vote. Thus, a should persuade c to enter the race. 2) sequential runoff . In the first round, c will be eliminated with only 17% of the vote. The second round result will lead to b winning with 58% of the vote . Therefore paying c is useless. Exercises ( 17) • Suppose three -way race where for opinion polls 40% a>b>c ; 31% c>b>a ; 29% b>a>c 1) Who will be the winner if an election were held today with sequential runoff ? • Suppose that by campaigning hard , a is able to steal 3% of c’ voters . 2) Who would win the election now ? Which May’s Theorem properties has been violated ? Exercises ( 17) • Suppose three -way race where for opinion polls 40% a>b>c ; 31% c>b>a ; 29% b>a>c 1) Who will be the winner if an election were held today with sequential runoff ? • Suppose that by campaigning hard , a is able to steal 3% of c’ voters . 2) Who would win the election now ? Which May’s Theorem properties has been violated ? Exercises ( 17) • Suppose three -way race where for opinion polls 40% a>b>c ; 31% c>b>a ; 29% b>a>c 1) Who will be the winner if an election were held today with sequential runoff ? b will be eliminated in round 1 with only 29% of the vote (to a and c’s 40 and 31%, respectively). In round 2 between c and a, a will secure 69% of the vote, winning the election . Exercises ( 17) • Suppose that by campaigning hard , a is able to steal 3% of c’ voters . • 2) Who would win the election now ? Which May’s Theorem properties has been violated ? • If a is able to steal a slice of 3% of the voters who support c, then the round 1 result changes. Outcome c will be eliminated in the first round (with only 28% of the vote to b’s 29 % and a’s 43%). Interestingly, in the second round of the vote now b will win with 57% of the vote, so a’s accumulation of extra supporters has lead to his defeat. • This is a violation of the monotonicity property employed in May’s theorem Exercises ( 18) • A small “society” of 9 people with the following preferences 3 people : ( w|zxy ) ; 4 people : ( xzy|w ) ; 2 people : ( y|zwx ) All outcomes to the left of | are « approval worthy » a) Which outcome will win if the society employs simple plurality voting ? What about if it employs b) P lurality runoff c) Borda counting d) Approval voting ? e) Would the society select a clear winner if it is used the Condorcet procedure ? Exercises ( 18) a ) Under simple plurality rule, outcome x will win with a plurality of 4 votes. b ) Under plurality runoff, w and x will proceed to the second round and w will defeat x in the second round, 5 votes to 4. c ) Under approval voting, outcome y will win with 6 votes. d) Under a Borda count (each top choice gets 4 points etc.), z wins with 27 total points. w, x and y have 20, 24 and 19 points, respectively. e) Under the Condorcet procedure, outcome z would win because its defeats each of w, x and y in head -to – head competitions. Exercises ( 19) Cox (1990) claims that if the number of candidates (m) is less than 2 times the number of votes per voter, then a centrist tendency is predominant. Preferences are single peaked and voters are honest: 1) What is a stable equilibrium for a first -past -the -post system when m=2; What voting model does this result reiterate ? 2) Is that same value an equilibrium when m= 3 or 4 Suppose same set up except now each voter has 2 votes (v=2) which are not cumulable (c=no) 1) If m=3, what is a stable equilibrium ? Is that same value an equilibrium when m=4 or 5 ? when m=2 and v=1 1) the ideal point of the median voter is a stable equilibrium as in the downsian model 2) The ideal point of the median voter is not an equilibrium when m = 3 or 4,because candidates will have an incentive to deviate slightly from the median position to secure a plurality of votes. when m=3 and v=1 r t when m=3 ,v=2 and c=0 1) If m = 3, v = 2 and c = no, then all of the candidates will again converge to the ideal point of the median voter. 2) Consider the case where each of the three candidates are already at the median voter’s ideal point. If one candidate were to deviate slightly to the left, then the other two would secure all of the votes of those to the right of the median, and a small sliver of those to the left of the median (up to the halfway point between the median and the new position of our deviating candidate). In addition, they would divide the second votes of the voters who most prefer the deviating candidate. 3) The deviating candidate would thus secure less votes than either of the other two, and lose. Therefore, concentrating at the median is an equilibrium. when m=3 ,v=2 and c=0 r z when m=5 ,v=2 and c=0 1) Suppose that all 5 candidates have positioned themselves at the median voter’s ideal point. One of the candidates could move slightly to the left of the other 4, and secure one of the votes from just less than 50% of the people (which is just less than 25% of the total available votes). The other four candidates would split the 75% of the remaining votes among them evenly, guaranteeing that the candidate who deviated wins one of the seats. 2) Therefore , all 5 positioning themselves at the median cannot be an equilibrium Exercises (20) Two pairs of lotteries over the three outcomes: x=$ 2.5 million ; y=$ 0.5 million; z=$0 First pair P1 vs P2; P1= (p1(x), p1(y), p1(z))=(0,1,0) P2= (p2(x), p2(y), p2(z))=(0.10,0.89,0.01) Second pair P3 vs P4 P3=(p3(x), p3(y), p3(z))=(0, 0.11, 0.89) P4=(p4(x ), p4(y ), p4(z))=(0.10,0,0.90) Empirically for most individuals P1>P2 and P4>P3. Is this behaviour consistent with the theory of expected utility ? No knowledge of the actual utility function is necessary to solve this problem. Exercises (20) P1>P2 implies 1u(y ) > 0.10u(x ) + 0.89u(y ) + 0.01u(z) P4>P3 implies implies 0. 10u(x ) + 0. 90u(z ) > 0. 11u(y ) + 0. 89u(z) • add 0.89u(z ) to both sides of the first expression, and then subtract 0.89u(y ) from both sides of the first expression . 1u(y) + 0.89u(z )-0.89u(y)> 0.10u(x)+0.01u(z)+ 0.89u(z) 0.11(y)+0.89u(z)>0.10u(x)+0.90u(z) 0.11(y )+0.89u(z)>0.10u(x)+0.90u(z ); (P1>P2) or 0. 10u(x) + 0. 90u(z) > 0. 11u(y) + 0. 89u(z ); (P4>P3) Both are not compatible . Inconsistency !!! Exercises (21) Consider the following two sets of individuals (“Societies”) and their group preference rankings, aggregated using the same voting rule 1) Individual preferences a: (x>y>z>w) b: (y>z>w>x) c: (z>w>x>y) Group preferences (x>y), (z>x), (x>w), (y>w), (y>z), (z>w) 2) Individual preferences a: (y>z>x>w) b: (y>w>x>z) c: (y>w>z>x) Group preferences (y>x), (y>w), (y>z), (x>w),(z>x),(z>w) Which of the Arrow’s Conditions (P, D, I or transitivity) is violated by the group preferences and why ? Exercises (21) Consider the following two sets of individuals (“Societies”) and their group preference rankings, aggregated using the same voting rule 1) Individual preferences a: (x>y>z>w) b: (y>z>w>x) c: (z>w>x>y) Group preferences (x>y), (z>x), (x>w), (y>w), (y>z), (z>w) Society 1’s group preferences violate transitivity. For example, x >G y >G w >G x; x >G y >G z >G x; and, x >G y >G z >G w >G x. Exercises (21) Consider the following two sets of individuals (“Societies”) and their group preference rankings, aggregated using the same voting rule 2 ) Individual preferences a: (y>z>x>w) b: (y>w>x>z) c: (y>w>z>x) Group preferences (y>x), (y>w), (z>y), (x>w),(z>x),(z>w) The rule that aggregates Society 2’s group preferences violates the Pareto principle (also known as unanimity) because all three members of the society individually prefer y to z , yet for the group z > y. Exercises (21) Consider the following two sets of individuals (“Societies”) and their group preference rankings, aggregated using the same voting rule 1) Individual preferences a: (x>y>z>w) b: (y>z>w>x) c: (z>w>x>y) Group preferences (x>y), (z>x), (x>w), (y>w), (y>z), (z>w) 2) Individual preferences a: (y>z>x>w) b: (y>w>x>z) c: (y>w>z>x) Group preferences (y>x), (y>w), (y>z), (w>x),( z>x),(z>w ) The voting rule violates condition I (independence of irrelevant alternatives) because in both societies the three individuals share the same ordinal rankings over x and w (x > w, w > x and w > x, respectively) yet x > w in society 1 and w > x in society 2. Exercises ( 22) Three individuals : i,j,k ; P = > i: q>s>r>t j: r>q>t>s k: t>r>s>q 1) If they vote honestly using a round -robin tournament , are there any group preference cycles ? 2) If k changes his mind and switches his preferences to t>s>r>q, are there any group preference cycles ? Exercises ( 22) 1) If they vote honestly using a round -robin tournament , are there any group preference cycles ? i: q>s>r> t j: r>q> t>s k: t>r>s>q i: q>s> r>t j: r>q>t>s k: t> r>s>q i: q> s>r>t j: r>q>t> s k: t>r> s>q i: q >s>r>t j: r> q >t>s k: t>r>s> q Exercises ( 22) 1) Value Restriction Theorem’s conditions are always respected therefore there are not group preferences cycles . The winner is r. i: q>s>r> t j: r>q> t>s k: t>r>s>q i: q>s> r>t j: r>q>t>s k: t> r>s>q i: q> s>r>t j: r>q>t> s k: t>r> s>q i: q >s>r>t j: r> q >t>s k: t>r>s> q Exercises ( 22) 2) If k changes his mind and switches his preferences to t>s>r>q, are there any group preference cycles ? i: q>s>r> t j: r>q> t>s k: t>s>r>q i: q>s> r>t j: r>q>t>s k: t>s> r>q i: q> s>r>t j: r>q>t> s k: t> s>r>q i: q >s>r>t j: r> q >t>s k: t>s>r> q Exercises ( 22) 2) There are two subsets of three alternatives that do not respect value restriction theorem . There group preference cycles . i: q>s>r> t j: r>q> t>s k: t>s>r>q i: q >s>r>t j: r> q >t>s k: t>s>r> q Exercises ( 23) Three individuals : i,j,k ; P = > i: q>s>r>t j: r>q>t>s k: t>r>s>q 1) If they vote honestly using a round -robin tournament , can i fashion a sequential agenda such that her top choice wins ? Identify (if they exist ) agendas that j and k could design to secure their top choices ) . 2) What about if k changes his mind and switches his preferences to t>s>r>q ? Identify (if they exist ) agendas that j and k could design to secure their top choices ) Exercises ( 23) Three individuals : i,j,k ; P = > i: q>s>r>t j: r>q>t>s k: t>r>s>q 1) If they vote honestly using a round -robin tournament , can i fashion a sequential agenda such that her top choice wins ? Identify (if they exist ) agendas that j and k could design to secure their top choices ) . Nobody can (or has interest in) fashion( ing ) a « convenient » sequential agenda as there is a C ondorcet winner , r, that will win always . Exercises ( 23) Three individuals : i,j,k ; P = > i: q>s>r>t j: r>q>t>s k: t> s > r >q 2) What about if k changes his mind and switches his preferences to t>s>r>q ? Identify (if they exist ) agendas that q, j and k could design to secure their top choices ) Exercises ( 23) Three individuals : i,j,k ; P = > i: q>s>r>t ; j: r>q>t>s ; k: t> s > r >q 2) What about if k changes his mind and switches his preferences to t>s>r>q ? Identify (if they exist ) agendas that q, j and k could design to secure their top choices ) In this case there is no Condorcet winner , therefore different agendas can drive to different results . Exercises ( 23) Three individuals : i,j,k ; P = > i: q>s>r>t ; j: r>q>t>s ; k: t> s> r>q 2) First you have to identify the best alternative for each individual and which alternatives are defeated by this alternative. These alterative must located in the last round of the agenda. Then you have to select an alternative that is defeated by the alternative defeated by the top choice . i: q; (….s q); (….t q); (r t s q) (r t q s) (r s t q) (r s q t) j: r; (…..q r); (….t r); (s t q r) ( s t r q) (s q t r) (s q r t) k: t; (…..s t) ; (q r s t) (q r t s) Other two agendas drive to s as winner (q t r s ); (q t s r) Exercises ( 24) Three individuals : i,j,k ; P = > i: q>s>r>t j: r>q>t>s k: t>s>r>q 1) Suppose that player j proposes the agenda ( tsrq ). Has player i any incentive to strategically misrepresent her vote if she assumes the others vote honestly ? 2) Suppose k proposes the agenda ( rqst ). Can j do better by misrepresenting his preferences than by voting sincerly ? Exercises ( 24) Three individuals : i,j,k ; P = > i: q>s>r>t j: r>q>t>s k: t>s>r>q 1) Suppose that player j proposes the agenda ( tsrq ). Has player i any incentive to strategically misrepresent her vote if she assumes the others vote honestly ? If i vote sincerly the outcome is r . If during the second round she votes for t instead of r then q ( her top preference ) will win . Exercises ( 24) Three individuals : i,j,k ; P = > i: q>s>r>t j: r>q>t>s k: t>s>r>q 1) Suppose k proposes the agenda ( rqst ). Can j do better by misrepresenting his preferences than by voting sincerly ? If j vote sincerly the outcome is t. If during the first round he votes for q instead of r then q ( his second choice ) will win . Exercises ( 25)The following table illustrates the probability of cyclical majority as the number of voters and/or the number of alternatives increase. a) Why these probabilities can overemphasize the real occurence of the phenomenon ? ( 1) .……………………………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………………………………… …………………………… …………………………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………………………………………………… b) Why, on the contrary, can they underestimate it ? (2) ………………………………………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………………………………………… Hume’s Marsh – Draining game • No matter what Farmer B does, Farmer A always gets a higher payoff if he chooses not to drain. The reasoning is precisely the same • No matter what Farmer A does, Farmer B always gets a higher payoff if he chooses not to drain. Stag Hunt game (Rousseau) • An alternative vision of the problem of social cooperation is provided by the Stag Hunt Game Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • What is the most preferred outcome ? Is there another outcome in which neither player has an incentive to alter his strategy ? Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • The most -preferred outcome for both players is (Stag, Stag), for which each player receives a payoff of 3. This outcome is an equilibrium inasmuch as neither player wishes to alter his strategy when he believes the other player will be playing Stag. • (Hare, Hare) is also a stable outcome or equilibrium because if A believes that B is going to play Hare, than A’s best response is also to play Hare. Likewise, if B believes A is going to play Hare, than B will play Hare, too. Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • Does either player end up doing better playing either Stag or Hare no matter what his partner chooses to do ( as in the marsh -draining game)? Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • Unlike the marsh -draining game,neither Stag nor Hare is always the optimal strategy regardless of the strategy employed by the other player. If a player believes his partner will play Stag then his best option is to play Stag. But if a player believes his parter will play Hare, than his best response is to play Hare. Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • How certain must A be that B will playing Stag to do the same ? Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • Let’s define pB as the probability that B plays Stag . Then A’s expected utilities associated with the two strategies are: • EUA[Stag] = pB ⋅ 3 + (1 − pB ) ⋅ 0 = 3pB • EUA[Hare] = pB ⋅ 1 + (1 − pB ) ⋅ 1 = 1 Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • A will wish to play Stag when EUA[Stag] > EUA[Hare] , namely 3pB > 1 or pB > 1/3 Achieving the most -preferred outcome in this game then requires that both players believe that the other player will play Stag with at least probability 1/3. One interpretation of this is that the equilibrium depends on each player’s conjecture about the other’s behavior . Another interpretation is that the players must trust one another to play a certain outcome (at least up to a point) in order to secure the socially -optimal outcome. Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Chicken game ( Rebel Without a Cause, 1955) • Another famous coordination Game. Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • What is (are) the most preferred outcome (s) ? What are the «pure» equilibria outcomes ? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • Does either player end up doing better playing either Go Straight or Swerve no matter what his competitor chooses to do ( as in the marsh -draining game)? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • Does either player end up doing better playing either Go Straight or Swerve no matter what his competitor chooses to do ( as in the marsh -draining game)? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game (« Rebel Without a Cause», 1955) • How certain must A be that B will playing Swerve to play the Go Straight ? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game (« Rebel Without a Cause», 1955) Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 • Let’s define pB as the probability that B plays Go Straight . Then A’s expected utilities associated with the two strategies are: • EUA[Go straight] = pB ⋅ -5 + (1 − pB ) ⋅ 3 = 3 -8pB • EUA[Swerve] = pB ⋅ -1 + (1 − pB ) ⋅ 0 = -pB Chicken game (« Rebel Without a Cause», 1955) Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 • A will wish to play Go Straight when • EUA[Go Straight] > EUA[Swerve] , namely • 3 -8pB > -pB or 3 >7pB or pB <3/7 Gangs’rewarding cooperation • In order to prevent the prisoner’s dilemma outcome , criminal organizations can also reward the “cooperation” ( do not confess) for instance by looking after an individual’s family while the criminal is in prison. Gangs’rewarding cooperation • Suppose that a bonus of is given to a criminal who cooperates but whose partner defects, while a payoff of is given to a criminal who cooperates and whose partner also cooperates. a) Rewrite the payoff matrix b) For what values of and is cooperation an equilibrium ? c) For what values is it the only equilibrium ? Gangs’rewarding cooperation • Mutual cooperation is an equilibrium if ≥ 1. For what values is it the only equilibrium ? • Mutual cooperation is the only equilibrium if ≥1 Apartment cleaning • 4 friends (X, Y , Z ,W) live together in a college apartment and must work together to clean common areas . Outcome is dichotomous and has the feature of the following collective action problem . Assume that B (utility coming from cleaniless )>C ( cost of cleaning ) Apartment cleaning • What are the possible equilibrium outcomes when all 4 friends must contribute ? Which do you think is likely ? Apartment cleaning • What are the possible equilibrium outcomes when all 4 friends must contribute ? Which do you think is likely ? • There are two outcomes which are equilibria: 1) everyone contributes or 2) no one contributes. If everyone contributes, each individual secures a benefit B and pays cost C. Thus, their net payoff is B -C > 0. 1) With everyone contributing, if one person decides to not contribute, than the apartment is not cleaned. Those contributing then get net payoff -C, while the person who didn’t contribute gets a payoff of 0. Because B -C > 0, the now non -contributor is worse off than she had been when she contributed along with all of her apartment mates . 2) If no one is contributing, than each player earns a payoff of 0 . No player will wish to unilaterally start contributing because that will only lead to them paying the cost of contribution without securing any benefit, hence the net payoff goes from 0 to -C, and 0 is preferable. Apartment cleaning • What are the possible equilibrium outcomes when only 2 of the 4 friends must contribute to clean the apartment (k=2)? Can you predict which outcome will occur without further information ? Apartment cleaning • What are the possible equilibrium outcomes when only 2 of the 4 friends must contribute to clean the apartment (k=2)? Can you predict which outcome will occur without further information ? • If only two members are required to clean the apartment, than any combination of the two apartment mates contributing is an equilibrium. XY; XZ; XW; YZ; YW; WZ • Imagine for example, suppose X and Y contribute and Z and W don’t. Z and W certainly don’t want to start contributing because they are already getting B without having to pay C. Nevertheless X and Y still prefer B -C to 0, which is what will occur if either one of them decides to not contribute. There are 6 possible ‘cooperative ’ equilibria. • But there is also one ‘non -cooperative’ equilibrium as no one wants to be a sucker and start contributing on their own, because one person cleaning is insufficient to fully clean the apartment. Thus, there are 7 possible equilibria, and it is hard to predict beforehand which will occur . • However , compared to the previous condition, now each roommate has ½ probability to enjoy the cleaniless without any effort Apartment cleaning • How might the prediction change ( when k=2) if B increases or C increases ? What about if B is different for different members of the group ? Apartment cleaning 1)How might the prediction change ( when k=2) if B increases or C increases ? 2) What about if B is different for different members of the group ? 1) Higher B or lower C means that the incentives to coordinate on a cooperative equilibrium are greater . 2) we might expect to see high -B individuals exploited by low -B members, who free ride confident in the knowledge that the very high benefits secured by high -B individuals will motivate them to coordinate on cleaning the apartment. Typology of goods Excludability Yes No Non Rivalrous Yes No Cable or satellite TV «Premium» version on line Journal A Pizza The Global Positioning System (GPS) Public beaches Knowledge Street lighting National Parks One individual public good provision game with mixed strategies • Suppose that there are n individuals who desire a collective good that yields benefit B to all n individuals . Provision of the good requires only one individual (k=1) to expend C to provide it (B>C). Show that there are n possible sets of «pure» strategy equilibria ( each player i plays « contribute » or « Don’t contribute » with probability 1) One individual public good provision game with mixed strategies • The n possible pure strategy are all the same: one individual contributes and no other player does. • If one individual is contributing, no one else wishes to add on their own contribution because the group benefit B has already been secured by all, and adding a contribution would only waste C units of utility. • The net payoff for the sole contributor is B -C; however, we know this is a positive quantity which the contributor prefers to receiving no B and paying no C ( that would be equal to 0) , which is what will occur if he withdraws his contribution. • There is one ‘single -contribution’ equilibrium for each of our n individuals, and hence n pure strategy equilibria. One individual public good provision game with mixed strategies • Now suppose that all players are playing an indentical mixed strategy . In other terms they probabilistically choose whether to play C (oop .)or D( on’t ) . Call p the probability that any one player plays C. a) Show that for any player i, if he does not contribute , the probability that the good is supplied by some one else is 1 -(1 -p) n -1 One individual public good provision game with mixed strategies if A represents some event occurring and A’ represents that event not occuring , then Pr (A’) = 1 -Pr(A) 2. If A and B are two independent events, then Pr (A and B both occur) = Pr (A)* Pr (B). The probability that a player contributes is p, the probability of that player not contributing is (1 − p). Each player makes their decision independently , so the probability that every player but i does not contribute is (1 − p)(1 − p)…(1 − p) = (1 -p) n -1 If (1 -p) n -1 is the probability that everyone but i does not contribute, then 1 − (1 − p) n−1 is the probability that at least one person (excluding i for the moment) contributes. One individual public good provision game with mixed strategies • Equate the expected utility for i of contributing with the expected utility of not contributing . • Solve the expression you found for p • Show that p is decreasing in C, increasing in B and decreasing in n. One individual public good provision game with mixed strategies • Equate the expected utility for i of contributting with the expected utility of not contributing . • The utility for i of contributing is : EUi [Contribute] = B − C; • while the expected utility for i of not contributing is EUi [Don’t] = B[1 − (1 − p) n−1 ] One individual public good provision game with mixed strategies • When these two expected utilities are equal, i is ambivalent about whether to play Contribute or Don’t, which opens up the possibility of playing a probabilistic mixture of Contribute or Don’t, otherwise known as a mixed strategy. • A player is only willing to play this probabilistic mixture of strategies when the payoffs associated with each strategy are exactly equal. One individual public good provision game with mixed strategies • B − C = B[1 − (1 − p) n−1 ] • B(1 − p) n−1 = C • p = 1 − (C/B ) 1/n− 1 • B > C, therefore C/B < 1. As C increases, the second term gets larger so p gets smaller. • An intuitive interpretation of this is that as the costs of contribution increase, each person is less willing to contribute, holding all other factors constant. • Similarly, as the benefits of contribution (B) increase, individuals are more likely to contribute, reflecting the extra gains from contribution. Finally, as the number of individuals increases, each person is less likely to contribute. • This makes sense, because as more individuals contribute with some probability p the more likely it is the good will be provided (only one needs to contribute, after all so each person can relax a little bit. Externalities • A factory located in a small village produces a good with increasing marginal costs MC(q) = 12 + q; so the first unit costs 13 , the second 14 etc. This firm can produce at most 15 and no fractional amount can be produced . The market price for the good is p=20$ and firm’s level of production does not affect this price . Assume that the factory owner maximizes profit and her utility is measured in dollar . Profit is calculated by summing up differences between the price and the marginal cost of each unit produced . • The factory is noisy and interfere with the practice of a neighboring doctor . For every extra unit produced the doctor loses $2 worth of profits . • The doctor’s welfare depends only on his profits , which are 50 -2q Externalities 1) How many units of the good will the factory produce if it ignores the externality imposed on the doctor in its profit maximization ? What will be the aggregate social utility ( factory’s and doctor’s total profits ) ? Externalities (1) • If the factory ignores the external effects of its production, then it will produce up to the point where the marginal revenue of an extra unit equals the marginal cost of an extra unit. The marginal revenue for each unit is $20, and is invariant to the level of production. The marginal cost increases steadily, and will equal $20 when q = 8, which yields a profit of 7+6+5+4+3+2+1+0 = $28. • At this level of production, the doctor’s profits are 50 − 2q = $34. Therefore, aggregate social utility is 28 + 34 = $62 unit revenue cost profit 1 20 13 7 2 40 27 13 3 60 42 18 4 80 58 22 5 100 75 25 6 120 93 27 7 140 112 28 8 160 132 28 9 180 153 27 10 200 175 25 11 220 198 22 0 5 10 15 20 25 30 0 2 4 6 8 10 12 Titolo del grafico marginal revenue marginal cost profit Externalities 2) Identify the level of production that is socially most preferred , in other terms that maximizes aggregate social utility. Externalities (2) • Social utility, S, has been defined as the sum of the factory’s and doctor’s profits. • S(q) = 20 − σ = 1 12 + + 50 − 2 . • This is maximized where q = 6. • We need to consider the marginal profit for an extra unit of production for the factory owner against the marginal cost of that extra unit imposed on the doctor. For example, if the factory increases q from 0 to 1, this garners the factory $7 units of profit while imposing a cost of only $2 on the doctor. • When q = 6, the extra two dollars of profit for the factory are exactly cancelled out in the social utility function by the two dollars of loss in the doctor’s profits. • At this level of production, S = 120 -93+38 = $65 -10 0 10 20 30 40 50 60 70 0 2 4 6 8 10 12 Titolo del grafico Marginal profit Marginal d’s cost Social utilityunit revenue marginal revenue marginal cost cost profit Doctor’s tot utility Marginal profit Marginal d’s cost Social utility 1 20 20 13 13 7 48 7 2 55 2 40 20 14 27 13 46 6 2 59 3 60 20 15 42 18 44 5 2 62 4 80 20 16 58 22 42 4 2 64 5 100 20 17 75 25 40 3 2 65 6 120 20 18 93 27 38 2 2 65 7 140 20 19 112 28 36 1 2 64 8 160 20 20 132 28 34 0 2 62 9 180 20 21 153 27 32 -1 2 59 10 200 20 22 175 25 30 -2 2 55 11 220 20 23 198 22 28 -3 2 50 Externalities 2) Propose a government taxation scheme that will lead to socially preferred outcome … Obviously a tax of $2 per unit of production levied against the factory would lead to the socially optimal amount of production. The factory would only produce up to 6 units (after this point, the marginal profit of an extra unit turns negative) Donation of time • 5 civic -minded patrons of a public library contemplate donations of time to its annual fundraiser . Each individual i bases his or her decision of how much time to donate, x i, on the following utility function : Where namely the total amount of time given by all library patrons and is the cost of losing x i of one’s leisure time Donation of time • What is the socially optimal amount of time donated ? Determine this by summing the utility functions of five individuals , and finding the q that maximizes this function . Donation of time • What is the socially optimal amount of time donated ? Determine this by summing the utility functions of five individuals , and finding the q that maximizes this function . • First we need an expression for society’s utility, (S ). We can find this by adding up the utilities of the individuals who we will index by j 2 (1; 2; 3; 4; 5): Donation of time • We can maximize with respect to q to find the socially optimal net contribution (let’s call it q*). Because all of our individuals are identical, we can then divide this by 5 to find the socially optimal individual contribution x* i At the maximum of this function, this derivative will equal zero. We can solve this for q to find that The socially optimal individual contribution is thus Donation of time • Imagine that an individual assumes that everyone else will contribute no time. How much time will this individual donate ? Is it at the socially optimal level ? Donation of time • Imagine that an individual assumes that everyone else will contribute no time. How much time will this individual donate ? Is it at the socially optimal level ? • If some individual i believes that no one else will contribute, then he will behave as if ; his expected utility will be ;Maximizing this with respect to x i, ; x i=1 < 1.710 , the socially optimal individual contribution < 8.550, the “social” optimal level. Donation of time • Imagine that each individual assumes that the other 4 members will donate 0.8 units of time. Does this individual donate more or less time than before ? Why this ? Donation of time • Imagine that each individual assumes that the other 4 members will donate 0.8 units of time. Does this individual donate more or less time than before ? Why this ? If i assumes that the other individuals will collectively supply .8 units he will believe that ; Maximizing with respect to x i, we get this expression which equals zero where EUi is maximized ; x i=.2 Donation of time • Imagine that each individual assumes that the other 4 members will donate 0.8 units of time. Does this individual donate more or less time than before ? Why this ? The level of contribution .2 turns out to be a symmetric equilibrium , where each individual gives .2 towards the cause. This results in only 1 unit total of time donated, which is far less than the socially optimal level of 8.550 . Each individual gives only .2 , rather than the 1.710 units which would maximize the group’s welfare. The individual’s belief that the others will collectively provide .8 units is confirmed in equilibrium. Thus , this belief is “rational”. Dimension by dimension decision making process • 5= SQ • Committees of members with gatekeeping powers • Specific jurisidictions attached to committees • Rules of amendment once a committee has sent a bill to the full legislative body SQ Dimension by dimension decision making process • Which partioning of legislatures into commitees and which structure of jurisdictions for those committees can give rise to a stable , predictable equilibrium under: 1) Closed rule 2) Open rule with germaneness rule in effect (related to the substance of the original bill ) SQ Dimension by dimension decision making process Closed rule • Dimension X = comm. 1,2,3 • Dimension Y = comm. 4,6,7. • If SQ=5 then 2 on X dimension is approved by 1,2,3, 7 • On Y dimension SQ=5 is the median voter therefore Committee will keep the gate closed . SQ Dimension by dimension decision making process O pen rule (with germaneness ) • Dimension X = comm. 1,2,3 • Dimension Y = comm. 4,6,7. • If SQ=5 then on dimension X Floor would approve 1 Comm. will open the gate. • On Y dimension Floor would approve 4 (worse than 5 for the comm.) Committee will keep the gate closed . SQ Dimension by dimension decision making process Would a change in the committee membership invalidate the previous equilibria ? SQ Dimension by dimension decision making process Closed rule =Open rule • Dimension X = comm. 1,3,5 • Dimension Y = comm. 4,6,7. • If SQ=5 then 1 on X dimension is approved by 1,7,2,3 • On Y dimension SQ=5=6 is the median voter therefore Committee will keep the gate closed . SQ Dimension by dimension decision making process Under what circumstances would a stable equilibrium not exist? SQ Dimension by dimension decision making process 1) If the committees had multi – dimensional jurisdictions. Imagine committee 1,2,3 with gatekeeping power and closed rule SQ Dimension by dimension decision making process 1) If the committees had multi -dimensional jurisdictions. Imagine committee 1,2,3 with gatekeeping power and closed rule .. There are a wide range of (x,y ) pairs that each member of the committee (and player 7 or 6, needed to form a majority) would prefer to 5’s ideal point, but in the multi -dimensional spatial setting it is impossible to predict which of these will be the committee’s proposal. But at least there is a range of plausible proposals. Dimension by dimension decision making process 1) If instead, the committee had multidimensional juridiction and was composed of 1, 2, 3, 4 and 7, then it is impossible to even state a plausible range of proposals. Similarly, if committee jurisdiction is multi -dimensional and the full house operates under an open rule, then chaos is likely to prevail. Dimension by dimension decision making process 2) The second circumstance is when a germaneness rule is not in effect (but it is in effect the open rule) .Under this scenario, even if committee jurisdictions are limited to single issue dimensions, any proposal made by the committee will be subject to the chaos of the multidimensional spatial world once it reaches the full house. Multidimensional decision making process • Three equivalent blocs of voters (1,2,3) • 3 has gatekeeping power and legislature operates under open rule . • Status quo=q Multidimensional decision making process • If 3 open the gates and propose p to the whole house could it achieve final passage of that bill ? In general would the committee be guaranteed final passage of a bill that it prefers to q ? Multidimensional decision making process • P oint p is strictly preferred to q by both 2 and 3 (a majority) . However 3 cannot guarantee passage of a bill that it prefers to q under an open rule… Multidimensional decision making process • The committee proposes p. When p reaches the full legislature, it can be amended under the open rule. 1 might propose some alternative r which is strictly preferred by 2 to p and which 1 also prefers. • The proposal r is then a plausible alternative but in fact, this process of amendments could continue ad libitum and it is impossible to predict what will happen. • In an open -rule setting the committee can’t guarantee an outcome which is preferred to q Multidimensional decision making process • Suppose that there is a rule which grants the members of the committee , 3, an after -the -fact veto ( it can reinstate the status quo q). If the committee opens the gates proposing the p, would it be guaranteed final passage of a bill that it prefers to q ? Multidimensional decision making process • Any final legislation must be strictly preferred by the committee to q .For example, if 1 and 2 were to propose a point like r again, it might pass the whole house but would then be vetoed by the committee resulting in the policy remaining at the status quo, q. • In fact, any point that 1 prefers to q would be vetoed by the committee, therefore 1 is unlikely to be part of any coalition with 2 to amend the status quo. • A wide array of points that 2 and 3 can agree are preferred to q, and this range of points are one set of reasonable predictions for the outcome . Specifically, any point on the dashed line connecting 2 and 3’s ideal points which falls in the preferred -to -q sets is a plausible outcome. Multidimensional decision making process • Granting the committee an after -the -fact veto provides the committee a measure of control over the eventual outcome. • This feature grants committee’s real power, tempering the otherwise chaotic nature of the multi -dimensional spatial setting with an open rule. In this way, specialized committee’s become a vehicle for legislator’s with special interests to secure influence over outcomes in those areas. Krehbiel model • A persistent feature of American political life is the legislative gridlock (high policy stability ) • Khrebiel insist on the importance of the real rules of the U.S. law making 1) Congress can override a presidential veto if a 2/3 of the members vote to do so 2) Most bills can only escape ( for ending the filbustering ) the Senate with a vote of cloture (3/5 of senators ) Krehbiel model • Given president p c = median voter of the Congress v= the ideal point of the pivotal member of Congress need to override a presidential veto (2/3th) f= the ideal point of the pivotal member of the Senate necessary to close the debate (3/5th) Krehbiel model Can the Congress secure the implementation of any preferable law when the status quo is …? SQ SQ SQ SQ The possibile pivots are in fact 4 ( v, v’ and f and f’) but when the position of the president is known than f’ and v’ are not influent . So we have to consider only v and f and c. v ’ f’ 2/3 2/3 3/5 3/5 Krehbiel model • Given president p c = median voter of the Congress v= the ideal point of the pivotal member of Congress need to override a presidential veto (2/3th) f= the ideal point of the pivotal member of the Senate necessary to close the debate (3/5 ° ) Outcome =c Outcome =(cx* 01.2  1,5Q 2 + p[f(Q -1,2 )] + (1 -p)0 2) If p=0.2 and f=5 for Q>1.2  1,5Q 2 + 0.2[5(Q -1,2 )] + ( 1 -0.8)0 = 1,5Q 2 + Q – 1,2 Niskanian Bureaucracy 1) If p=0.2 and f=5 for Q>1.2  1,5Q 2 + 0.2[5(Q – 1,2)] + (0,8)0 = 1,5Q 2 + Q – 1,2 for Q≤1,2  EC(Q )= 1,5Q 2 The legislature’s demand constraint remains binding, and so Q** = 2. Niskanian Bureaucracy 1) If p=0.2 and f=5 for Q>1.2  1,5Q 2 + 0.2[5(Q – 1,2)] + (0,8)0 = 1,5Q 2 + Q – 1,2 for Q≤1,2  EC(Q )= 1,5Q 2 The legislature’s demand constraint remains binding, and so Q** = 2. 2,16 Niskanian Bureaucracy 1) If p=0.5 and f=10 for Q>1.2  1,5Q 2 + 0,5[10(Q – 1,2)] + (0,5)0 = 1,5Q 2 + 5Q – 6 for Q≤1,2  EC(Q)= 1,5Q 2 The legislature’s demand constraint is not binding anymore . Niskanian Bureaucracy We set the cost constraint B=TC; 1,5Q 2 + 5Q – 6= 8Q -2Q 2 3,5Q 2 – 3Q – 6= 0 Then we have to solve the equation for Q Q= 1,806 Niskanian Bureaucracy We set the cost constraint B=TC; 1,5Q 2+ 5Q – 6= 8Q – 2Q 2 3,5Q 2- 3Q – 6= 0 The we have to solve the equation for Q Q= 1,806 .Now the monitoring system makes the cost constraints more binding than the demand constraints Principal – agent game • A principal delegates some authority to an agent and can choose whether or not to audit that agent’s effort in any period • An audit is costly to the principal , but he does not have to pay the agent if he detects shirking • The principal earns 4 if his agent works ; she earns 0 if the agent shirks • The principal pays the agent 3 to work but if she audits and catches the agent shirking he does not have to pay the agent • It costs the agent 2 to do his work • The audit costs 1 to the principal Principal – agent game Principal Agent Audit Don’t Audit Work (3 -2), (4 -3-1) (3 -2), (4 -3) Shirk 0, -1 3, -3 Principal Agent Audit Don’t Audit Work 1, 0 1, 1 Shirk 0, -1 3, -3 • Does either individual have a strategy that is optimal no matter what the other individual plays ? • Are any of the four cells equilibria ? • What percentage of time must the principal inspect to make the agent indifferent between working and shirking ? • What percentage of time must the agent work to make the principal indifferent between auditing and not auditing ? Principal – agent game Principal Agent Audit Don’t Audit Work 1, 0 1, 1 Shirk 0, -1 3, -3 • Does either individual have a strategy that is optimal no matter what the other individual plays ? Are any of the four cells equilibria ? • Neither individual has a strategy which is optimal no matter what the other person plays (there is no dominant strategy .) • If the agent works, the principal prefers not to audit, but if the agent does not work, he of course prefers to audit. If the principal does not audit the agent prefers to shirk, but if the principal audits, the agent prefers to work. None of the four cells represents a pure strategy equilibrium . In other terms, given a certain strategy of a player it is not true that conditional on the other player’s choice of strategy, the player has no incentive to play a different strategy. Principal – agent game Principal Agent Audit Don’t Audit Work 1, 0 1, 1 Shirk 0, -1 3, -3 • What percentage of time must the principal inspect to make the agent indifferent between working and shirking ? • What percentage of time must the agent work to make the principal indifferent between auditing and not auditing ? pA = the probability of an audit or inspection. The expected utilities of the principal’s two strategies ( Audit or No Audit): EU[Work ] = pA ⋅ 1 + (1 − pA ) ⋅ 1 = 1 EU[Shirk ] = pA ⋅ 0 + (1 − pA ) ⋅ 3 = 3 − 3pA: Thus, the agent is indifferent between working and shirking when 1 = 3 − 3pA or pA = 2/3 . Principal – agent game Principal Agent Audit Don’t Audit Work 1, 0 1, 1 Shirk 0, -1 3, -3 • What percentage of time must the principal inspect to make the agent indifferent between working and shirking ? • What percentage of time must the agent work to make the principal indifferent between auditing and not auditing ? pW the probability that the agent works. The expected utilities of the agent’s two strategies are then: EU[Audit] = pW ⋅ 0 + (1 − pW ) ⋅ −1 = −1 + pW EU[ Don’tAudit ] = pW ⋅ 1 + (1 − pW ) ⋅ −3 = −3 + 4pW : The principal is therefore indifferent between auditing and not auditing when -1+3 =3pW ; pW = 2/3 Principal – agent game Principal Agent Audit Don’t Audit Work 1, 0 1, 1 Shirk 0, -1 3, -3 • These strategies (the agent randomly chooses to work with probability pW = 2/3 and the principal randomly chooses to audit with probability pA = 2/3 ) are a mixed strategy equilibrium. • They share the defining property of an equilibrium with pure (i.e. non – probabilistic) strategies: conditional on the other player’s strategy remaining the same, neither player wishes to alter his or her strategy. Each players’ strategy leaves the other player indifferent between his two strategies, and therefore content to play a probabilistic mixture himself. Principal – agent game Principal Agent Audit Don’t Audit Work 1*(2/3*2/3) , 0 *(2/3*2/3) 1*(2/3*1/3) , 1*(2/3*1/3) Shirk 0*(2/3*1/3) , -1*(2/3*1/3) 3*(2/3*1/3) , – 3 *(1/3*1/3) • The average payoff under mixed strategy equilibrium for agent is 1 and for the principal is -1/3 Principal Agent Audit Don’t Audit Work 4/9, 0 2/9, 2/9 Shirk 0, -2/9 3/9, -3/9 Ferejohn (1986) about accountability • Suppose the median voter’s V ideal point in 0 and an elected leader’s ideal point in 1 in on a single -dimensional issue space ( valence issue , corruption ) V L 0 1 • L’s utility for any outcomes is equal to p ( 0≤p≤1) and T for each term in office . Only two terms in office are possible . • In the first term the total payoff is p+T . • In the second term the total payoff is λ ( p+T ) where λ is a discount factor <1 Ferejohn (1986) about accountability • If L is reelected for a second term , what policy will be implemented ? • Assume voters use a « retrospective voting strategy » of the form : reelect if p ≤ r, and vote out otherwise . a) Come up with two expressions for L’s utilities, one if he is reelected and one if he is not , assuming for each case that L sets p as high as possible consistent with the desired electoral outcome . b) Show that for voters the optimal voting rule has r=1 – λ – λT or 0 depending on the values of λ and T c) How does voter utility in equilibrium change with λ and T ? Ferejohn (1986) about accountability • If L is elected for a second term then he will implement p = 1 in that term. This is because he can no longer be held accountable by the electorate and so freely selects his most -preferred policy without facing any negative consequences . • L has two electoral strategies to consider. 1) First , he can attempt to satisfy voters by choosing a p ≤ r in the first round. The overall payoff attached to this strategy is r + T + λ (1 + T) = r + λ + (1 + λ )T . 2)Alternatively , L can forget about reelection and attempt to milk everything possible out of a single term in office by choosing p = 1 in the first round. This results in an overall payoff of 1 + T. Ferejohn (1986) about accountability • Voters can use these possible payoffs to determine an optimal voting rule, i.e. a value for r that just guarantees `good behavior’ in the first term at minimal cost. The politician will choose his `seek re -election’ strategy only if r+ λ +( 1 + λ )T ≥ 1+T . In other terms if r ≥ 1 – λ – λ T. The lowest r at which voters can ensure that the politician behaves himself in the first term is r = 1 – λ – λ T • Depending on the values of λ and T, it may be that r = 0, i.e. any politician will prefer to behave himself in the first term to guarantee the payoffs in the second term . This is more likely as λ gets larger (meaning the politician doesn’t discount future payoffs heavily) and as T gets larger (meaning there is a big payoff associated with simply being in office). • Voters always get a payoff of 0 in the second round, therefore we only need to consider their payoff of p = r = 1 – λ – λ T in the first round . Ferejohn & Weingast model of Court’s behaviour • XH= median voter in House • XS= median voter in Senate Supreme Court ( XSc , median voter ) is hearing an argument about some statute ( at XQ) passed in a previous session of Congress . XH XS XSc XS/XH XH/XS New legislation XQ New legislation XQ XQ Ferejohn & Weingast model of Court’s behaviour • Assume that XSc =XQ Would the Supreme Court alter the bill when XQ is ? XH XS XSc XS/XH XH/XS New legislation XQ New legislation XQ XQ XQ Ferejohn & Weingast model of Court’s behaviour • Assume that XSc =XQ Because xQ is between xH and xS , there will be no way for one house to move the law closer to its ideal point which doesn’t make the other house worse off. Anticipating this, the Supreme Court will leave the law unchanged and secure its ideal point, xQ , in equilibrium XH XS XSc XS/XH XH/XS New legislation XQ New legislation XQ XQ XQ Ferejohn & Weingast model of Court’s behaviour • Assume that XSc =XQ Would the Supreme Court alter the bill when XQ is ? XH XS XSc XS/XH XH/XS New legislation XQ New legislation XQ XQ XQ Ferejohn & Weingast model of Court’s behaviour • Assume that XSc =XQ • The House and the Senate will be able to agree on a variety of proposals which both houses strictly prefer to xQ . Proposals between xH and either xH + jxH − xQj (that is, proposals which are up to equally far from xH as xQ is,but on the right side) or xS , whichever is smaller. These proposals constitute the bargaining range. XH XS XSc XS/XH XH/XS New legislation XQ New legislation XQ XQ XQ XH+|XH -XQ| Bargaining range Ferejohn & Weingast model of Court’s behaviour • Assume that XSc =XQ • if the Supreme Court proposes XSc • the outcome will be in the range [XH , xH + |XH − XSc |] or [ xH,xS ], whichever is shorter. • What is the Supreme Court’s optimal proposal? It will be XSc = xH . • Anything less than xH raises the possibility of a final bargained outcome between the Senate and House which is greater than or equal to xH . This strategy guarantees an outcome equal to xH , because the House will have no interest in compromising with the Senate to move the status quo closer to the Senate’s position XH XS XSc XS/XH XH/XS New legislation XQ New legislation XQ XQ XQ XH+|XH -XQ| Bargaining range Ferejohn & Weingast model of Court’s behaviour • Assume that XSc =XQ • These results suggest that Supreme Court justices (and judges on lower courts who may be asked to interpret or amend existing law) might have an incentive to change the law, if their main goal is as little change in the law as possible. XH XS XSc XS/XH XH/XS New legislation XQ New legislation XQ XQ XQ XH+|XH -XQ| Bargaining range Example government formation according to different hypotheses • Suppose some country with PR ends up with the following distribution of 100 Parliamentary seats among 5 parties from left to right A (15) B(28) C(11) D(33) E(14) 1) Find all minimum winning coalitions (MWC) 2) Find the smallest MWC 3) Find the MWC with the fewest members 4) Find all MWCs for which the parties are adjacent in the political space Example government formation according to different hypotheses • Suppose some country with PR ends up with the following distribution of 100 Parliamentary seats among 5 parties from left to right A (15) B(28) C(11) D(33) E(14) 1) Find all minimum winning coalitions (MWC) The set of minimum winning coalitions is: ABC (54 members), ABE (57), ACD (59), ADE (62 ), BCE (53), BD (61), and CDE (58). Example government formation according to different hypotheses • Suppose some country with PR ends up with the following distribution of 100 Parliamentary seats among 5 parties from left to right A (15) B(28) C(11) D(33) E(14) 2) Find the smallest MWC The smallest MWC in terms of number of parliamentarians is BCE with 53 . Example government formation according to different hypotheses • Suppose some country with PR ends up with the following distribution of 100 Parliamentary seats among 5 parties from left to right A (15) B(28) C(11) D(33) E(14) 3) Find the MWC with the fewest members 4)Find all MWCs for which the parties are adjacent in the political space The MWC with the fewest parties is BD. The only coalitions with adjacent parties are ABC and CDE. Cabinet formation • Two dimensions ranging from 0 to 10 • X dimension = Finance • Y dimension = Defense • Three parties A, B, C ; no party has a majority of seats , two parties are sufficient to have a majority of seats Cabinet formation • What is the set of possible minority and majority governments ? • Is there a stable majority government coalition ? • Is there a minority government that is a stable equilibrium ? Cabinet formation • What is the set of possible minority and majority governments ? • Is there a stable majority government coalition ? • Is there a minority government that is a stable equilibrium ? Cabinet formation • What is the set of possible minority and majority governments ? • Is there a stable majority government coalition ? • Is there a minority government that is a stable equilibrium ? Cabinet formation • What is the set of possible minority and majority governments ? • Is there a stable majority government coalition ? • Is there a minority government that is a stable equilibrium ? Chicken game ( Rebel Without a Cause, 1955) • Another famous coordination Game. Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • What is (are) the most preferred outcome (s) ? What are the «pure» equilibria outcomes ? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • Does either player end up doing better playing either Go Straight or Swerve no matter what his competitor chooses to do ( as in the marsh -draining game)? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • Does either player end up doing better playing either Go Straight or Swerve no matter what his competitor chooses to do ( as in the marsh -draining game)? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game (« Rebel Without a Cause», 1955) • How certain must A be that B will playing Swerve to play the Go Straight ? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game (« Rebel Without a Cause», 1955) Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 • Let’s define pB as the probability that B plays Go Straight . Then A’s expected utilities associated with the two strategies are: • EUA[Go straight] = pB ⋅ -5 + (1 − pB ) ⋅ 3 = 3 -8pB • EUA[Swerve] = pB ⋅ -1 + (1 − pB ) ⋅ 0 = -1 Chicken game (« Rebel Without a Cause», 1955) Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 • A will wish to play Go Straight when • EUA[Go Straight] > EUA[Swerve] , namely • 3 -8pB > -1 or 4 >8pB or pB <1/2 Fees and cooperation • A cultural association, composed of 100 members, has to pay as soon as possible 1000$ of annual rent for the room where its weekly meetings take place. Each member enjoys on from these meetings a utility equal to 15$ per year. All this is common knowledge. Some members suggest the association chair to fix the fee 15> f > 10 just have some margin in case of few member’s defection; Others insist on keeping the fee as low as possible, namely f=10. Which advice should the chair follow and why ? • 2 0 members decide to voluntarily donate to the chair 280 $ to cope with such emergency and this donation is common knowledge. • Should the association chair change the previous fee ? Which is the final fee ? Fees and cooperation • A cultural association, composed of 100 members, has to pay as soon as possible 1000$ of annual rent for the room where its weekly meetings take place. Each member enjoys on from these meetings a utility equal to 15$ per year. All this is common knowledge. Some members suggest the association chair to fix the fee 15> f > 10 just have some margin in case of few member’s defection; Others insist on keeping the fee as low as possible, namely f=10. Which advice should the chair follow and why ? Chair should fix the fee as low as possible . In this case unless all members pay the rent won’t meet the required amount . In other words as the unanimity is required the cooperation will be very likely . Fees and cooperation • 20 members decide to voluntarily donate to the chair 280 $ to cope with such emergency and this donation is common knowledge. Should the association chair change the previous fee ? Which is the final fee ? Now the members that do not donate and that have still to contribute are 80 and the required amount is 720. Therefore the chair has to fix the fee equal to 9. Committe membership and closed rule • The parliamentary floor has to decide the composition of two committees (with 3 members ) that control dimension Y and dimension X and have gatekeeping power . The Parliament members are 1,2,3,4,5. Only one Mp can ( and has to) be member of both committees and each committee enjoys the closed rule . • Given the following position of SQ and the positions of MPs , which committes ( and controlling which dimension ) will be formed ? Committe membership and closed rule SQ 1 2 3 4 5 Committe membership and closed rule SQ 1 2 3 4 5 Committe membership and closed rule SQ 1 2 3 4 5 Committe membership and open rule . • What happens if the rule is open with germaneness ( amendments are possible only on the dimension that is considered by the committee ) ? Vote trading Each legislator has one vote on any issue coming before the body. He cannot aggregate the votes in his possession and cast them all, or some large fraction of them, for a motion on a subject near and dear to his heart (or those of his constituents). What prevent the vote trading to be an efficient solution to this type of problem ? Filling the matrix ..2) Write down payoffs to create the “ Stag Hunt game ” ( assurance game ) ( 1) Player B Cooperate Do not cooperate -1 Player A Cooperate Do not cooperate Constitutional and Statutory Interpretation Supreme Court ( XSc , Supreme Court median voter) is hearing an argument about some statute ( at XQ) passed in a previous session of Congress formed by House and Senate ( XH and XS, median voters of the two Chambers) . Assume that XSc =XQ and that in order to change a law a majority in both chambers is necessary, while it is necessary a majority of 2/3 in order to change the constitution. Assume also that 1/3 of the House members is on the left of XQ. Draw where the status quo will be located after the Supreme Court ’s statutory interpretation (1) and after a constitutional interpretation (1) Constitutional and Statutory Interpretation Supreme Court ( XSc , Supreme Court median voter) is hearing an argument about some statute ( at XQ) passed in a previous session of Congress formed by House and Senate ( XH and XS, median voters of the two Chambers) . Assume that XSc =XQ and that in order to change a law a majority in both chambers is necessary, while it is necessary a majority of 2/3 in order to change the constitution. Assume also that 1/3 of the House members is on the left of XQ. Draw where the status quo will be located after the Supreme Court ’s statutory interpretation (1) and after a constitutional interpretation (1) Multidimensional decision making process A is an agenda setter that enjoys a closed rule . Where is the policy outcome if a) the decision rule is the majority rule b) The decision rule is the unanimity rule Multidimensional decision making process A is an agenda setter that enjoys a closed rule . Where is the policy outcome if a) the decision rule is the majority rule b) The decision rule is the unanimity rule Accountability In the country Referendumland citizens have to vote about the MPs salary . They have already decided that each Mp can be in office only for two terms . The current salary is 6000 Euro per month The political group that sponsorized the referendum proposes to reduce such a salary . New salary would be 3000 Euro. • According to Ferejohn’s model how should citizens vote in order to increase accountability ? Coalitional drift • What is ? • Which is its relationship with the Bureaucratic drift ? p q r . . . 1 2 3 Multidimensional decision making process ( again ) p q r . . . 1 2 3 Three equivalent blocs of voters (1,2,3) 1 ( the committee )has gatekeeping power and legislature operates under open rule . Status quo=q p q r . . . 1 2 3 1 proposes to 3 p that is for both better than q. However such a alliance is broken by 2 that proposes r that is better for 2 than p p q r . . . 1 2 3 Suppose that there is a rule which grants the members of the committee , 1, an after – the -fact veto ( it can reinstate the status quo q). If the committee opens the gates proposing the p, would it be guaranteed final passage of a bill that it prefers to q ? Which coalition will be formed ? p q r . . . 1 2 3 If 1 can threat to reinstate q then 3 could bargain an alternative between p and the border of the indifference curve of 1 as all these alternatives are better than q. However the coalition between 1 and 3 is not the most plausible .. p q r . . . 1 2 3 In fact 2 is available to concede to 1 slightly morethan 2, at most w that is sligltly closer to 1 than p. . w Paradox of voting • Illustrate the calculus of voting according to an « instrumental » approach and how and why it should be changed according to Riker & Ordershook
Political Anaylisis ( Rationality behaviour instition exam) I have an exam tomorrow There will be two parts and each one has 30 min time limit. ( please check the attachment) need someone who have kno
Ch . 2 Rationality . The model of choice Preferences • Individuals have preferences (wants ). • For the positive (or rational choice ) approach preferences are given and are fixed in the short run . • Do preferences correspond necessarily to material well being ? No • Preferences belong to the interior world.. therefore .. • We make often assumptions about an individual’s preferences • Are individuals assumed to be always self -interested ? Maybe but they are not assumed to be necessarily selfish .. External environment and.. • Complementing interior world of the individuals is an external environment . This environment is filled with uncertainty about a) Preferences of the others b) Random events ( lack of control and often lack of knowledge ) People « behave » in pursuit of their preferences , sometimes they choose directly what they want , more often they choose « instruments » Beliefs and instrumental rationality • As individuals are uncertain about the external environment , the effectiveness of « instruments » are only imperfectly known • Person’s Beliefs = hunches (conjectures ) an individual has about the efficacy of a certain instrument or behaviour for meeting a preference . • Beliefs can have vary various origins and the can change  Learning process . • (Instrumental ) rationality = acting in accord both with one’s preferences and one’s beliefs . Rational Choice and other theoretical approaches • Rational choice is a form of methodological individualism . • Marxist approaches for instance assume social classes as actors • In International quite often the nation -states are the actors . More in general Theories without actors: •System analysis •Structuralism •Functionalism (Parsons) Theories with non rational actors: •Relative deprivation theory •Imitation instinct (Tarde) •False consciouness (Engels) •Inconscient pulsions (Freud) •Habitus (Bourdieu) “NON RATIONAL CHOICE THEORIES Rational behaviour • What really means « acting in accord both with one’s preferences and one’s beliefs” ? Imagine 1 actor i and 3 obiects (x, y, z) over which i has preferences . Obiects are alternatives . If for i xP iy then : x is better than y according to i’s preferences If for i xI iY then : i is indifferent between x and y i is rational if he/ she chooses in accord with his / her preferences . However it is necessary that i is able to put alternatives in a ranking order in terms of preference to choose the top ranked one . Logical properties of Rationality • Property 1 Comparability : Alternatives are said to be comparable in terms of preference ( and the preference relation complete) if for any two possible alternatives (x and y) either xP iy , yP ix , or xI iy . • Property 2 Transitivity : for three possible alternatives (x, y and z) if xP (I) iy and yP (I) iz then xP (I) iz . Money pump example • One can create a “money pump” from a person with intransitive preferences . Imagine that person i has the following preference ordering ( P = x>y>z>x ); she holds x. I can persuade her to exchange z for x provided she pays 1$; then I can persuade her to exchange y for z for 1$ more; again I can persuade her to pay 1$ to exchange x for y. She holds x as at the beginning but she is $3 poorer Limits of Rational Choice approach Individual are assumed to be rational if they have the capacity to order the alternatives and to chose for the top of the order . However …. When the stakes are low , uncertainty is high, and individual choices are of little consequence to the chooser then inconsistencies are likely to be common and behaviour is likely to be more random than rational . Digression about realism of Rational Choice Theory and in general of formal models • From Fiorina (1994) • “Until we compare a model to the real situation , we do not really know what are the “obviously important features” or the “most appropriate features” Often we think we know — we have hunches (hypotheses), but these are always subject to modification or rejection” • The most striking illustration of this quandary is drawn from the natural sciences. Digression about realism of Rational Choice Theory and in general on formal models • Rossby waves — wobbles in the flow of air currents around the earth — are named after a Swedish meteorologist who explained them. His crucial demonstration consisted of a simple physical model . • Rossby placed a pan of water on a rotating turntable to simulate the Coriolis force produced by the earth’s rotation and wrapped a heating element around the pan to produce hotter temperatures at the “equator” than at the “poles.” Surprisingly , photos of aluminum flakes suspended in the water showed waves similar to those in the atmosphere. Rossby’s two -variable explanation came to be the accepted one. Digression about realism of Rational Choice Theory and in general of formal models • Rossby’s model incorporates several assumptions that are certainly empirically questionable. lt postulates that 1) the earth is flat , 2) air and water are the same, 3 ) mountain ranges, deserts,and oceans are irrelevant a 4) the boundary of the system ( closed versus porous) is unimportant. Nevertheless by generating an analogue of Rossby waves, the model changed what meteorologists viewed as “ obviously important features” and the “most appropriate features” of the phenomenon. The success of the model implied that many of these were not important or relevant for the phenomenon, despite the hypotheses — or perhaps more accurately, the intuitions — of many experts . Maximization paradigm Individual are assumed to be rational if they have the capacity to order the alternatives and to chose for the top of the order …in other terms They are assumed to maximize something Consumers  max Contentments Producers  max profit Elected politicians  max votes next election Bureaucrats  max their budget Environmental uncertainty and beliefs As already said the individual usually does not choose directly the outcome ( the alternative) . He/ she mostly chooses an instrument that affects what outcome actually occurs . More precisely A rational individual chooses the instrument or action he/ she will lead to the best outcome . believes What is a belief ? (in rational choice language ) A belief is a probability statement relating the effectiveness of a specific action (or instrument ) for achieving various outcomes . Condition of certainty : probability of a specific action to achieve an outcome = 1 (100%) Condition of risk if the individual can assign probabilities to the effectiveness of a specific action based upon « obiective » frequencies Condition of uncertainty if the individual can hardly assign probabilities and in any case only on the basis of subjective impressions. Example • 3 outcomes x, y, z • 3 Actions A, B, C • For Individual i x>y>z Certainty : i knows that C leads y, B leads z and A leads x then i will choose A Risk : i knows that A leads x with 50% and z with 50% B leads y with 50% and z with 50% C leads y with 1/3 (33,333%), y with 1/3 (33,333%), z with 1/3 (33,333 %).. Uncertainty : i is not sure how tu put the probabilities (but at the end he/ she can assign them ) Example • When there is certainty for i is simple : he/ she has to pick the action or instrument that leads to his / her highest – ranked alternative • When there is not certainty the simple ranking order is not sufficient . Individual i has to assign a number to the different outcomes . A Utility number . In other terms he/ she has to evaluate also how much he/ she appreciates an outcome compared to the others • For x, y, z utility numbers are respectively u(x), u(y), u(z) Example Imagine u(x)=1; u(y)=0.2 , u(z)= 0. Remember that A leads x with 1/2 (50%) and z with 1/2 (50%) B leads y with 1/2 (50%) and z with 1/2 (50%) C leads y with 1/3 (33,333%), y with 1/3 (33,333%), z with 1/3 (33,333 %) Individual i will choose action that maximizes expected utility. For A the expected utility is for instance EU(A) = Pr A (X)* u(x)+ Pr A (y)*u(y)+ Pr A (z)*u(z) Example Remember u(x)=1; u(y)=0.2 , u(z)=0. EU(A) = Pr A (X)* u(x)+ Pr A (y)*u(y)+ Pr A (z)*u(z)= 0.5*1+0*0.2+0.5*0 =0.5 EU(B) = Pr B (X )* u(x)+ Pr B (y )*u(y)+ Pr B (z )*u(z )= 0*1+0.5*0.2+0.5*0 =0.1 EU(A) = Pr C (X )* u(x)+ Pr C (y )*u(y)+ Pr C (z )*u(z )= 1/3*1+1/3*0.2+1/3*0= 0.4 Action A maximizes expected utility Preliminary in Group Choice Ranking Andrew Bonnie Chuck 1 Museum of Fine Arts Walden Pond Red Sox 2 Walden Pond Red Sox Walden Pond 3 Red Sox Museum of Fine Arts Museum of Fine Arts The three friends do not unanimously share an alternative as their first preference . So Unanimity Rule does not have an outcome There is not also a majority in favor of an alternative that is first in ranking So absolute majority does not have an outcome ; Preferences are very heterogenous What about a round -roubin tournament ? With Round -Roubin Tournament Different Majorities support the winner alternative Votes Alternative 1 Alternative 2 Winning Alternative and Coalition Majority 1-2 Museum of Fine Arts Vs Walden Pond Walden Pond (Bonnie & Chuck ) 1-2 Museum of Fine Arts vs Red Sox Red Sox (Bonnie & Chuck ) 2-1 Walden Pond vs Red Sox Walden Pond (Andrew & Bonnie) W.P. wins but different majorities prefer WP to each of the other alternatives . In other terms there is not always the same group of actors who support always the same alternative against any other alternative We have implicitly assumed that each individual reveals his or her preference honestly . And if someone would misrepresent his /her preference and vote strategically ? Can he or she shift the outcome by shifiting his /her vote ? And if he/ she can then he/ she wants ? And if the actors are not «sincere» ? Votes Alternative 1 Alternative 2 Winning Alternative and Coalition Majority 1-2 Museum of Fine Arts Vs Walden Pond Walden Pond (Bonnie & Chuck ) 1-2 Museum of Fine Arts vs Red Sox Red Sox (Bonnie & Chuck ) 2-1 Walden Pond vs Red Sox Walden Pond (Andrew & Bonnie) Feasibility of misrepresentation : yes, everybody can misrepresent his / her preferences Desiderability of misrepresentation : Bonnie : No, she wins « honestly » her first preference Andrew: No, he is decisive ( Pivotal ) in the last voting . If he voted for Red Fox then he would allow the worst (for him ) alternative to win . Chuck : maybe . He could misrepresent his preference in the first voting . He could vote for his worst alternative, MFA… Strategic voting can create « stalemate » and intransitivity Votes Alternative 1 Alternative 2 Winning Alternative and Coalition Majority 1-2 Museum of Fine Arts Vs Walden Pond Museum of Fine Arts (Bonnie & Chuck ) 1-2 Museum of Fine Arts vs Red Sox Red Sox (Bonnie & Chuck ) 2-1 Walden Pond vs Red Sox Walden Pond (Andrew & Bonnie) By doing it the round robin tournament would have no winner !!! His preference for no voting outcome depends on what happens if the friends do not reach any agreement . It depends somehow on the desiderability of the «Status Quo» Even a certain change in preferences can produce the same outcome Ranking Andrew Bonnie Chuck 1 Museum of Fine Arts Walden Pond Red Sox 2 Walden Pond Red Sox Museum of Fine Arts 3 Red Sox Museum of Fine Arts Walden Pond Suppose that after a debate Chuck becomes convinced that MFA is better than WP. Round -roubin majority rule does not produce a winner anymore … Agenda procedure Votes Alternative 1 Alternative 2 Winning Alternative and Coalition Majority 2-1 Museum of Fine Arts Vs Walden Pond Museum of Fine Arts (Andrew & Chuck ) 1-2 Museum of Fine Arts vs Red Sox Red Sox (Bonnie & Chuck ) 2-1 Walden Pond vs Red Sox Walden Pond (Andrew & Bonnie) An alternative rule ( alternative institution ) that you find in the « real world» is the agenda procedure. For a given set of alternatives some individual ( or committee ) called – the agenda setter – is charged with assembling an order of voting for a larger group . The first two items on the agenda are voted on by majority rule , with the losing item eliminated and the winning paired with the next item on the agenda and so on. At the end the item that survives is the winner In the example suppose that Andrew is the agenda setter And « dictatorship » of the agenda setter Agenda I Agenda II Agenda III Museum of Fine Arts Red Sox Walden Pond Walden Pond Museum of Fine Arts Red Sox Red Sox Walden Pond Museum of Fine Arts For a set of k alternatives there are k! Ways to order an agenda. If the items are 3 ( as in the example ) then you have 3*2*1 =6 ways . However in this circumstance it does not matter which of the two items in the initial pair is first and which is second . Therefore the ways are 3. If Andrews knows his friends’ preferences and he believes that they will vote « honestly » once he chooses an agenda then … Agenda I = RS winner Agenda II = WP winner Agenda III = MFA winner By choosing Agenda III Andrew can produce his most –preferred alternative as the outcome of the group choice . Institutions matter !!! Group Choice and Majority Rule • From previous class we learned that even though each individua l in the group has preferences that are consistent (complete and transitive), this need not be true of the group’s preferences. In other terms A group of rational individuals can collectively produce irrational (intransitive) results. This puzzle has come to be known as Condorcet paradox Cycles a group G = {1, 2, 3 } 1 must choose by majority rule from among the three alternatives , {a, b, c} ( political candidates , public policies , etc.) (1 , 3), prefers a to b; (1, 2), prefers b to c; (2 , 3), prefers c to a . For members of a group with these preferences , majority rule produces alternatives that are said to cycle . More formally , a group preference relation, P, is said to be cyclical if it violates transitivity . aP Gb bP Gc cP Ga, Importance of the paradox H ow important is this puzzle of group intransitivity ? Is it a profound discovery ? First we can look at the likelihood of Condorcet’s paradox . We have to calculate how many preference orderings are possible Any one individual may rank order the three alternatives in 13 different ways Preference orderings (1) through (6) involve no indifference and are said to be strong. Orderings (7) through (12) involve some indifference, while (13) represents total indifference; these latter orderings are weak. Probability of the paradox Each member of the group thus may adopt any one of thirteen orderings, so that there are 13 x 13 x 13, or 2197, combinations of three individuals with preferences over three alternatives. Let us focus for sake of exposition only on strong orderings, therefore only on 6 x 6 x 6 orderings = 216 orderings Two types of cycles The cyclical group preferences are produced by a situation in which each alternative is ranked first by exactly one person, second by exactly one person, and third by exactly one person. This produces the “forward cycle ” ( aP GbP GcP Ga) and the backward cycle (cP GbP GaP Gc). For each cycle there are actually six different ways to produce it. Forward cycle Backward cycle 12 Ways to produce the cycles 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 a b c b a c b c a a c b c b a c a b b c a c b a c a b b a c a c b a b c c a b a c b a b c c b a b a c b c a 6 ways to produce forward cycle . And similarly 6 ways to produce backward cycle There are 12 orderings of 216 that produce a cycle ; less than 6%.. So Majority rule works almost prefectly … In fact no… We have considered only three alternatives and three voters • The good news of the small -group /few -alternatives situation does not extend to more general situations . • As the number of group members increases , and especially as the number of alternatives increases , the probability of cycles grows , becoming nearly certain as we approach the limit . • In general, then , we cannot rely on the method of majority rule to produce a coherent sense of what the group wants , especially if there are no institutional mechanisms for keeping participation restricted (thereby keeping n voters small ) or weeding out some of the alternatives (thereby keeping m alternatives small ). Questionable assumptions in the Probability calculation • In computing the entries of the Table we assumed that, for any size group (n), each of the strong preference orderings is as likely as any other to characterize the preferences of an individual. Moreover, one person’s “selection” of a preference ordering is entirely independent of some other person’s. • However most conceptions of society emphasize interdependence rather than independence among individuals. Individuals often choose to join groups because they have preferences in common with other group members. This would lead one to expect correlation, not independence, between the preferences of group members. • Therefore concerns about cycles in majority rule as reflected in the Table are probably a bit exaggerated . • Nevertheless as long as either n or (especially) m is large, the odds of majority preferences cycling is sufficiently large to be of concern. • Moreover, in circumstances quite common in politics ( “distributive politics”) majority cycles are inevitable. Majority Cycle in «Divide the Dollars » game. (or Distributive politics ) • Suppose that the state had made an earlier error of overcollecting fees from a small town. Now the money (1000 $ ) is back. The town’s three -person council must decide how to spend this “found” money. • The Politicians in the council represent the E ast, C entral and W est districts of town, respectively . Each politicians ( E, C, W ) believes that the more money he/she can land for the district, the better his or her chances are for reelection. • Given a Sharing Scheme [s(E), s(C), s(W)] where s(.) is not negative and s(E )+ s(C)+s(W)≤1000 •The most preferred distribution scheme for E: (1000,0,0); C: (0,1000,0); W:(0,0,1000) Majority Cycle in «Divide the Dollars » game. (or Distributive politics ) Distribution 1 Distribution 2 Majority Coalition Preferring 2 to 1 [333,33;333,33;333,33] [500;500;0] (E,C) [500;500;0] [700;0;300] (E,W) [700;0;300] [333,33;333,33;333,33] (C,W) There is no distribution scheme that wins against any other Divide the dollars is a «game» that produce cycles . Any distribution can be defeated by another with a majority vote. In this very important class of political activities ( Distributive Politics ) the only way to avoid preference cycles is to impose some forms of Anti -majoritarian restrictions : Agenda power , time limits , procedural rules etc. Arrow’s Theorem • Arrow’s Theorem asserts that Condorcet’s paradox is a problem for any reasonable method of aggregating individual preferences into group preferences (not only majority rule , not only round robin tournaments ). • Group of individuals : G=(1, 2,..n) with n≥3 • Set of Alternatives : A=(1,2,..m) with m≥3 • Typical individual =i; • Typical alternative=h or j or k • The individuals in G are assumed to posses preferences over the alternative of A ( Ri for i ∈ G) and they are rational , namely their preferences ’ orderings are complete and transitive Arrow’s Theorem • Condition U ( universal domain). Each i ∈ G may adopt any complete and transitive preference ordering over the alternatives in A • Condition P ( Unanimity or Pareto Optimality ). If every member of G prefers j to k (or is indifferent ) then the group preference must reflect a preference for j over k ( or an indifference ) • Condition I ( Independence of Irrelevant Alternatives ). If alternatives j and k stand in a particular relationship to one another in each group member’s preferences , and this realationship does not change , then neither may the group preference between j and k, even if individual preferences over other (irrelevant ) alternatives in A change • Condition D ( Nondictatorship ). T here is no distinguished individual i* ∈ G whose own preferences dictate the group preference , independent of the other members of G. If Conditions U, P, I and D are satisfied , then there exists no mechanism for translating the preferences of rational individuals into a coherent group preference Arrow’s Theorem • Condition U ( universal domain). Each i ∈ G may adopt any complete and transitive preference ordering over the alternatives in A • Condition U means that we are interested in designing a mechanism of group choice that responds to the preferences of group members without any restrictions. Thus , if A = a, b, c, then Ms. i may select as her preferences anyone of the thirteen preference orderings given in Display 4.2 . • Condition P (Unanimity or Pareto Optimality ). If every member of G prefers j to k (or is indifferent) then the group preference must reflect a preference for j over k ( or an indifference) • The rationale for condition P is driven by a concern for linking group preference to individual preferences . If the group preference ordering ranked alternative k ahead of alternative j, even though every member of G had the opposite preference , then the choice mechanism would be perverse. It would certainly be the case that the group preference ordering was not a «positive» reflection of individual preferences Arrow’s Theorem • Condition I ( Independence of Irrelevant Alternatives). If alternatives j and k stand in a particular relationship to one another in each group member’s preferences, and this realationship does not change, then neither may the group preference between j and k, even if individual preferences over other (irrelevant ) alternatives in A change • Suppose an expert group of American historians in 2010 sought to rank American presidents. Some in this group ranked Thomas Jefferson ahead of Andrew Jackson while others had the opposite view. The decision rule combined these various views into a group preference , say for Jefferson over Jackson. Suppose all initially had Barack O bama below these two. Suppose, however , that Obama’s brilliant leadership during the financial crisis of 2009 caused some members of the group to elevate Obama in their respective preference orderings . Condition I states that it still should be the group’s assessment that Jefferson ranks ahead of Jackson — that in the comparison between Jefferson and Jackson, the group’s (changed ) assessment of Obama is irrelevant and therefore should not affect this comparison . Arrow’s Theorem • Condition I ( Independence of Irrelevant Alternatives). If alternatives j and k stand in a particular relationship to one another in each group member’s preferences, and this relationship does not change, then neither may the group preference between j and k, even if individual preferences over other (irrelevant ) alternatives in A change Borda Count is a voting system that does not respect Condition I 5 voters rank 5 alternatives [ A ,B ,C ,D ,E] and give scores. 1 ° alternative=4 ; 2 ° alternative=3 ..5 ° alternative = 0 3 voters rank [ A >B >C >D >E]. 1 voter ranks [ C >D >E>B >A ]; 1 voter ranks [ E>C >D >B >A ]. Borda count C =13, A =12, B =11, D =8, E=6. C wins. Now, Imagine that : a) the voter who ranks [ C >D >E>B >A ] instead ranks [ C >B >E>D >A ]; b) voter who ranks [ E>C >D >B >A ] instead ranks [ E>C >B >D >A ]. They change their preferences only over the pairs [ B ,D ], [ B ,E] and [ D ,E] Nevertheless The new Borda count: B =14, C =13, A =12, E=6, D =5. B wins. The social choice has changed the ranking of [ B ,A ] and [ B ,C ]. The changes in the social choice ranking are dependent on irrelevant changes in the preference profile. In particular, B now wins instead of C , even though no voter changed their preference over [ B ,C ]. Arrow’s Theorem • Condition D ( Nondictatorship ). There is no distingushed individual i* ∈ G whose own preferences dictate the group preference , independent of the other members of G. Condition D says that if j is preferred to k by some specific person — say Ms . i*_and if k is preferred to j by everyone else, then it can not be that the group preference is jP G k . There cannot, be some privileged person in the group (Ms. j*) whose preferences become the group’s preferences, no matter what the preferences of the other members of the group, even if this person is an expert, a philosopher -king, or a megalomaniac. Arrow’s Theorem • If Conditions U , P, I and D are satisfied , then there exists no mechanism for translating always the preferences of rational individuals into a coherent group preference In other terms and more dramatically any scheme for producing a group choice that satisfies U, P, and I is either dictatorial or incoherent : the group is either dominated by a single distinguished member or has intransitive preferences. This restates the Arrow Impossibility Theorem in terms of the great trade -off it implies : “ there is, in social life, a trade -off between social rationality and the concentration of power ”. Arrow’s Theorem •If Conditions U , P, I and D are satisfied , then there exists no mechanism for translating always the preferences of rational individuals into a coherent group preference • Arrow’s Theorem result d oes not mean that any particular mechanism for aggregating preferences is always either inconsistent or unfair (in the three voter /three — alternative situation, the method of majority rule yielded coherent group preferences in 204 of 216 configurations ) • It means that mechanisms for aggregating preferences (including majority rule ) cannot guarantee group coherence in all situations . That is , the Rationality Assumption is violated on some occasions . • The important fact is that is possible for clever , manipulative , strategic individuals to exploit this fact . Arrow’s Theorem • The method of Majority Rule (M MR ) requires that , for any pair of alternatives , j and k, j is preferred by the group to k ( written : j P G k) if and only if the number of group members who prefer j to k exceeds the number of who prefer k to j. • MMR is actually composed of several essential building blocks or properties and these properties are all special cases of the general conditions in Arrow’s Theorem . Majority Rules and May’s Theorem Consider , some “ reasonable ” conditions on preference aggregation methods that embody notions of fairness • Condition U (universal domain). All complete and transitive preference orderings over alternatives are admittable . • Condition A (Anonymity ). Social preferences depend only on the collection of individual preferences , not on who has which preference . • Condition N (Neutrality ). Interchanging the ranks of alternatives j and k in each group member’s preference ordering has the effect of interchanging the ranks of j and k in the group preference ordering . • Condition M ( Monotonicity ). If an alternative j beats or ties another alternative k — that is , j R G k — and j rises some group member’s preferences from below k to the same or a higher rank than k, then j now strictly beats k — that is , jP G k . May’s Theorem. A method of preference aggregation over a pair of alternatives satisfies conditions U, A , N and M if and only if it is a Majority Rule (MMR). May’s Theorem • Condition A (Anonymity ). Social preferences depend only on the collection of individual preferences , not on who has which preference . It is a condition that requires only that we know what an individual’s preferences are , not who the individual is holding them. Condition A requires that each individual’s preferences be fed into the group decision -making machinery with his or her name omitted — as, for example is done with a secret ballot. May’s Theorem • Condition N (Neutrality ). Interchanging the ranks of alternatives j and k in each group member’s preference ordering has the effect of interchanging the ranks of j and k in the group preference ordering . Neutrality is for alternatives what anonymity is for individuals . Condition N says that it does not matter how we label alternatives ; all that matters is the alternatives’ respective ranks in individual preference orderings. May’s Theorem • Condition M ( Monotonicity ). If an alternative j beats or ties another alternative k — that is , j R G k — and j rises some group member’s preferences from below k to the same or a higher rank than k, then j now strictly beats k — that is , jP G k . M onotonicity requires that the method of group choice respond “ non perversely ” to changes in individual preferences ; The first feature of condition M requires that if one alternative is strictly preferred by the group to another ( j P G k ), and then rises in someone’s preferences it still is strictly preferred; that is, the method of preference aggregation does not respond to this change in a perverse or negative manner. The second feature of condition M states that if two alternatives, j and k, are judged to be socially indifferent ( j IG k), and j then rises in an individual’s preferences from below k to above k, j would now be strictly preferred to k by the group — j P G k; in effect, this says that the decision procedure must be sensitive to changes in individual preferences (a knife -edge property). May’s Theorem Hence , MMR cannot assure group choice May’s Theorem and Arrow’s Theorem Restrictions on Arrow’s Conditions • Conditions P, I, D are conditions of fairness or appropriateness therefore it is not easy to change them according to an acceptable normative criterion . • Condition U it is not, a fairness criterion, nor a criterion of appropriateness. It is a domain requirement. • It expresses the desire that the group decision mechanism work in all conceivable environments that the mechanism have the widest possible domain .. • Is it really desiderable ? In other therms do we really have to prefer no restrictions on the admittable preferences ? Restrictions on Arrow’s Conditions • The most famous domain restriction was proposed by Duncan Black . Black’s Single -Peakdness Theorem . Consider a set A of alternatives from which a group G of individuals must make a choice . If , for every subset of three alternatives in A one of these alternatives is never worst among the three for any group member, then this is sufficient consensus so that the method of majority rule yields group preferences P G that are transitive . Example of Abortion Positions Ranking Group 1 Ranking Group 2 Ranking Group 3 Ranking Group 4 Pro Abortion 1 3 3 2 Pro Life 3 1 2 3 Suprem Court position 2 2 1 1 It is easy to see that S .C . position is not ever the worst . Collective choice in this case is transitive . Final outcome is S .C . position . T he theorem says that majority rule works perfectly well, even when group members hold wildly divergent views on what the group ought to do . as long as a minimal degree of consensus, captured by single – peakedness obtains . But what single -peakedness means ?? (next Chapter) Sen’s Value restriction theorem • Sen’s Value -Restriction Theorem. The method of majority rule yields coherent group preferences if individual preferences are value restricted . • A group’s preferences are value restricted if , for every collection of three alternatives under consideration , all members of the group agree that one of the alternatives in this collection either is not best, not worst, or not middling (with all members agreeing on which quality the alternative in question was not ) • This Theorem is obviously more general than Black’s one . Examples Positions Ranking Group 1 Ranking Group 2 Ranking Group 3 A ( is not ever first) 2 3 3 B ( winner ) 1 1 2 C 3 2 1 Positions Ranking Group 1 Ranking Group 2 Ranking Group 3 A( is not ever middling ) 1 3 3 B 3 1 2 C (winner ) 2 2 1 Spatial M odels of Majority Rule • If individual preferences happen to arrange themselves in particular ways — that reflect a consensus of a specific sort — then group decisions (certainly those made by majority rule) work out quite nicely. • Single -peaked preferences is one kind of consensus that facilitated coherence in majority – rule • But what single -peakedness means in geometric terms ? • Imagine a bank’s board of directors ( G=1,2,3,4,5) that must decide each week on the week’s interest rate for thirty -year home mortgages . In effect , the relevant interest rates are points on a line , one endpoint being 0 percent and the other being some positive number , say 10 percent . This interval is written as [0 , 10]. In this group of individuals each of whom has a most -preferred point on the line and preferences that decline as points further away in either d irection are taken up. • The board is meeting on Monday morning to decide the interest rate to charge for home mortgages the coming week. Each individual iin G has a most -preferred point (ideal point), labeled x, located on the [0, 10] interval representing his or her most -preferred interest rate. Director 1 has a most -preferred interest rate (x1) of just less than 4 percent, director 2’s (x2) is just more than 4 percent, and so on. • On the vertical axis is represented the “Utility” to measure preferences. For each individual a utility function is graphed . It represents the director’s preferences for various interest rate levels in the [0, 10] interval. Each utility function, labeled ui for Mr. or Ms. i, is highest for i’s most -preferred alternative, x, and declines as more distant points are considered. For instance Ms. 5 most prefers an interest rate a little higher than 8 percent, with her preference declining either for higher or lower rates. • The preferences of these individuals are single -peaked, which is defined as follows: Single -P eakedness Condition. The preferences of group m embers are said to be single -peaked if the alternatives under consideration can be represented as points on a line and each of the utility functions representing preferences over these alternatives has a maximum at some point on the line and slopes away from this maximum on either side . The Single -Peakedness Condition is connected with the Black’s condition requiring the existence of an alternative that is not ever the worst one. F or any three interest rate levels you choose, one of those is not worst for any of the five bankers!! Let us focus on Ms. 5, who most prefers a fairly high interest rate: x5 = 8,25%. Consider an alternative rate, y = 7%. The set of points Ms. 5 prefers to y is described by the set labeled P 5(y) This is Ms. 5’s preferred -to -y set: if y were on offer, then P(y) describes all the points she would prefer to it, given her preferences. As the figure shows, P 5(y) is computed by determining the utility level and then identifying all the interest rates on the horizontal axis with utility levels greater than the utility for y. The set of points Ms. 4 prefers to y is described by the set labeled P 4(y) and it is narrower than the Ms. 5’s preferred -to -y set. It is also included in the previous one. Therefore we can say that in this specific case the intersection between P 4(y) and P 5(y) is equal to P 4(y) and it represents the alternatives that both Ms. 4 and Ms.5 prefer to y. If they were the only directors and they had to decide by unanimity then any final outcome should be included in this intersection In Figure 5.3 you can see the preferred to -y sets of all five bank directors (note that y, in this figure, is just below 6 percent). The set of points a majority prefers to y is called the majority winset of y, written as W(y). Let M be the set of majorities in our group of bankers G; 10 majority coalitions have 3 members, 5 Majority coalitions have 4 members . Then there is the coalition of whole . For each of the 16 majority coalitions , consider the common intersection of preferred -t.o . y sets ( if there is any ); these are the points that this particular majority prefers to y. Determine this set for each of the majority coalitions. Then take the union of these 16 sets. This is W(y), namely the majority winset of y. In fact the only majority coalition that has this intersection set of point in the example is the coalition of {3,4,5} . Therefore in Figure 5.3 points shared by 3, 4 and 5 are the only one that can defeat y. W(y) Majority winset • If some alternative, x, has an empty winset (written : W(x) =ø, where ø means “ empty in set notation ) then it is a clear candidate for the group choice . W(x) =ø means there is no other alternative that any of the sixteen majority coalitions prefers to x. It is an « equilibrium ». We know that under the Arrow’s condition there is not necessarly an alternative x for which W ( x) is empty . ( Cycling Majorites ) Black Median Voter Theorem • If members of group G have single -peaked preferences, then the ideal point of the median voter has an empty winset . • Therefore the ideal point of the median voter is the final outcome approved by majority rule Consider any arbitrary point in the feasible set of interest rates, [0, 10], to the left of x3 (ideal point of the median voter ) — say the point labeled α in Figure 5.3’. Notice that α is preferred to x3 by members 1 and 2, since x3 is not in either P1( α ) or P2( α ) but x is preferred to α by members 3, 4 and 5. Thus, x3 is majority.preferred to α .The same is true for any arbitrary point to the left of x3. For any such point, we know at the very least that members 3, 4, and 5 will prefer x3 to it . Consider any arbitrary point α to the right of x3 . Members 4 and 5 may prefer it to x, but members 1, 2, and 3 hold the opposite preference, so that x3 is majority -preferred. The argument is exactly the same as with the previous position of α since we selected an arbitrary alternative to the right of x3. W e now know that the ideal point of the median voter (in this example x3) is preferred by a majority to any arbitrary point to the right or to the left of it, that is, to all remaining points. Hence, it has an empty winset [W(x3)=ø ] and is the majority choice. 3 hidden assumptions behind the median voter result • Actors in the example are odd. Black’s Median – Voter Theorem is true whether the group size is odd or even however when a group has an even number of members, then more than one alternative can have the property that it cannot be beaten. • Full partecipation is assumed . No Abstensionism . • Those exercising the franchise do s o sincerely (no sophisticated or strategic behaviour ) More than one dimension • Many social situations cannot be reduced to one dimensional affairs . • For instance in the game of “divide the dollars ” if the game were played by a group of three individuals , then it is necessary to have two dimensions in which to represent outcomes . The first dimension gives the amount that player 1 receives , while the second dimension gives the amount that player 2 receives . (Subtract the sum of these two numbers from the total number of dollars to be divided and you get the amount that player 3 receives .) One and two dimensions • In the left graph Utility is function of the quantity of only one « good ». • In the right graph Utiliy is function of the quantities of two « goods » • Technically there is also a second or third dimension , (Utility) . Two dimensional graphs • In the figure on the right (5.5) instead of adding a third dimension in order to graph actor’s preference function, we overlay “slices” of his utility function onto the policy space, producing the set of nested circles called indifference curves . Each circle is a slice of the policy hump in the other figure 5.4 It is a locus of policy outcomes among which the legislator is indifferent (since all the points on a circle lie on the same slice and hence at the same height on the utility function of Figure 5.4). Ideal point Ideal point Two dimensional graphs • Since distance from an ideal point is a measure of preference, points on a circle centered on her ideal, being equidistant from that ideal, are equally preferred by her. The logic is the same in comparing a point on one circle to that on another. An actor prefers a point on a circle with a smaller radius to one on a circle with a larger radius because this means the former point is closer to her ideal than is the latter point. Ideal point Two dimensional graphs • The circle through y centered on our actor’s ideal point contains all the points of actor indifference to y. Therefore all the points inside the circle, being closer to her ideal, are actually preferred by her to y. That is, we can call the points inside the circle our legislator’s preferred.to -y set, a natural generalization of the same concept in the one -dimensional Winset in two dimensions The Figure displays three legislator ideal points and each legislator’s indifference curve through y (the curve plus all points inside it comprising each legislator’s preferred -to.y set, labeled P(y)). The shaded intersection P1(y)∩P2(y) gives the points preferred by both Iegislators 1 and 2 to y; P3(y) ∩ P1(y) are the points preferred by 1 and 3 to y; and, finally, P2(y) ∩ P3(y) give those points for 2 and 3. Since two out of three is a majority, the union of these three “petals” is the winset of y, W( y). Each petal gives the points that a specific majority coalition prefers to y. y A median voter in two dimensions If we line up in a row the ideal points of the three legislators then the winset of the middle legislator’s ideal, W(x2) is empty . Mr. 2’s preferred -to -x2 set is obviously empty. Ms. 1’s and Ms. 3’s indifference curves through x2 and centered on their respective ideal points are tangent to one another. They do not overlap at all. Hence W(x2) =ø , since there are no points preferred to x2 by a majority X2 is a majority equilibrium outcome . Median voter theorem applies also in this specific case A median voter in two dimensions Unfortunately this outcome is very fragile. If x2 shifts slightly from the perfect alignement then W(x2) is not empty anymore. A median voter in two dimensions Let us consider a set of m (odd ) voters whose ideal points are x1, x2, . . .,x. These ideal points are distributed in a radially symmetric fashion if the following conditions hold: (1) There is a distinguished ideal point, labeled x*; (2) The n -1 remaining ideal points can be divided into pairs (since n is odd, n -1 is even and this is possible); (3) The two ideal points in any pair, say xi and xj plus x* all lie on a line with x* “between” xi and xj . In Figure 5.8, x2 is the distinguished point, x1 -x3 and x4 -x5 are the pairs of remaining ideal points, and x2 lies on a line “between” the ideal points in each pair. A median voter in two (or more) dimensions Plott’s Theorem : too restrictive conditions • Plots condition of radial symmetry is highly sensitive to small perturbations of voter ideal point locations. • Imagine the ideal points of 1.000.000 voters radially distributed around the ideal point of the 1.000.001st voter. So, W (x1.000.001) =ø, in accord with Plott’s Theorem. Now suppose two new voters move into the community, and their ideal points are not radially symmetric about x1.000.001. • This small perturbation in the voting situation completely destroys the previous equilibrium. • Departures from the radial symmetry are important and somehow « devastating ». Plott’s Theorem : too restrictive conditions • Plots condition of radial symmetry is highly sensitive to small perturbations of voter ideal point locations. • Imagine the ideal points of 1.000.000 voters radially distributed around the ideal point of the 1.000.001st voter. So, W (x1.000.001) =ø, in accord with Plott’s Theorem. Now suppose two new voters move into the community, and their ideal points are not radially symmetric about x1.000.001. • This small perturbation in the voting situation completely destroys the previous equilibrium. • Departures from the radial symmetry are important and somehow « devastating ». McKelvey’s Chaos Theorem • In multidimensional spatial settings, except in the case of a rare distribution of ideal points (like radial symmetry) that hardly ever occurs naturally, there will be no majority rule empty — winset point. • Instead there will be chaos — no Condorcet winner, anything can happen and whoever controls the order of voting can determine the final outcome. 29 Cycling majorities ( McKelvey Theorem) Spatial models of elections Downs • Suppose A one -dimensional ideological continuum, [0, 100]. The continuum is scaled by the proportion of economic activity left in the hands of the private sector, so that the left endpoint reflects a fully socialized economy, while the right endpoint is identified with a totally private -enterprise economy. • Political competition is a contest between politicians for capturing control of government by appealing to voters with offers of alternative plans, platforms, programs, visions. These appeals are identified with different points on the left -right ideological continuum . Spatial models of elections Downs • Downs assumed politicians seek to maximize their vote, (their vote Plurality ,namely the difference between their vote and that of their closest competitor or their probability of winning). • Politicians are focused exclusively on winning elections by promising policies , platforms, and programs that attracted voters. In this spatial context, a candidate is represented by some location on the ideological continuum, some point in the [0, 100] interval. This is his or her political position. • Downs assumed voters as singlemindedly interested in policy : the goods and services produced by government (or left to the private sector); the form and content of government regulation of the private sector; the distribution of tax, unem ployment , and inflation burdens ; government policies on social issues like abortion and divorce ; and matters of war and peace . • Voters care mightily about these matters and base their assessments of candidates accordingly . Voters , however , are heterogeneous in their tastes so, just as there are left -wing and right -wing politicians , there are left.wing and right -wing voters . Spatial models of elections Downs • E ach voter is identified with some point in the [0, 100] ideological space — the voter’s ideal point — and his or her preferences are assumed to decline for points more and more distant from this ideal. In other terms the set of voters may be represented by single – peaked preferences. Spatial models of elections Downs • Figure 5.10 displays an electorate of five different voter “types” with 125 voters of each type). A voter of type i(i= 1,2,3,4,5) has ideal point xi and preferences declining in distance from xi. • Imagine two -candidate competition. Where will two single -minded seekers of election locate themselves? • Let’s fix the position of L, the leftist candidate, at l. What position, r, should R, the rightist candidate, adopt so as to maximize his votes? Spatial models of elections Downs • Downsian rule is that each voter votes for the candidate whose location is closest to his or her ideal point. In other terms each voter vote for the party platform that is supposed to increase his/her utility. • Candidate R should locate it self infinitesimally close to the right -hand side of L. So doing he/she gets all the votes to the right of l and, since l is to the left of the midpoint of the voter distribution that means that R gets more than half of all the Votes. • R gets 375 votes from voters of types 3, 4, and 5; L gets the 250 votes of types 1 and 2. r Spatial models of elections Downs • L knows that her l position will divide the electorate into two groups and she will get the smaller group. Therefore she will try to make this smaller group as large as possible by moving (locating )his/her position toward (in) the ideal point of the median voter, since the groups to the left and right would then be equal in size. • If L and R will locate themselves close as much as possible (respectively from left and from hand side) to the median voter ideal point then each location is optimal against the other’s and the election ends in a virtual tie. r Spatial models of elections Downs • We have the same result also if we hypothesize that candidate start the campaign simultaneously fro the opposite sides of the ideological spectrum r l r Spatial models of elections • Empirical evidence suggest us that parties do not converge all the time. Suppose that there is a third candidate — call her (T) — who may enter the race if she thinks she has a chance. Well, if L and R locate at the median m* (l= r = m* ). If when there are more than two candidates, the one with the most votes (not necessarily a majority) wins the election, then T can locate close on one side or the other of the median, win nearly all the votes on that side, and thus defeat L and R, who end up splitting the remaining votes Spatial models of elections r t • On the other hand, if the positions of L and R are sufficiently widely dispersed, then T can enter between them at some position t. She will get the votes of voters whose ideal points lie in the interval [(l + t)/2, (t + r)/2]. The left boundary of this interval is the midpoint between the positions of L and T, whereas the right boundary is the midpoint between the positions of T and R. If the voters in [(l + t)/2, (t + r)/2] are more than voters in [0, (l + t)/2] and ,[0, (t + r)/2] then T will win. Spatial models of legislatures in one dimension • The legislature is supposed to be a set of n individuals, where n is an odd number, and where everyone casts a vote. It makes decisions by majority rule. • In one dimension the legislature must choose a point on a line. Each legislator, i, has an ideal point x i, and single – peaked preferences. The median voter is legislator m with ideal point xm . • In this circumstance xm can defeat any other point on the dimension in a majority contest (Black’s Theorem). • There is always a status quo in place , labeled x0. This is the current policy at the time of legislative choice . It remains in place if the legislature chooses not to change it . Spatial models of legislatures • Most legislatures have a division -of -labor arrangement known as a committee system. • A committee is a subset of the n legislators . The median ideal point of the committee members is labeled xc; as majority preferences in the entire legislature are identical to the preferences of the legislature’s median voter xm , majority preferences inside a committee are a copy of the preferences of the committee’s median member xc. • Because of these identities, much of analysis need only consider x0, xm , and xc Three possible decision – making regimes • Pure majority rule . There is a status quo, and any legislator can offer a motion to change it. A motion, once proposed, is pitted against the status quo. If it wins it becomes the new status quo. The floor is always open for some new motion (against the old status quo, if it survived, or the new status quo, if the previous proposal prevailed). This procedure of motion making and voting continues until no member of the legislature wishes to make a new motion. Three possible decision – making regimes • The closed -rule committee system . A (previously appointed) committee first gets to decide whether the legislature will consider changes in the status quo; it has gatekeeping agenda power and can decide whether to open the gates to enable policy change or not. Second, if the gates are opened, only the committee gets to make a proposal (monopoly proposal power). Third, the parent legislature may vote the committee’s proposal either up or down. If it passes, then it becomes the new status quo; if it fails, then the old status quo prevails. The proposal is closed to amendments. Three possible decision – making regimes • Open -rule committee system . A (previously appointed) committee first gets to decide whether the legislature will consider changes in the status quo; it has gatekeeping agenda power and can decide whether to open the gates to enable policy change or not. Second, if the gates are opened, only the committee gets to make a proposal (monopoly proposal power). Third, differently from the closed rule under an open rule, once the committee has made a proposal, the parent legislature may open the floor to amendments to the committee’s pro posai. Once the committee has opened the gates and made a proposal , it concedes its monopoly access to the agenda. • . • In all « regimes » we have considered final proposal must defeat status quo x0. Therefore it is crucial identifying the set of policies that defeat x0. In the figure the policies the median legislator prefers to x0 are in the set Pm(x0). Graphical representations of outcomes Pure Majority rule • In this graph x0 (status quo) is fixed and Xm (Floor median voter ) changes his / her position. Final outcome will correspond always to the position of xm . Graphical representations of outcomes Pure Majority rule • In an alternative graphical representation , more useful , xm is fixed and x0 (status quo) changes its position. Obviously also in this graph final outcome will correspond always to the position of xm • In this regime median voter in the commitee will propose a bill that is the best one for the majority in the committee and it is better for xm (the majority in the Floor ) than the status quo x0. • 2xm -xc is a point far from xm as much as xc Graphical representations of outcomes Closed rule • When x0 is on the left (or coincide with) of 2xm -xc (x0≤2xm -xc) or on the right (or coincide with) xc then xc can propose its ideal point since for xm x0 is worse . • xc will be the final outcome Graphical representations of outcomes Closed rule • When x0 is between 2xm -xc and xm (2xm -xc2xc -xm) then xc will propose a bill ( will “open the gate”) and the final outcome will be xm Graphical representations of outcomes Open rule • When x0 is between xm and 2xc -xm (xm≤x0≤2xc -xm) then xc will “keep the gate closed” as any proposal will become xm and xm is farer than x0. The final outcome will be x0. • When x0 is between xc and 2xc -xm x0 is a suboptimal equilibrium. It would be better for xm and xc agree on a change but xm cannot credibly commit not to amend the proposal of xc and to shift it to xm Graphical representations of outcomes Open rule • When w0 is the suboptimal equilibria area the power of the Floor (amending power) is against its interest. Its strenght is also its weakness. Graphical representations of outcomes Open rule Spatial models of legislatures in more than one dimension Pure Majority rule • In a pure majority -rule regime, according to the results of the McKelvey Chaos Theorem we know that W(x)≠ø for any x in the policy space. • In other terms anything can be beaten. In particular, any status quo, x0, has a nonempty winset . As long as a motion is made from that set, the status quo will be replaced. • However also x1 Є W(x0), in turn, has a nonempty winset of its own,W (x1) ≠ø. A motion x2 Є W(x’) will replace x1. And so on . With a Pure Majority regime an existing status quo is continually replaced. 57 Cycling majorities ( McKelvey Theorem) Multidimensional space and decision making dimension by dimension Pure Majority Rule • As in the pure majority rule anyone is free to make a motion to change the status quo • However decision making takes place one dimension at a time, in some pre -set order. • The group (the parliament) continues to focus on amending the status quo on the first dimension until no more amendments are offered , then turns its attention to the next dimension etc. Multidimensional space and decision making dimension by dimension Pure Majority Rule • There are three legislators with idealpoints xa = (xa1, xa2) ; xb = (xb1, xb2); xc=(xc1,xc2) • For any status quo (not pictured), x0=(x01, x02), motions are entertained, first on dimension 1 and then on dimension 2. Multidimensional space and decision making dimension by dimension Pure Majority Rule • The final outcome is the multidimensional median, xm = (xc1,xa2) since legislator c is median on the first dimension and legislator a is median on the second. • It does not mean that W( xm )=ø. For instance there are points to xm’s northeast, that both a and c prefer to xm • It means that on any dimension — say, for instance the first — holding policy fixed on the other dimension, no movement away from xc1 would be supported by a majority. • The only points preferred by a majority to xm require changes on both dimensions at once. Multidimensional space and decision making dimension by dimension Pure Majority Rule • In order to have a decision making dimension by dimension it is necessary a specific institutional structure that “ unpackages ” the potential proposals according to the different dimensions that characterize them. • In the Parliaments Committee system plays at least partially this role . Multidimensional space Open Rule • Imagine, that xc is an agenda setter and the status quo is x0. If her proposals are subject to amendment by the parent legislature (xc plus xa and xb ) , then we are back to the chaos theorem results. Multidimensional space Closed rule • Under a closed rule regime she can make a take it or leave it proposal as in this regime the floor ( xc, xa , xb ) cannot amend. The petal -shaped shaded regions W(x0) namely the win set of x0, the status quo. • Xc will choose the alternative that is still approved by a majority in the floor (it belongs to W(x0) ) and it is the closest one to her ideal point • The proposal is at the tangency between one of the petals of W(x0) and the smallest indifference curve of xc and will be supported by a and c Rational foresight • Rational actors , seeking to enhance the prospects of the purposes they pursue, must think strategically. And one of the fundamental principles for thinking strategically is “looking before you leap”. Rational foresight • Three people legislature ( i, j, k). Each member would like a pay rise but realizes that constituents will not pleased with a representative voting to increase his or her own salary. • The best of all possible worlds for legislator i(a or b or c) is for the other two legislators to vote yes, making the pay rise to pass, and voting nay to keep his/her good reputation in front of the costituents . • The preference ordering on the outcomes are the same for all legislators: 1. Pn =Pass and vote nay 2. Py =Pass and vote yes 3. Fn =Fail and vote nay 4. Fy =Fail and vote yes Each legislator must make a public declaration on the motion to raise pay. They are called following alphabetical order: 1 °: i 2 °: j 3 °: k Suppose you are legislator i. How shouldyou vote? Game solution by backward induction yes no yes yes no no yes yes no no no yes yes no i j j k k k k Py Py Py Py Py Pn Py Pn Py Fy Fn Fn Pn Py Py Fn Fy Fn Fn Fn Fy Fn Fn Fn i: j: k: Backward induction yes no yes yes no no yes yes no no no yes yes no i j j k k k k Py Py Py Py Py Pn Py Pn Py Fy Fn Fn Pn Py Py Fn Fy Fn Fn Fn Fy Fn Fn Fn i: j: k: i: j: Py Py Py Pn Pn Py Fn Fy yes no yes yes no no yes yes no no no yes yes no i j j k k k k Py Py Py Py Py Pn Py Pn Py Fy Fn Fn Pn Py Py Fn Fy Fn Fn Fn Fy Fn Fn Fn i: j: k: i: j: Py Py Py Pn Pn Py Fn Fy i: Py Pn Backward induction Backward induction • Strategic thinking and rational foresight entail a logic that takes advantage of the sequential structure to the decision making. • They involve thinking forward by reasoning backward. • T he method is called backward induction • Imagine three individuals who must make a collective decision each by revealing an individual choice in turn. • L et’s us suppose that Mr. I and Mr. III each have three options available in their “action sets” — {a1, a2, a3} and {c1, c2, c3}, respectively — whereas Ms. II has only two options in hers — {b1, b2). • Imagine that the individual move according to this sequence I, III, II • Given the best moves for II , in the last step of the sequence , which are the best moves for III ? • It depends obviously by the alternative decided by I. However if I chooses a1 is c3, if I chooses a2 is c2, if I chooses a3 is c2. • What will I choose ? a1=>c3=>b2 : 8 a 2=>c2=>b1 : 7 a 3=>c2=>b1 : 3 The final equilibrium strategy is (a1, c3, b2) Gibbard – Satterthwaite Theorem • Strategic behaviour can be considered a controversial behaviour as it means also misrepresenting your real preferences . Somehow « lying » • Are there decision -making procedures that encourage only honest, nonstrategic behavior — procedures that are basically strategy -proof ? Gibbard – Satterthwaite Theorem • Suppose there are n group members, G = {1, 2,. . ., n} , who must choose from a set of m alternatives A= {a1 , a2 ,. . ., an } • the final choice will be one of the alternatives in A and that it will depend (somehow) on the preferences expressed by the group • The social choice is F(Q1 , Q2,. . . , Qn, A) Є A where A is the set of alternatives and Qi is a preference ordering of the alternatives A expressed by member i , and F is some decision process that trasforms these expressed preferences into an outcome in A. Gibbard – Satterthwaite Theorem • Let’s suppose that the true preference orderings of the members of G over the alternatives in A are P1, P2… Pn , although no outside observer has any way of knowing or validating this . • We say that Ms. i is sincere only if, in the group decision setting where she is asked to reveal her preferences , her revealed preference ( Qi) is identical to her true preference (Pi); (Qi)=(Pi) • if Q1 ≠ P1, then she is said to be sophisticated. So Pi reflects her true tastes, Q1 is what she chooses to reveal, and she is sincere only if the two are the same • A sophisticated individual is someone who may misrepresent her true preferences and , when she does so, she is said to manipulate F, the social -choice procedure Gibbard – Satterthwaite Theorem • Assume a group G of at least three individuals and a set A of at least three alternatives. Also assume that any member of G may have, as his or her true preferences, any preference ordering over A (universal domain). Then every non dictatorial social -choice procedure, F, is manipulable for some distribution of preferences. Sophisticated voting • Example “Killer amendment” • In legislative politics, there is something known as a killer amendment. It is an amendment to a bill which, if successfully attached to the bill, will cause the bill to be defeated, even though the bill would have passed if it had not been amended. • Such amendments require sophisticated voting in which an enemy of the original bill votes for the killer amendment , even though she doesn’t like the amendment per se. She does so because she appreciates that the now amended bill will be defeated , whereas the unamended bill would have passed . • Conventional legislative decision making is , as the Gibbard – Satterthwaite Theorem suggests , often vulnerable to manipulation . Chauncey Depew and the seventeenth amendment • C.D. simply by offering a cleverly formed amendment delayed for 10 years the adoption of the 17 ° Amendment of U.S. Constitution about the direct election of senators • Proposed Constitutional Amendment from the House:” The Senate of the United States shall be composed of two Senators from each state, elected by the people thereof, for six years; each Senator shall have one vote. The electors in each state shall have the qualifications requisite for electors of the most numerous branch of the state legislature”. 2/3 of Senators are required to emend the U.S. constitution • Almost all Democrats were in favour of the Const.Amend . ; Northeastern senators (mostly Repubblican ) were against. Preference ordering before the Depew killer amendment Preferenc e ordering Northern Dem.(25) Southern Dem.(35) Northeastern Rep.(20) Midwestern and Western Rep.(20) 1 ° Constituti onal Amendm ent (C.A.) C.A. Status quo C.A. 2 ° Status Quo Status Quo C.A. Status Quo C.A Vs SQ = 80:20 Chauncey Depew’s killer amendment • Proposed amendment to the Constitutional Amendment from the House:” The Senate of the United States shall be composed of two Senators from each state, elected by the people thereof, for six years; each Senator shall have one vote. The electors in each state shall have the qualifications requisite for electors of the most numerous branch of the state legislature. (and) the qualifications of citizens entitled to vote for U.S. Senators and Representatives in Congress shall be uniform in all states, and Congress shall have the power to enforce this article by appropriate legislation and to provide for the registration of citizens entitled to vote, the conduct of such elections, and the certification of the result . ”. • In effect Depew proposed to come back to the strict federal control of elections in southern states (even by the Army) as just after the civil war, during the so called Reconstruction Era. Sequence of voting 1. Amendment to the proposal vs Proposal 2. Proposal vs Status quo Senate Depew propose amend . Not propose amend. yes No (original Const.proposal Senate yes no Senate yes no Depew amendment Status Quo original Const.proposal Status Quo original Const.proposal Preference ordering after the Depew killer amendment Preference ordering Northern Dem. (25) Southern Dem. (35) Northeastern Rep.(20) Midwestern and Western Rep.(20) 1 ° Depew amendment C.A. Depew amendment Depew amendment 2 ° Constitution al Amendment (C.A.) Status Quo Status quo C.A. 3 ° Status Quo Depew amendment C.A. Status Quo Depew amend. Vs CA = 65: 35 Depew amend. Vs SQ= 65:35 (not enough..not 2/3) CA Vs. SQ = 80 : 20 ( more than 2/3); There is a cycle Preference ordering Northern Dem. (25) Southern Dem. (35) Northeastern Rep.(20) Midwestern and Western Rep.(20) 1 ° Depew amendment C.A. Depew amendment Depew amendment 2 ° Constitution al Amendment (C.A.) Status Quo Status quo C.A. 3 ° Status Quo Depew amendment C.A. Status Quo Strategic voting • It the Anglo.American electoral arrangements, known as plurality voting systems any number of parties may compete for a single office, with the candidate of the party winning the most votes (not necessarily a majority of the votes ) declared the winner. • The reason this arrangement nearly always reduces to two -party competition is that individual voters are loath to waste their votes, individual contributors are loath to waste their campaign resources, and individual political managers are loath to waste their electioneering skills on hopeless candidacies. They tend to desert such candidacies , even if they would rather see that candidacy succeed because it is preferable in their view. • People who want to make the most of their strategic endowments (votes, dollars , organizational skills) will prudently deploy them where they think they might make a difference (say, in helping to choose the lesser of evils) , rather than deploying them where they serve only to express a preference but have no effect on the final outcome. Strategic voting and Sophisticated voting • Sophisticated voting is made possible by backward induction on a fixed agenda • Strategic voting involves weighing two different lotteries. Heresthetics • A political strategy by which a person or group sets or manipulates the context and structure of a decision -making process (without changing people’s underlying preferences ) in order to win or be more likely to win (William Riker) • The logic of heresthetic can be understood as the introduction of a new issue, or the redefinition of an old one, in order to destroy a currently winning coalition and replace it with some other. Heresthetics • Heresthetics is an art, not a science. There is no set of scientific laws that can be more or less mechanically applied to generate successful strategies. Instead, the novice heresthetician must learn by practice how to go about managing and manipulating and maneuvering to get the decisions he or she wants. Practice is, however, difficult to engage in, especially since one must win often enough to become a political leader before one has much opportunity to practice . (Riker 1986) Heresthetics and the example of slavery issue • For much of the first half of the nineteenth century . American national politics were dominated by the Jeffersonian -Jacksonian coalition. By 1820, after the Federalist Party had disappeared, this coalition was virtually unopposed . The coalition was united principally by the issue of agrarian expansionism and found its greatest strength in the middlle Atlantic states, the South, and the states of the North West Territory. Opposition politicians searched and searched for issues that might split this governing coalition . Their substantive opposition to the agrarian expansionism of the Jeffersonian – Jacksonians consisted in their desire for public policy to encourage commercial development. Slavery issue • It needed to find a new issue that would split the currently dominant governing coalition, one that would divide Mid – Atlantic from Rim South, Northwest from Deep South. • The slavery issue was the answer. Slavery worked not because of its moral content nor even because so many people were animated by abolitionist agitation. Slavery worked as a strategic maneuver because it divided the members of an existing winning coalition, some of whom tolerated slavery and some of whom opposed it. Once the northern elements of the Jeffersonian -Jacksonian coalition came to fear that support of slavery on which their southern coalition partners depended would be their own personal undoing, the coalition could no longer hold. Lincoln at Freeport or the best example of an heresthetical device (in the large) • For a person who expects to lose on some decision, the fundamental heresthetical device is to divide the majority with a new alternative (preferred to the previous winning alternative). If successful, this maneuver produces a new majority coalition, composed of the old minority and a portion of the old majority that likes the new alternative better. • Practically the heresthetician redefine the situation choice. The Lincoln’s trap • 2 elections (a, b) , 2 competitors (Douglas and Lincoln) : • a) candidates for Illinois legislatures who on their turn were pledged to vote for Lincoln or Douglas for the U.S. Senate • b) two years later presidential elections . • For both competitors ( L. and D.) b>a The Lincoln’s trap • Main political actors 1) Democratic party that was divided between Northern D. who were weakly against slavery and Southern D. who strongly defended it . 2) Republican party. Douglas was a Northern D who tried unsuccesully ( because a supreme Court’s sentence ) to push back in the localities the issue of the slavery . In order to become senator in Illinois he had to win the support of Northern D. However he had also to please Southern D to win two years later the presidential nomination for the Democratic Party. The Lincoln’s trap • Lincoln’s question during electoral debates for Illinois legislatures : «Can the people of a United States Territory , in any lawful way,against the wish of any citizen of U.S., exclude slavery from its limits prior to the formation of a state constitution ?» If Douglas answered yes he could become senator as Illinois Democrats agree but he would have lost the support of S.Democrats for the presidential nomination If Douglas answered no he would hurt his chances in Illinois … The Lincoln’s trap • He answered yes and was reelected to the Senate . However in the nominating convention in 1860 he was nominated only after Southerns withdrew in order to nominate a third candidate for themselves . • Lincoln won the Republican nomination and was elected president by a plurality . Voting methods • Suppose we have a group of 55 individuals, choosing among five alternatives, {a, b, c, d, e}. • Of the 120 possible complete and transitive strict preference orderings an individual might adopt as his or her own, there are only six distinct orderings, or “opinions,” represented in this particular group. • Imagine, six different ways for the group to arrive at a choice among the five alternatives. Voters are supposed to vote sincerly Each group is assumed to cast votes for all the alternative above the line a b c d e d & e Discussion on Voting methods A voting method may be thought of in terms of (1) the inputs required (2) what the procedure does to those inputs (3) the output or outcome produced. Inputs required 1. Plurality voting makes the simplest demands on voters: each person must simply name an alternative (his or her first preference if a sincere voter, something else otherwise). 2. In runoff plurality election in round one, the same data as in a simple plurality contest; in round two, relative preference between the two highest vote getters in round one is needed. 3.Sequential round off, Borda and Condorcet require whole preference ordering . 4. Approval voting requires as much information about preferences as each voter wants to reveal The Quandaries of the procedures (1) Winner (a) with Plurality loses to any other alternative in pairwise comparison ; Condorcet winner loses in a plurality contest , Runoff and sequential eliminate alternative that could defeat any other in pairwise comparison , Borda and Approval are very vulnerable to strategic behaviour and Condorcet does not produce always a winner . The Quandaries of the procedures (2) The problem of the Endogeneity : Which motions are moved or which candidacies are activitated depends strictly on the voting method . For instance one of the drawbacks of approval voting it is that encourages a larger number of alternatives to be brought forward than many other voting methods The importance of the output The nature of the output affects the way we think about the voting methods . For instance if we choose someone that will choose among alternatives not necessarily all available then a method that consider our whole preference ordering makes sense . Philosphical puzzle • If the “wish of the group,” or the “collective will,” or the “public interest” in other terms the output of group deliberation — is to be ascertained from the inputs that the individual group members bring to the voting method (and those inputs vary from method to method), then how are we to give meaning to “wish of the group,” “collective will,” or “public interest”? Electoral Systems • Electoral systems may be thought of in terms of the degree to which their “core value” is representation or governance. • Representation when an electoral arrangement places priority on the degree to which the elected reflect or represent the beliefs and preferences of the electors. Proportional systems=> Representation • Governance when an electoral arrangement yields elected representatives capable of governing , of acting decisively . Plurality systems => Governance • A n arrangement that emphasizes representativeness may make governance more difficult, and viceversa . Cox’s Classification • 5 bit of information are sufficient to classify all electoral systems • v = number of votes per voter • p = when v>1 if the voters must cast all votes or can partially abstain • c= when v>1 if the voters may cumulate their votes or they must distribute them • k = the number of legislators to be elected per district , district magnitude • f= electoral formula Plurality systems • FPP: UK and U.S. . In a society very fragmented representation of many groups can be sacrified . Plurality systems • SNTV : Japan ( until 1993 in the lower chamber ). It is a plurality system with « proportional effects », that encourage parties to carefully plan the candidacies . Plurality systems • LV : It is similar to SNTV but voters have more than 1 vote and less than k and can decide how many votes use. Plurality systems • CV : Now voters have k votes and they can cumulate their votes over only a candidate. Cox’s Equilibrium results • There are two equilibrium tendencies (in a one dimension world). • A central tendency is one in which the candidates tend to converge on the median voter’s ideal point location. Centripetal forces in which electoral competition drives candidates toward one another, • A dispersed tendency is one in which the candidates tend to distribute themselves along the dimension, adopting distinctive policy positions. Centrifugal forces that drive candidates away from one another in order to differentiate themselves. • The number of competing candidates is the crucial variable Cox’s Equilibrium results • When cumulation is not allowed (c= no), if the number of candidates competing for election (k) is small enough relative to the number of votes (v) per voter [ in Display 7.3], then centripetal forces will dominate (in the sense that equilibria will be centrist); • When the number of candidates (k) is large enough, then centrifugal forces become strong enough to create a certain amount of dispersal in equilibrium. • When cumulation is allowed (c= yes), then “centrifugal forces will always dominate. Proportional system • The proportional system does not simply allocate to the party the seats proportionally to its votes • First of all, the legislature’s size is typically fixed in advance and not possible to translate electoral proportions evenly into seat proportions . The level of proportionality will depend on the size of the Parliament and the are always some “remainders”. • Most PR schemes differ primarily on how they handle the problem of allocating the “fractional seats .” • In many country there are explicit representation threshold under which a competing party does not receive any seat . • There is also an implicit representation threshold depending on the district mangnitude and the way to calculate the remainders . Representation vs Governance Proportional vs Plurality • Duverger’s Law, • 1) First -past -the post , single -member district systems are strongly associated with two -party (or two -candidate ) competition. Third parties and third candidates (or both) will ordinarily be loath to enter the race because they have so little chance of winning; in turn , they have so little chance of winning because neither voters , nor campaign consultants, nor campaign contributors are likely to waste their votes, time, and money, respectively, on hopeless candidacies . (strong argument and large evidence) • 2) PR systems are associated with multiparty competition . ( no strong argument but ample evidence) Representation vs Governance Proportional vs Plurality • Districts in which there are, by design, a very small number of winners , only one in first -past -the -post ; exactly k in k -past -the -post, discourage independent political entry and encourage cooperation, coordination, coalition , and merger -like political activity before elections. • In districts where there are many possible winners, as in most PR systems (especially those with a very low threshold) and even in those k -past -the -post systems where k is quite large, independent political entry is encouraged and various forms of cooperation , coordination, coalition, and mergerlike political activity are either discouraged altogether or deferred until after elections. Electoral Formulas: Quotas and Divisors • All PR systems either employ quotas or divisors to determine how many seats each party wins. • In quota systems, the quota indicates the number of votes that guarantees a party a seat in a particular electoral district. Electoral Formulas: Quotas • Hare quota – Valid Votes/Seats. – Examples: Benin, Liechtenstein, Colombia, Brazil, Peru • Droop quota – Valid Votes/(Seats+1) – decimal part. – Examples: Slovakia, Luxembourg • There are other quotas such as the Imperiali quota (n=2), the Hagenbach -Bischoff (n=1), and the Reinforced Imperiali quota (n=3). Electoral Formulas: Quotas • Example: Hare quota • Imagine that there are 100,000 voters in a district that is electing 10 members. – The Hare quota is 100,000/(10) = 10,000 – Thus, each party wins a seat for every 10,000 votes it wins. Electoral Formulas: Quotas • Several ways of allocating the unallocated seats – Largest remainder method (Costa Rica, Colombia, Honduras) – Highest average method (Brazil) – Modified highest average method (Luxembourg) Electoral Formulas: Divisors • All PR systems either employ quotas or divisors to determine how many seats each party wins. • A divisor, or highest average, system divides the total number of votes won by each party in a district by a series of numbers (divisors) to obtain quotients. • District seats are then allocated according to which parties have the highest quotients. Electoral Formulas: Divisors • D’Hondt – 1, 2, 3, 4 … – Examples: Finland, Spain, Bulgaria, Cape Verde, Netherlands • Sainte -Lague – 1, 3, 5, 7 … – Examples: Latvia • Modified Sainte -Lague – 1.4, 3, 5, 7 … – Examples: Norway 1953 -88, Sweden 1952 -69 Electoral Formulas • The different electoral formulas influence how proportionally votes are translated into seats. Representation vs Governance Proportional vs Plurality • First -past -the -post systems typically, and other “small” k – past -the -post systems often, resolve many conflicts before legislative politics commences. • PR and “large” k -past -the -post systems, on the other hand, defer this kind of conflict resolution until the legislature convenes. • Thus , parliamentary political conflict tends to be more muted and centrist in legislatures elected by FPP; indeed, there is typically a single majority party that can get on with the business of implementing its agenda. • Legislatures elected by PR reflect rather than resolve political conflict in advance, depending upon post – election parliamentary politics — coalition government, for example, to discover the means for resolution . Cooperation and the problem of order • “ Problem of order ” — can be solved either by • C ausing individuals to internalize pacific attitudes and intentions, in the form of altruism, religion, or other moral principles • Endowing Leviathan (Hobbes) with the power to root out predatory behavior and otherwise regulate social life for peaceful ends. • But cooperation may emerge and be maintained, even though people have not internalized pacific or otherwise touchy -feely attitudes, and even though there is not the heavy sword of Leviathan hanging over them. Just for self -interest … Cooperation and the problem of order • Individual behavior typically involves bearing some costs in order to secure some benefits. A rational individual weighs benefits against costs. The former are enjoyed exclusively by her; the latter are borne exclusively by her. It is an individual optimization problem covered in any standard economics course. • When a collection of individuals is pursuing some objective Individual group members must bear the burdens — club dues, effort, investments of time, perhaps — but the benefits are often not exclusively private. Cooperation and the problem of order • David Hume tells the story of two farmers whose respective fields abut a common marshland. If the marsh were drained, common benefits would be generated — for instance, the destruction of a mosquito habitat. • Farmer A’s individual effort in draining the marsh, itself a burden, would produce this benefit not only for himself, but also for Farmer B. Each farmer is certainly desirous of the benefit, but is loath to pay the price especially if he can get the other guy to do all the heavy lifting Hume’s Marsh – Draining game • Each of Hume’s farmers valued the drained marsh at 2 utiles . • If either were to take the project on by himself, the cost to him (in terms of the things he would have to forgo in order to take on the bother of the job) would be 3 utiles . Thus, if there were only one farmer available to take on the task, then draining marsh won’t be very convenient. • If each farmer worked“cooperatively ” with the other, then it would cost each only 1 utile. In this case each farmer would enjoy 2 utiles ’ worth of drained marsh at a cost of 1 utile. • However the best deal of all would be for the marsh to be drained entirely by the other farmer. Hume’s Marsh – Draining game • No matter what Farmer B does, Farmer A always gets a higher payoff if he chooses not to drain. The reasoning is precisely the same • No matter what Farmer A does, Farmer B always gets a higher payoff if he chooses not to drain. Hume’s Marsh – Draining game • B’s incentives reinforce A’s inclination not to drain, and vice versa, ad infinitum. Do not cooperate – Do not cooperate as strategy equilibrium. • Rational individuals , « Irrational society» ? Another Hume’s parable • Your corn is ripe today : mine will be so tomorrow . It is profitable for us both that I shou’d labour with you today, and that you shou’d aid me tomorrow . I have no kindness for you, and know that you have as little for me . I will not, therefore, take any pains on your account ; and should I labour with you on my account, I know I shou’d be disappointed, and that I shou’d in vain depend upon your gratitude . Here then I leave you to labour alone : You treat me in the same manner . The seasons change ; and both of us lose our harvest for want of mutual confidence and security . Two – persons cooperation with repeat play • Hume’s example is a one -shot deal. • Most societies, however, including the one consisting of Hume’s two farmers, are more enduring. • They do not usually materialize for that one opportunity of securing a cooperation dividend; nor do they immediately disintegrate thereafter. • Rather, this week it’s the marsh that needs draining, next week it’s the common fence between the farmers’ fields that needs patching, the week after there is the two -man job of replacing a roof on one farmer’s barn, and the week after that it’s the other farmer’s pond that needs to be sealed. • In short, societies consist of a series of repeated (or even Continuous) encounters, not one -shot plays of a game Two – persons cooperation with repeat play • What happen with two marsh (in sequence) that need draining ? • Each farmer will know that the second play of the strategic interaction will be the last and it will be a one -shot affair. So, in that last interaction each farmer will play his noncooperative strategy. • Backing up to the play before that (the first play), the farmers will realize that, effectively, that one is the last play, since the second play will be determined no matter what happens in the first play. • So once again , each will rationally play his “do not cooperate” strategy . • If a strategic interaction like Hume’s marsh -draining game is played repeatedly , if the number of repeat plays is finite and commonly known to the members of the society, then each encounter will be played out as though it were a one -shot affair . Repetition in this instance is no more than a string of one -shot games, and the cooperative dividend in each case is lost. Two – persons cooperation with repeat play • However Societies are ongoing and continuous. Farmers A and B may not live forever, but they don’t know when their microsociety will come to an end. Thus, they don’t know when the last play for a cooperation dividend will arise. • If each farmer assumes the string of opportunities for cooperation will be very long, then each may be willing, on the first occasion , to take a chance. The worst that could happen is that he will be exploited that one time, learn his lesson, and simply refuse to cooperate on subsequent occasions. Given the symmetry of the situation, both farmers may take a chance in that first encounter, resulting in an outcome Cooperate – Cooperate and a payoff of 1 utile. • On the next occasion,each farmer will remember that the previous encounter had elicited cooperation from the other, thereby encouraging each to try it again . A cooperation path. Axelrod tit for tat strategy • The first time, cooperate . The next time, cooperate if your colleague cooperated the last time . But don’t cooperate if he didn’t the last time, and don’t cooperate again until he changes his wicked ways. That is, cooperate conditionally after the first play of the game . • Each of two ruggedly individualistic , rational egoists has, by virtue of being embedded in an ongoing social relationship, found it in his interest to Cooperate with his counterpart. Axelrod tit for tat strategy • However If the relationship had gotten off to a bad start — with one or more of the farmers not being “nice” at the outset — then at each play each farmer will “punish” the other for failing to cooperate the time before . A non cooperation path. Alternative Mechanisms to induce cooperation Internalized values • Imagine the previous Hume ´s game but the story is the well known story of the prisoner dilemma Alternative Mechanisms to induce cooperation Internalized values • However suppose the two prisoners are Mafia soldiers who have sworn omertà (conspiracy of silence ) in dealing with anyone outside the family . The payoff matrix in the display does not seem to capture this internàlized value very well . Alternative Mechanisms to induce cooperation Internalized values • If one of the Mafia soldiers implicates his partner, but the other does not , then his payoff is now – β , a very large and nasty negative number. • A game -theoretic analysis of this situation suggests that there are two possible outcomes for this game . Alternative Mechanisms to induce cooperation Internalized values • Either both will squeal and receive 0 utiles each, or neither will squeal and receive 1 utile each. • Each of these results is an “ equilibrium point» in the sense that in each of these outcomes neither player has an incentive to change his strategy if he believes the other guy isn’t going to change his . • Internalized values , can produce cooperation, even in one -shot games. But they do so by changing the game. Alternative Mechanisms to induce cooperation External enforcement • The idea of third -party enforcement of agreements reached by two contracting (or Cooperating ) individuals is the principal mechanism upon which the entirety of neoclassical economics rests . • In economic contexts it is normally assumed that contracting individuals are assured that their contracts will be enforced by a judge , court , or sheriff and , moreover , that these agreements are enforced costlessly and in an error -free fashion . • Third -party enforcement , in this instance , is precisely the kind of assurance that is required in order to consummate many trades (that is , to capture the dividends of cooperation ) Alternative Mechanisms to induce cooperation External enforcement • In the Mafioso Dilemma, for example, imagine the payoffs are as they originally were in Prisoner ´s dilemma. • Mafioso A says to mafioso B, “I won’t squeal.” Mafioso B says to mafioso A, “ Neither will I.” These promises are credible because there is a third -party enforcer, Don Corleone, who imposes sanctions on those who break their promises. Alternative Mechanisms to induce cooperation External enforcement • The presence of Don Corleone effectively transforms Prisoner ´s dilemma into the Mafioso non dilemma ( MnD ). • As long as mafioso A and mafioso B believe they are playing MnD then Don never has to display his might. • To those ignorant of the Don’s existence, the two prisoners might be thought simply to be “honest” men who keep their promises. • Similar logic is behind Hobbesian Leviathan. The problems of the enforcement • The enforcement is costly • The enforcement is imperfect • The incentives of enforcers can be inappropriate (“Who will guard the guardians ”) Collective Action The groups • Much of what we do in life we do in groups, so much so that we often commit the fallacy of false personification and talk as though these groups had a life of their own. • The very ubiquitousness of groups in pluralistic political systems explains why, for most of the first half of the twenteth century, the study of politics was the study of groups. Political outcomes were seen as the result of struggles among groups. Collective Action The groups • Arthur Bentley wrote almost like a physicist about the “parallelogram of forces” that constituted group interactions and infighting.’ • He thought of the status quo in any policy domain as a point on a page. Change occurs as this point gets pushed around by various forces impinging on it. Bentley treated each group as consisting of a “direction” and a “magnitude.” Direction indicated what changes in the status quo a group wanted; magnitude measured the group’s political power of struggles among groups. • Axiom of political science at that time: „Common interest , however defined and however arrived at, leads naturally to organizations coherently motivated to pursue that common interest ; politics is all about how these coherently motivated organizations support and oppose one another .“ SQ Collective action as multiperson cooperation • Groups of individuals pursuing some common interest or shared objective consist of individuals who bear some cost or make some contribution on behalf of the joint goal. • We can think of this as an instance of two -person cooperation writ large. • Each of a very large number of individuals has, in the simplest situation , two options in his or her behavioral repertoire: “contribute ” or “don’t contribute” to the group enterprise . • If the number of contributors is sufficiently large, then a group goal is obtained. If the group goal is obtained , then every member of the group enjoys its benefits , whether he or she contributed to its achievement or not . • A goal like this as a public good , since once it is provided it becomes available to all,whether they participated in its provision or not . Collective action as multiperson cooperation . Complications • 1) Dichotomous or continous outcomes ? • 2) In a multiperson situation, we need to specify how many contributors are necessary to attain the group goal. We need to specify the relationship between each of the individual contribute/don’t contribute choices and the final outcome (what economists call the production function ). • 3) Group’s goal or group’s goals ? When there is more than one goal.. Collective action as multiperson cooperation . Complications When unanimity is required • Suppose each individual in the group evaluated the group goal equal to B utiles (where B stands for “benefit”); Assume that B> O. • Suppose further that the utility value of contributing to the group project is — C utiles (C stands for “ cost”), where C> O, too. • If C > B, then no one will contribute , no matter how many others do. • if B> C, then it is quite likely that every member of the group will contribute. there are two possible equilibrium outcomes when B> C. a) Everyone has chosen not to contribute. If Ms. j, decided to contribute, then her payoff would be — C ( the cost she bears) and there would still be no compensating benefit B (since the latter is produced only if everyone contributes ). So, it is possible for a group to be stuck in an “ equilibrium trap” in which no one is contributing, even though everyone would be better off if everyone contributed. b) Everyone has chosen to contribute. Ms . j, like every other group member, makes the following calculation: If I don’t contribute, then I get a payoff of 0. If I contribute, and so does everyone else, then I get a payoff of B — C> O. If I contribute , but someone else does not, then I get — C. Everyone else makes the same calculation. Collective action as multiperson cooperation . Complications When unanimity is required All other i group member Do not contribute Contribute Ms. J Do not contribute 0, 0 0, -C Contribute -C,0 B-C The “ everyone contribute ” event is what Thomas Schelling has called a focal point Collective action as multiperson cooperation . Complications When unanimity is not required • Suppose that a unanimous contribution rate is no necessary. Of the n persons in our group, suppose that only k contributors are necessary for the group objective to be obtained (where k is a number greater than zero but less than n). Collective action as multiperson cooperation . Complications When unanimity is not required • If fewer than k — 1 of J’s her colleagues are contributing, or more than k — 1, then it doesn’t pay for Ms. j to contribute. In either of these cases she obtains a higher payoff by not contributing (0 instead of — c and B instead of B — C, respectively). • Only if exactly k— 1 others are contributing is Ms. j’s contribution essential, in which case she would obtain B — C > 0 instead of 0 by contributing. Ms. j does not have a clearcut course of action , because she does not know in advance whether less than k — 1 , more than k — 1, or exactly k — 1 of her colleagues are going to contribute . Collective action as multiperson cooperation . Complications When unanimity is not required • There are two rational, or equilibrium , outcomes possible. Either no one contributes or exactly k do. In either of these cases, no group member will have reason to reconsider his or her action. • If no one is contributing , then Ms. j (or any other member for that matter ) would be foolish to decide to contribute. • If exactly k people are contributing, then those not contributing have no need to contribute while the k that are contributing know that each and every one of them is absolutely essential. • Exactly k individuals will need to believe that they, and only they, are likely to contribute…difficult. • This poses a much more complicated problem for group members than the case where unanimity was required . The “tipping point,” k, is a crucial determinant of whether or not this group is able to get its act together . Collective action as multiperson cooperation . Complications When unanimity is not required • Example . Suppose n = 100. and a) suppose k = 95. Most group members , Ms . j included, are bound to think that, if they are going to achieve the group goal, then a lot of them are going to have to contribute. The group cannot stand too many defectors. In other terms when k is relatively high then is high the pressure on people to contribute. As k gets smaller, the pressure reduces . b) What about k= 25 ? p artecipation is very unlikeley . When k gets very small, the “ rational” forecast is that the group objective will simply not be accomplished… However other factors should be considered . In every group there are always some people who “do the right thing” no matter what . For one thing, they may secure utiles directly from the participation itself for another they may have internalized a value system that encourages contribution to group life. Collective action as multiperson cooperation . Complications When unanimity is not required A combination of strategic and psychological pressures that encourage contribution rise as k gets large relative to n . • Holding n fixed the likelihood that there will be sufficient contribution declines as k declines. (!!) However for very small k there may be a small increase in the likelihood of group success, since there is often someone who will contribute for non strategic reasons. • General tendencies toward the “no one contributes” get more pronounced as n gets large, as C gets large, • As n gets large, holding k fixed it is not equivalent to holding n fixed and letting k get smaller. As n gets large the psychological identification with the group, an identification that may well affect the benefit utiles one enjoys upon achieving a group goal, becomes more tenuous.. • As C gets large, holding n, k, and B — C fixed, there will be a tendency for the group to fail to secure its objective . Even if B — C remains fixed. • As B — C gets large, holding everything else fixed , the importance of the group goal grows and people are prepared to take psychological risks in these circumstances. Collective action as multiperson cooperation . Complications When there is more than one goal • Imagine a social club in which each member might enjoy an evening at the ballet together or a night at the opera together . They are indifferent between these two activities as long as they do it together . Ms. j again, has two choices: go to the opera house or go to the ballet theater. If every group member makes the same choice, then they each realize a benefit of B utiles . Failure to coincide on a common choice yields a positive payoff smaller than B, say b utiles (where b β and for Ms.k β>B • Cooperation problems among group members are exacerbated as members not only have to overcome the difficulties of coordination associated with nonuniqueness but also must overcome inherent intragroup differences of opinion about where to go. • The situation is even worse if the group is evenly divided among people with different preferences. It is necessary some sort of institutional solution . Olson ´ s Logic of Collective Action • The pluralist assumption of that common interests among individuals are automatically transformed into group organization and collective action, was problematic. Individuals are tempted to “free -ride ” on the efforts of others, have difficulty coordinating on multiple objectives ( nonuniqueness ), and may even have differences of opinion about which common interest to pursue (conflict of interest ). • One cannot merely assume that groups arise and are maintained ; rather, formation and maintenance are the central problems of group life and politics generally. Olson ´ s Logic of Collective Action • Olson claims that this difficulty is severest in large groups for three reasons . 1) Large groups tend to be anonymous. No group identity. 2) In the anonymity of the large -group context, it is especially plausible to claim that no one individual’s contribution makes much difference. 3) In large group the problem of enforcement is very hard to cope with. In a large, anonymous group it is often hard to know who has and who has not contributed, and, because there is only the most limited sort of group identity, it is hard for contributors to identify, much less take action against, slackers. As a consequence , many large groups that share common interests fail to mobilize at all — they remain latent. Olson ´ s Logic of Collective Action Small groups as privileged groups • The same problems can plague also small groups but to a smaller extent 1) Small groups are more personal, therefore their members are more vulnerable to interpersonal persuasion . 2) In small groups, individual contributions may make a more noticeable difference (k is large relative to n) so that individuals feel that their contributions are more essential. 3) Contributors in small groups often know who they are and who the slackers are. “Punishment” is easier to effectuate . 4) Small groups often engage in repeat play and , therefore, can employ tit – for -tat strategies to induce contributions. • These small groups are privileged because of their advantage in overcoming the free -riding, coordination, and conflict -of interest problems of collective action. • It is for these reasons that small groups often prevail over, or enjoy privileges relative to, larger groups: producers over consumers, owners of capital over owners of labor, a party’s elites over its mass members. Olson ´ s Logic of Collective Action Exploitation of powerful members • The asymmetry between large and small applies within groups as well . The inequalities among group members may help the group achieve its goals. • A large powerful group member is likely to “make a difference” in many situations, therefore he or she will be under intense pressure to contribute . Indeed the enterprise may succeed, even if one of the weaker members opts not to contribute. Example of Nato . Exploitation of the stronger by the weaker Powerful member Contribute Do not contribute Weak member Contribute A -0.5c , A -0.5c -c,0 Do not contribute A, A -c 0, 0 With A>0; 0 u(K), where u is Ms.j ´s utility function and J and K stand for the election of Jackson and Kendall, respectively. Put equivalently, if J, should win rather than K then Ms.j gets a benefit B= u(J) -u(K) >0. Theories of voting and collective actions (2) • If Ms. j were the only voter, then the participation would depend on a) the benefit of picking the winner, B, exceeds the costs of doing so , C (B>C) , then she should do so, picking J.; b) If , on the other hand , C> B, then she shouldn’t participate since the utility difference between the candidates is not worth the cost Ms. j would bear to make the choice. • Obviously Ms. j is not the only voter so she must take the intentions and capabilities of others into account . Theories of voting and collective actions (3) There are five circumstances that Ms. j(or any other voter) needs to consider. These involve how the other n -1 voters (excluding Ms. j) behave in the aggregate, and may be partitioned into five “states of the world” (S1 through S5). These “states” are the outcomes that would take place if Ms. j abstained (did not participate) . • S1: J. loses to K. by more than one vote. • S2: J. loses to K. by exactly one vote. • S3: J. and K. tie. • S4: J. beats K. by exactly one vote. • S5: J. beats K. by more than one vote . L= lottery when the election ends in a tie . u(L)= 1/2u(K) + 1/2u(J) Theories of voting and collective actions (4) Ms.j won ´t ever vote for K. However the crucial question is : Does the pay off from voting for J. exceed the payoff from Abstention ? An answer requires us to compare the first and third rows of the display . Unlike the comparison of rows 1 and 2, however, in some states “vote for Jackson” gives the larger payoff while in others “abstain” is more attractive Theories of voting and collective actions (5) • We must incorporate into the analysis Ms. j’s beliefs about the likelihoods of the various states. We represent . j’s beliefs by probability numbers . Let us set u(J)=1 and u(K)=0 • Ms. j believes S1 occurs with probability p1, S2 with probability p2, S3 with probability p3, S4 with probability p4 , and S5 with probability p5 (where each probability number is 0 or larger and together they sum to 1; p1+p2+p3+p4+p5=1). • When the electoral behaviour „ produces “ tie than j´s utile = 0.5*0+0.5*1=0.5. Theories of voting and collective actions (6) We have to calculate the expected utility EU (vote for J.) = p1 (-C ) + p2 ( 1/2 -C ) + p3 ( 1 -C ) + p4(1 -C ) + p5(1 -C ) • = – C (p1+p2+p3+p4+p5) +p2(1/2) +p3 +p4+ p5 • = -C+p2(1/2) +p3+p4+p5 EU (abstain) = p1 (O)+p2 (O)+p3 (1/2 )+p4 + p5 = 1/2p3+p4+p5 Theories of voting and collective actions (6) • Ms . j should vote J rather than abstaining • if and only if EU ( J.) > EU (abstain ). • -C+p2(1/2) +p3+p4+p5 > (1/2)p3+p4+p5 • p 2(1/2)+p3(1/2) > C; p2+p3>2C • In words, she should vote if the sum of the probabilities that she either makes a tie (voting for J. in S2) or breaks a tie (voting for J. in S3) exceeds twice the cost of voting . Theories of voting and collective actions (7) • What the inequality p2+p3>2C tell us ? a) The costlier it is to participate, the less likely Ms. j will be to do so. This follows because the inequality of p2 + p3 > 2C is more difficult to satisfy as C gets large. if C is sufficiently large (specifically if C > 1/2), she should never participate as p2+p3 cannot be larger than 1. b) Ms . j should be most disposed to participate if she believes the election is going to be close. This follows because p2 + p3 — the likelihood of “making or breaking a tie” — is a definition of a close election c) However in real elections how large are p2 and p3 ? … here the paradox of voting … Theories of voting and collective actions (7) Paradox of voting • Suppose C were very small relative to U(J.) – u(K.) = 1; say, C = 1/1000. So the inequality p2 + p3> 2C now says that Ms. j should vote if and only if her probability of making or breaking a tie (p2 + p3) were larger than 2/1000. • The likelihood of making or breaking a tie in mass election, for instance in the United States, where there are approximately 125,000,000 prospective voters is infinitesimal , and certainly much smaller than 2/1000. • So it is very unlikely that a sensible person, like Ms. j, will believe that P2 + p3 exceeds 2/1000. Most sensible persons, according to this analysis should not vote . Theories of voting and collective actions (7) • One interpretation of the results this strictly instrumental analysis is that instrumental calculations are simply insufficient to induce participation in large -scale elections. • According to Riker and Ordeshook there is an experiential as well as an instrumental basis for voting — voting has consumption as well as investment value . 1) For one thing , individuals in democratic societies possess a belief system or ideology in which great stock is placed in participation . 2) There are often punishments inflicted on nonparticipators . In some societies , a fine is imposed . In others there are “ watchdogs ”. 3) Individuals participate in electoral activities not only to avoid feelings of guilt or to dodge the “ punishments ” inflicted by others , but also because it can be „ fun “. Now the equation is not only pB >C but pB+D >C where D is somehow „ consumption “ utility , utility coming from experential behavior . A typology of goods • Excludability: the “owner” may exclude others from enjoying the good • Jointness of Supply (non rivalry): one person’s use does not diminish the supply available to others. Who produces public goods ? • Because a public good is nonexcludable it may be enjoyed without paying a price for it. • However producer will be loathe to provide a good if he cannot elicit payment for it. And even if there were some imperfect method by which a potential producer could extract a return from providing a public good, the amount supplied would likely be very much less than it would be if payment could be extracted directly. • As a result, everyone is worse off. Who produces public goods ? Possible solution to the under -production of public goods • Example of peasant farmers and Lord’s protection in middle age => An institutional solution by coercion • Example of Lighthouses in England : If a monopolist controlled the waterfront of the harbor , then he could jointly price lighthouse services together with docking privileges in a manner that captured a return for the former. • Example of a special case of «public good »: OPEC, oil price and the role of Saud Arabia • In general the most common solution of all — the quintessential political solution — is the public supply of public goods. Public Supply • In many parts of the world, lighthouses, the protective services of the police and army, judicial services, public utilities like water, sewage, and power, and provision for public health , roads, and other infrastructure are commonly provided by government. • T he argument used to justify the public supply is often that because they are public goods private market actors will not provide them (at least not in sufficient quantities ) because they cannot be assured of adequate compensation . • The state may use its authority to require payment, either out of general revenue raised by taxation or from user charges of various sorts. Public Supply Paradox • Public provision does not just happen . Political actors must be persuaded to act. • If the provision of a public good distributes some benefit widely, and if the enjoyment of that benefit is unrelated to whether a contribution has been made toward mobilizing politicians to act, then why would any individual or interest group lobby the government for public goods ? Is it not more convenient “free riding” ? • The “public supply” solution to the provision of public goods seems becoming a problem of collective action Public Supply Production and Consumption • We have to distinguish the consumption of a public good and its production . When designating a good as public or private, we are talking about consumption properties (excludability and rivalry) • In fact, in nearly every instance public goods are produced with substantial private input . There are “interest groups” that receive private goods . They are privileged groups and they are more able to solve collective action problems than consumers’ groups in order to lobby on the political actors. Example of the highways. • In short, the politics of public supply is as much about the production of public goods as it is about their consumption. Public Supply Not necessarily efficient and fair • The production of public goods is a problem because of the very nature of these products. • Private incentives are insufficient to encourage their production voluntarily. Some sort of political fix is required , (grants of monopoly privilege, waivers of antitrust laws, public subsidy of private production, and outright public provision). • Each entails the grant of extraordinary privilege or authority to some individual — the lord of the manor, the firm granted a monopoly , a public -sector bureaucrat — whose incentives may not be aligned properly to the social objectives being sought. • The lord of the manor wants prestige and glory, not public defense ; the firm wants profits, not a national highway system ; the bureaucrat wants turf and budget authority, not scientific discoveries . The public good is the incidental by -product of, not the motivation for, their behavior . Externalities • An externality is a special kind of public good. It is typically the unintended by -product of voluntary activity that is imposed on others. Thus, an externality is jointly supplied and , because it cannot be easily avoided, non excludable (unavoidable). • Externalities can be “negative” (more frequently) or “positive” • The most popular and widely used solutions are taxes (against negative externalities) and subsidies ( in favor of positive externalities) Externalities Taxes and subsidies are imperfect • Two major shortcomings associated with taxes and subsidies. 1) Setting the appropriate level for taxes and subsidies can be extremely difficult . Lack of knowledge. 2) Some goods are taxed or subsidized to deter negative or encourage positive externalities many other goods are taxed or subsidized because political machinery for taxing and subsidizing exists in the first place, and comes under the influence of those who benefit from its policies, quite independent of any consideration of externalities. Externalities Regulatory Regime • Regulation is a “ pratical ” approach to the control of externalities. It entails the creation of a governmental bureaucracy — an agency, bureau, or commission — charged with setting standards, prices, fees, or practices in consumption or production activities that generate externalities . Statutory authority usually spells out the purposes to which this bureaucratic control should be put and the discretion the bureaucratic entity has in pursuing those purposes. • Through administrative procedures, or the civil and criminal court system, the agency has an ability to enforce its commands . Externalities Respecification of property rights • Example of pollutions permits and the creations of a market of “pollution” rights. • It is crucial in this case establishing a reasonable amount . Obviously the initial distribution of « rights » is a very controversial and political decision . The C ommons • Jointness of Supply (non rivalry): NO • Excludability: NO • A commons is, by definition, owned by everyone (in common ), and therefore is the responsibility of no one. • Ex. A field owned by a village and used by its residents’ herds as a grazing commons. Each villager gets to graze his or her cattle “ for free.” If a villager is contemplating adding a head to his herd , he will take into account his costs of doing so, but this calculation will not include the cost of grazing. If the commons is large, and the village demands on it minimal, this will not pose serious problems. But even if demands on the commons grow , no villager has an incentive to restrict his use of this “ free” resource. • The commons will be overgrazed and ultimately destroyed, inasmuch as its capacity to regenerate itself will have been disabled. The C ommons • Ex. village with a common grazing field . • 100 villagers are each grazing two cows on the commons, a number which the commons can support and still regenerate itself. • Each villager considers adding one cow to his herd. It is profitable given the free grazing privileges. If one villager were to proceed, the commons would be damaged only marginally since the herd size will only have increased from 200 to 201 . • However if all the villagers proceed, there will be a 50 percent increase in grazing, an amount exceeding the carrying capacity of the commons. So, if everyone proceeds each will be worse off since they will have destroyed their field. But if any one villager proceeds, he will be better off and all the others will hardly be affected at all. Thus individual incentives and social necessity clash. • Overgrazing the commons is the large -number analog of the Prisoners’ Dilemma and Hume’s marsh -draining game The C ommons • C ommons problems arise because of imperfectly specified property rights. • Political arrangements affect both the solutions selected to deal with commons problems and the likelihood of success. • According to Elinor Ostrom successful Commons are those with design features possessing : 1) clearly defined boundaries; 2) congruence between rules for using the commons and local needs and conditions; 3) individual rights to formulate and revise the rules for operating the commons; 4) monitoring arrangements in which the monitors are ultimately responsible to the community; 5) graduated punishments for violation of rules; 6) low -cost arenas for resolving disputes; 7) relative freedom of users of the commons from external governmental authorities. Conclusions Institutions • Responses to regularly recurring problems are often institutionalized. Collective action comes to pass in the political community because standard procedures are established that provide political actors with appropriate incentives to take the action necessary to provide a public good or control an externality. • Politics is omnipresent in human society, and its more routine aspects are dealt with by institutions. • Institutions, not just individual preferences, matter for collective results. Components of a framework for studying institutions • Division of Labor and Regular Procedure • Specialization of Labor • Jurisdictions • Delegation and Monitoring Division of Labor and Regular Procedure • Actors meeting regularly in an institutional context evolve procedures by which the various bits of business are divided into manageable units and then sequenced into a specific order. They do so, most obviously, to bring order to their deliberations. It is an efficiency -enhancing aspect of an institution. • A set of procedures and a division of labor define a strategic context in which individuals may think about whether to participate at all, in what parts of the sequence to participate, and, finally, precisely how to participate. They may condition these choices on what will already have transpired, as well as on what they know is coming up. Procedures embodying a division of labor are strategy enhancing. Division of Labor and Regular Procedure • A set of procedures and the division of labor empowers the organization’s members vis a vis the leaders . Reliable and publicly known procedures provide a check against arbitrary and capricious behavior by institutional leaders . • A particular divison of labor or set of procedures is not “rigid”. There are always mechanisms by which to change the way the institution’s business is conducted. Standard operating procedures may be amended , so that some particular piece of business , and all subsequent business of a similar sort, is conducted in a different manner. Or procedures may be temporarily suspended to handle a specific matter, after which the original procedure is reinstated. The former represents a permanent change in the institutional status quo.The latter is a “short -circuiting” device by which some specific piece of business is conducted in a different manner, but the rules governing future business of the same sort revert to the earlier institutional status quo. Specialization of Labor • By allowing different members to do different things, in light of their different interests and talents, an institution is able to capitalize on the rich endowment of “human capital” contained in its membership. • However not all institution encourages specialization . Supreme Court versus Congress . Usually there is an evolutionary pattern from non specialization to specialization . Jurisdiction • Each jurisdiction is a bundle of activities, and members of the institution assigned to a specific jurisdiction become jurisdictional specialists. They usually enjoy a considerable “power” in their jurisidiction compared to the institution’s member who are not jurisdictional specialists. • The « specialists » often ignore what is happening in other sectors of the same institutions . • The division and specialization of labor , institutionalized into jurisdictional arrangements , is induced both by external and internal pressures . Delegation and Monitoring • The delegation of authority and resources to specialist subunits exploits the advantages of the division and specialization of labor but risks jeopardizing collective objectives of the group as a whole . • T he division and specialization of labor allows an institution to decentralize its operations. This , in turn, facilitates the delegation of authority and resources to specialists who , because they have disproportionate influence over events in their respective bailiwicks, also have incentives to develop their expertise further. • The very act of delegating, however , generates a problem of control in which specialists may have opportunities to pursue private objectives at odds with the public purposes of the institution (what organization theorists refer to as the moral hazard problem). Institutions normally institute mechanisms both to monitor subunit performance and to control behavior wildly at odds with institutional objectives. Legislatures • All the people to whom the legislator is accountable (including to himself or herself) are legislator’s constituents. Different legislators will give different weight to personal priorities and the things desired by campaign contributors and past supporters. • Legislators come to the legislature with political purposes that motivate them to want to pursue specific public policies ( instrumental behavior ). • In a representative democracy the specific public policies that representatives want to pursue are heterogeneous (preference heterogeneity ). a) First , owing to their different constituencies , legislators will give priority to different realms of public policy. b) Secondly they are heterogeneous in the opinions they hold on any given issue. • Diversity in priorities and preferences among legislators is sufficiently abundant that the view of no group of legislators predominates ( diversity ). Legislative consensus must be built Legislatures • Cooperation is required and repetition of the interaction ( tit for tat strategies )is not sufficient . It is necessary an « institutionalization » of cooperative practices . • Underlying Problems a) Majority Cycles b) Matching influence and Interest c) Information d) Compliance Majority Cycle • Because there is typically preference heterogeneity and diversity, no specific program of policy distinguishes itself as an obvious course on which to proceed. • if issues are multidimensional in nature (as almost always ) then heterogeneity and diversity mean there is no equilibrium to majority voting. Procedures are required to cut through all this instability. • However procedures that give asymmetric advantages to some at the expense of others would not be tolerated if they extended across all issues. • A legislator can hope that whatever procedural practices are instituted to bring orderliness to legislative decision making allow him or her to have some say in those issue areas he or she cares about most. Matching influence and interest • Legislatures are highly egalitarian institutions. Each legislator has one vote on any issue coming before the body. He cannot aggregate the votes in his possession and cast them all, or some large fraction of them, for a motion on a subject near and dear to his heart (or those of his constituents). • In principle the problem could be alleviated by a system of vote (promise of vote) trading. However there are two drawbacks a) The system allowing negotiating and bookkeeping would be extremely complex with relatively large assemblies b) Vote -trading agreements, like other deals among politicians, are not enforceable contracts. • Party leaders can somehow facilitate the trade but cannot be the solution Information • In order to vote intelligently, legislators must know the connection between the instruments they vote for and the effects they desire. • Policy -relevant information is a public good and as any other public good tends to be undersupplied. • Institutional arrangements that provide incentives to some legislators to produce, evaluate and disseminate this knowledge for others will permit public resources to be utilized more effectively. Compliance • If legislators wish to have an impact on the world around them, especially on those matters to which their constituents give priority, then it is necessary to attend to policy implementation as well as policy formulation • Compliance will not “just happen” and, like the production and dissemination of reliable information at the policy formulation stage, the need for oversight of the executive bureaucracy is an extension of the cooperation that produced legislation in the first place. It must be institutionalized. A dynamic and conflictual response of the previous problems • Legislative institutions emerge and evolve in response to the previous problems. • Institutional solutions are not “static” both because the problems change from time to time and because the solutions arrived at are not always the best ones. Learning and adaptation to changes in the policy environment surely occur in institutional life. • Institutions incorporate conflict. Legislatures are not only forums in which teams of kindred spirits try to solve common problems. They are also battlefields on which the deployment of public authority is determined. Jurisdictional division and Specialization of Labor Dimensions Jurisdictions Committee Mps d1, d2 J1 c1 A d1,d2 J1 c1 B d1,d2 J1 c1 C d1,d2 J1 c1 D d1,d2 J1 c1 E d3 J2 c2 F d3 J2 c2 G d3 J2 c2 H d3 J2 c2 I d3 J2 c2 L d4,d5 J3 c3 M d4,d5 J3 c3 P d4,d5 J3 c3 Q d4,d5 J3 c3 R d4,d5 J3 c3 S Jurisdictional division and Specialization of Labor • Two crucial political processes a) The assignment of issues to the committees ( under which policy dimensions will be evaluated ?) b) The committee assignment (of Mps ) : the process determining who gets on what committees is significant because it designates which members will have extraordinary influence over the issues falling into the jurisdiction of each committee Delegation and Jurisdictional Authority • Division of labor makes sense if committees enjoy specific forms of authority in their respective jurisdictions . • Committee jurisdictions and authority, like nearly all other aspects of structure and procedure, are created by the parent legislature. • Committees may be thought of as agents of the parent body to whom jurisdiction -specific authority is provisionally delegated: a) gatekeeping power ( however a majority of the parent legislature can approve a discharge petition) b) proposal power; While any member is entitled to make proposals , the real proposal power is in the committee majorities ’ hands . Committees are lords of their jurisdictional domains , « setting the table » for their parent chamber . c) interchamber bargaining power d) oversight authority. In all of the dimensions of public policy falling within a committee’s jurisdiction, there is always some status quo policy in place. Monitoring and Control • Committees are quite consequential players in their respective policy jurisdictions . If unchecked they could easily take advantage of their authority . What prevents committees from exploiting their before – the -fact agenda power and their after -the -fact bargaining and oversight authority? • There are two “ordinary” ways in which the parent chamber keeps committees in check ( and can reduce the “agency loss”) a) committee assignments b) amendment control rules a) Committee assignments • The key issue is the degree to which a committee is representative of the entire chamber. • In representative committees the majorities are likely to behave much in accord with majorities in the larger chamber from which they were drawn. • Outlier committees, on the other hand, have a composition significantly different from the parent chamber. In these situations the distribution of preferences in the committee is very different from that of the full legislature. C Fx x Representative committee C Fx x Outlier committee a) Committee assignments • Why doesn’t the legislature ensure committee responsiveness to preferences in the chamber by appointing only representative committees? • Many committee jurisdictions consist of policy areas of interest to relatively small numbers of legislators. Most of legislature is highly concerned with decisions coming out of the more generalist committees. Therefore 1) legislators were allowed freely to flow to the specialist committees to which they give priority 2) the composition of generalist committees would be more carefully monitored by the parent body, with greater effort expended to make these committees representative. b ) Amendment control rules • Whenever a committee brings a proposal to the floor, the legislative chamber must decide how it will deliberate on the proposal. In many legislatures there are predetermined standing rules governing the disposition of legislation. • In the U.S. House, however, it is customary to craft a rule specific to the legislative proposal in question. This rule, which is called an amendment control rule, regulates the amount of time devoted to debate, how that time is divided between proponents and opponents, and what amendments are in order if any • Specialist committees usually open rules • Generalist committees usually restrictive rules (even closed rule ) b ) Amendment control rules • However 1) some protection from wholesale change in the larger legislature is sometimes afforded the proposals of outlier -specialist committees 2) while, on the other hand, the chamber grants itself some ability to amend the proposals of representative -generalist committees. Why ? b ) Amendment control rules 1) Krehbiel (information and Legislative organization) argues that most legislative activity takes place in a context of great uncertainty about the relationship between proposed legislative solutions and social problems. Information about these cause -and -effect relationships helps to reduce this uncertainty. 2) The system of protection of legislative products coming from a specialized committee ( that gives “policy” advantages to its members) is an incentive system that encourages the development of legislative expertise. 3) No committee is representative of the Floor in every respect. Leadership and Coordination • In the case of legislatures, the burden of layers of institutional complexity is reflected in the need for coordination and leadership. • One cannot turn twenty committees with a hundred subcommittees loose without some means of guaranteeing that their work is supervised by the parent chamber. The crucial roles of the committee chairs Two solutions in U.S. Congress : a) Seniority system b) Majority party nomination Bureaucracy and intergovernmental relations • Budget -maximizing Bureaucrats and passive legislative sponsor ( Niskanen model) • Bilateral Bargaining • Principals and agents • Agency slippage Budget – maximizing Bureaucrats and passive legislative sponsor • William Niskanen (1971) proposed that we consider a bureau or department of government as analogous to a division of a private firm, and conceive of the bureaucrat just as we would the manager who runs that division. He stipulates that a bureau chief or department head be thought of as a maximizer of his or her budget (just as the private -sector counterpart is a maximizer of his or her division’s profits). Budget – maximizing Bureaucrats and passive legislative sponsor Arguments to support this assumption : • The bureaucrat’s own compensation is often tied to the size of his or her budget; there may be enhanced opportunities for career advancement • Large budgets mean also nonmaterial personal gratification. • Individuals try to secure as large a budget as they can in order to succeed in the missions to which they have devoted their professional lives. Budget – maximizing Bureaucrats and passive legislative sponsor Imagine a government bureau whose chief is interested in eliciting as large a budget as his legislative overseers will appropriate. • The legislature, we assume, is interested in the bureau ’s output but this preference for bureau output increases at a decreasing rate (and may even turn down after it reaches some level). • In pecuniary terms, the legislature (or more precisely, its median voter) is prepared to pay more for the first unit of output than the tenth, and decidedly more for the first than the hundred and tenth. • Niskanen assumes, quite conventionally, that production process in public bureaus displays diminishing returns to scale, so that per -unit costs are increasing. Willingness to pay curve Quantity of Bureau output Total cost (increasing at increasing rate) B=aQ -bQ 2 TC=cQ+dQ 2 Budget – maximizing Bureaucrats and passive legislative sponsor It is assumed by Niskanen that a public bureau must always cover its costs. • This means that the bureau can only agree to produce values of Q in the range in which the legislature’s willingness -to -pay curve (B)lies above the bureau’s total cost curve (TC). In Figure 13.1, this is indicated by the interval [L, H] • A budget -maximizing bureaucrat, constrained to cover production costs, will choose the value of Q* that is associated with the highest point on the B curve in the range [L, H]. In other terms he/she will maximize aQ -bQ 2 given the condition aQ -bQ 2 ≥ cQ+dQ 2 ; • However the socially optimal output is Q0 , namely when it is maximized B -TC; aQ -bQ 2 – cQ -dQ 2 Budget – maximizing Bureaucrats and passive legislative sponsor Q*= Quantity of Bureau output that maximizes the «Budget». This quantity depends on TC, something that is unknown for the legislature. The «profit» for bureaucrats are nice offices, posh furnishings, more staff of boss around, new desktop computers, resources to send managers to conferences in pleasant location Q 0= Quantity of socially optimal Bureau output that maximizes the difference between B e TC Budget – maximizing Bureaucrats and passive legislative sponsor Q*= Quantity of Bureau output that maximizes a demand costrained «Budget». Q **= Quantity of Bureau output that maximizes a cost costrained «Budget». Q0= Quantity of socially optimal Bureau output that maximizes the difference between B e TC Budget – maximizing Bureaucrats and passive legislative sponsor • Niskanen demonstrates, as the previous figures make clear, that the socially optimal level of bureaucratic production is less than either the demand -constrained level or the cost -constrained level. • Bureaus with a propensity to seek budgets as large as possible also have a propensity to produce too much . Niskanen concludes, therefore, that public bureaucracies are too big, their budgets too large, and their social output more than society wishes Variations on Niskanen model (1) • If Bureaucrats are able to camouflage their costs of production from the legislators or taxpayers who provide their budget (something Niskanen assumes), then instead of maximizing B, they seek to maximize B – TC (allowing them to spend the difference on making life on the job comfortable). • This will lead to a bureau output of Q0, precisely the socially optimal level of production identified by Niskanen . • In this instance, the bureau operates at the “right” level, but does so inefficiently. Society , so to speak, has to bribe the bureau to operate there by allowing the bureau to keep its profits. • In this case bureaus are not too large — they do not produce “too much” output — but they are too expensive. The root of the problem should reside in the secrecy that protects the bureau’s costs of doing business Variations on Niskanen model (2) • Suppose that bureaucrats are interested in “ the quiet life ”. • Maybe people seek out appointment to the civil service for its security (twenty or thirty years of steady work, no heavy lifting, a secure pension with the promise of an early and comfortable retirement). • Such people tend to be risk averse, interested mainly in a modest pace, incremental change, and a commitment to routines and standard operating procedures. ’ • It seems likely that bureaucrats defined in this way will underproduce relative either to Niskanenian bureaucrats (Q*or Q**) or to the social optimum (Q ° ). Bilateral Bargaining ( Miller and Moe ) • Miller and Moe suggest that instead of an instance of a monopoly bureau facing a passive legislative sponsor — Niskanen’s formulation — the reality in American politics is much more that of bilateral monopoly: a single customer (the legislature as represented by one of its specialized committees) bargaining with a single supplier (the bureau), each of which has information (about willingness to pay and production costs, respectively) that may or may not be known to the other party. • Miller and Moe are saying that Niskanen failed to model the legislature as an active player in the making of policy. Bilateral Bargaining ( Miller and Moe ) • Instead of the legislature revealing information from which the bureau chooses in order to maximize budget size (or slack or quiet life), Miller and Moe have the bureau forced into revealing the intimate details of its operation, allowing the legislators to make choices in pursuit of their own objectives. • Government bureaus are too big — producing too many units of their product — or too small — producing too few — will depend on a complicated set of considerations. Principals and agents • In both Niskanen’s model of a budget -maximizing bureau with a passive legislative sponsor, and Miller and Moe’s model of a budget -maximizing bureau with a proactive legislative sponsor, it is presumed that the structure of intergovernmental relations is fixed in advance • In the United States the executive bureaucracy, piece by piece, was created by legislative enactment. The Bureaucracy is created by the Congress and sustained by the Congress. • There is an alternative to both the Niskanen and Miller -Moe formulations that emphasizes the subordinate role of the bureaucracy and the superior position of elected politicians . Principals and agents • Principals retain agents to act in their interest, agents whose specialized knowledge and skills make them more effective than Principals. • In these relationships, the principal faces the problem of controlling its agents. • Some methods work ex ante ( Screening, selecting ) others ex post ( oversight , third part testimony , monitoring , self reporting) Principals and agents • Principal -agent relationships arise because there are genuine advantages for individuals to specialize in their activities and trade with one another (in – kind or for cash) rather than depend on self – sufficiency. • However an agent does not work out of the generosity of her spirit. She works for herself as well as working for you, and this can lead to potential conflicts of interest. While control mechanisms exist, they are not perfect. Principals and agents (« McNollgast ” ) • Legislative principals establish bureaucratic agents to implement the policies promulgated by Congress and the president. • The legislature wants a compliant agent and will do its best to introduce control mechanisms in the enabling legislation that creates the bureaucratic entity in the first place. • For “ McNollgast ” a piece of legislation creating a new agency or assigning some new mission to an existing agency creates a principal -agent relationship between an enacting coalition, consisting of legislators in the two houses of Congress and the president (them selves agents for constituents), and a bureaucratic entity. • The enacting coalition has coordinated around a policy objective, seeking not only its faithful implementation, but also an arrangement possessing “durability” (surviving beyond the enacting coalition) Principals and agents (« McNollgast ”) • Suppose the original enabling legislation that created the Environmental Protection Agency (EPA) required that new legislation be passed after ten years renewing its existence and mandate. • The issue facing the House, the Senate, and the president revolves around how much authority to give this agency and how much money to permit it to spend. • The House, quite conservative on environmental issues, prefers limited authority and a limited budget. • The Senate wants the agency to have wide -ranging authority but is prepared to give it only slightly more resources than the House • The president is happy to split the difference between House and Senate on the matter of authority but prefers to shower it with resources because of the election’s years. • Bureaucrats in the EPA want more authority than even the Senate is prepared to condone and more resources than even the president is willing to grant. • An enacting coalition — relevant majorities in the House and Senate (including the support of relevant committees) and the president — agrees on a policy, x , reflecting a compromise. Principals and agents (« McNollgast ”) The bureaucrats are not particularly pleased with x. However if they try to implement a policy more to their liking, y— then they risk the unified reaction of the enacting coalition. For any policy outside the triangle connecting the ideal points of the House, Senate, and president, there is some other policy they unanimously prefer to it. Principals and agents (« McNollgast ”) If, however, the EPA implemented some policy on or inside this triangle they might be able to get away with it. For any point inside the triangle, departure from it makes either the House or the Senate or the president worse off. But new legislation to punish an out -of -control agency and its existing leadership requires the simultaneous support of a House majority, a Senate majority, and the president. Principals and agents (« McNollgast ” ) The best EPA can do is implement the policy x’. It is the point closest to B, the agency’s ideal, that satisfies the political requirements to avoid political reversal. The difference between x and x’ is termed bureaucratic drift by McNollgast . Principals and agents (« McNollgast ”) • A variety of controls exist that might conceivably restrict the bureaucratic drift. a) congressional hearings in which bureaucrats may be publicly humiliated; b) annual appropriations decisions which maybe used to punish “out -of -control” bureaus ; c) watchdog agents, like the Government Accountability Office, to monitor and scrutinize the bureau’s performance., However all come after the fact and aren’t really credible threats to the agency as it strategically implements x ’ in the triangle. Principals and agents (« McNollgast ”) • The most powerful before -the -fact political weapon is the appointment process. The control of the location of B by the president and Congress (especially the Senate), through their joint powers of nomination and confirmation, especially if they can arrange for appointees who more nearly share the political consensus on policy, is a self – enforcing mechanism for assuring reliable agent performance . • A second powerful before -the -fact weapon is procedural controls. Not only the general rules and regulations of the Administrative Procedures Act. It is not uncommon for the procedures for an agency required by the enabling legislation to be tailored to suit particular circumstances . Principals and agents (« McNollgast ”) A political consensus on environmental matters (policy x in Figure 13.3) might be somewhat more sympathetic to businesses that have to bear the cost of retrofitting their plants to help maintain environmental integrity than is preferred by bureaucratic leaders in the EPA (as well as various environmental groups ). Principals and agents (« McNollgast ” ) Politicians anticipate the EPA preferences. They can write into the enabling legislation that the EPA must proceed on a case -by -case basis rather than by promulgating general rules affecting a wide range of cases . (x’’) Business interests are in a far better position to mobilize their supporters to participate in the public part of agency decision making a case at a time than are the environmental groups to mobilize their members. Environmental groups, on the other hand, would be better served if the agency could make decisions that were fewer in number but of greater impact and magnitude , since they would find it easier to mobilize for few big “battles” . Principals and agents (« McNollgast ”) • The McNollgast approach emphasizes an asymmetric advantage for politicians (in contrast to Miller -Moe and diametrically opposite to Niskanen ). • The enacting coalition can anticipate bureaucratic drift and make provisions in the enabling legislation to reduce it, if not eliminate it altogether: a) Nomination and confirmation of political appointees to lead bureaus b) S tipulation of specific administrative procedures that make it difficult for the bureau to disregard the will of the principals. Agency Slippage ( Drift ) • If the policy actually implemented by a bureaucratic agent departs from the policy formulated by his or her political principals, there is agency slippage or drift. • The three approaches ( Budget Max.,Bilateral Barg . Principal -Agent) highlight different ways this slippage manifests itself. Budgetary exploitation • Specialized bureaucratic agents are very knowledgeable about the intricate details of policy making in their jurisdiction, perhaps more so than even specialists on legislative committees. • For bureaucratic agents these details are the core of their very existence. This produces an effective informational advantage for the bureaucrat that can use it for tangible and non tangible benefits. Bureaucratic drift • When authority and resources if bureacrats are used to pursue other policy objectives, then we have bureaucratic drift. • This policy drift is protected from after -the -fact retribution as long as the bureaucrat is crafty enough to protect against unanimous opposition from the components of the enacting coalition. Bureaucratic capture • Bureaucratic drift can be advantageous for some political actors ( against the interests of some other political actors) • If we assume that there are 435 ideal points scattered about H and 100 ideal points scattered about S, then it is very likely that a number of representatives and senators sympathize more with the bureaucrat’s preferences than they do with either the political preference arrived at in their legislative chamber (H and S, respectively) or the policy compromise arrived at among House , Senate, and president (x). • It is likely that the committees with jurisdiction over this bureaucracy and policy area are populated with legislators who are more likely to share the bureaucrat’s preferences for larger bureaucratic output than the preferences of their political colleagues for more modest levels (i.e., an outlier committee). • These committees are in a position to protect a drifting bureaucracy (through their gatekeeping and other agenda -control powers). Behind these legislators are the interest groups and geographic constituencies whose well -being the legislators pursue. The traditional political science literature makes frequent allusion to “ cozy little triangles,” involving legislators on key committees, bureaucrats, and interest groups, as the dynamos behind policy implementation . Coalitional drift • The politicians not only want the legislative deals they strike to be faithfully implemented , they want those deals to endure. • A victory today, even one implemented in a favorable manner by the bureaucracy, may be undone tomorrow. What is to be done? • The legislative structure somehow leans against undoing the handiwork of an enacting coalition. But even this structure is “ unstable ” . Coalitional drift • Legislatively formulated and bureaucratically implemented output is subject to coalitional drift. Today’s enacting coalition may no longer be in power tomorrow. • To prevent shifting coalitional patterns among politicians to endanger carefully fashioned policies, one thing the legislature might do is insulate the bureaucracy and its implementation activities from legislative interventions. If an enacting coalition makes it difficult for its own members to intervene in implementation , then it also makes it difficult for enemies of the policy to disrupt the flow of bureau output at a later date. • This political insulation can be provided by giving implementing agencies long lives, their political heads long terms of office and wide -ranging administrative authority, other political appointees overlapping terms of office, and security to their sources of revenue • However the civil servants and political appointees of bureaus insulated from political overseers are empowered to pursue independent courses of action. Protection from coalitional drift comes at the price of an increased potential for bureaucratic drift . It is one of the great trade -offs in the field of intergovernmental relations. Leadership Non excludable conceptualizations • Leader as agent • Leader as agenda setter • Leader as entrepreneur • Reputation and leadership Leadership as agent • Leader is usually thought to be proactive , whereas followers are reactive . It is a partial truth . • Leaders must secure the support of the followers . Compensation of leadership is often performance based . • The problem for followers, a special instance of the problem facing principals more generally, is that the specialized skills that make a particular agent attractive as leader are also the skills by which that agent can exploit his or her followers for “private” gain. The “who will guard the guardians” ? • Before and after the fact control mechanisms: reputation. • Leaders are « prisoners » of their intentions and to the extent the latter depend upon follower support the master become servant and the servant master. Leader as agenda setter • Leader can be thought as the person in charge of the group’s agenda . • The leaders of groups in general are in the condition to trade off satisfying the preferences of their followers in order to secure some of their own objectives. • Refining the group’s agenda is one of the standard activities that a group delegates to a leader (or an agenda committee) In doing so, there is ordinarily an awareness in the group that a “pound of flesh” will be exacted by the leader -agent. However politically savvy choices, by the group in selecting the leader and by the leader in exercising her agenda authority, place limits on this “compensation ” • Examples of Speaker of House and legislative committees : when the dimensionality of the policy space is crucial Leader as entrepreneur • Entrepreneurs are leaders who, as often by self – appointment as by selection, perform necessary services to enable a group to accomplish some collective purpose. • An entrepreneur may be seen as an agent who chooses (or creates ) a principal , rather than the more typical arrangement in which a principal hires an agent . • This conception of leadership emphasizes the artistry of seizing opportunities . Entrepreneurship emphasizes not only those actions that occur within a fixed structure but also the actions that create or transform structure. • Political entrepreneurship is not for risk -averse. Reputation and leadership ( Calvert ) • Every politician leader faces the problem of how most effectively to sanction uncooperative behaviour by followers. Sanctions may be costly, but for the most successful leaders the mere threat of such sanctions is usually sufficient . • Calvert assumes that in prosecuting the group’s agenda of objectives the leader must maintain a minimal level of group support. There are always “alternative ” leaders .. • Sanctions by the leader against group members are costly, because they are administered against members whose support is desired by the leader. The leader must assiduously balance the benefits, in terms of group success, against the costs, in terms of threats to her leadership , of employing sanctions. • “Any given punishment increases by a small amount the probability that the leader will face a full -scale rebellion, deposing him or impairing his ability to lead in the future. In this sense, the leader faces a kind of budget constraint on his ability to impose sanctions.” Reputation and leadership ( Calvert ) • T he precise nature of the benefits to the leader (in terms both of achieving group goals and obtaining private “compensation”), and of her costs of employing sanctions and bearing the associated risks, is assumed to be private information . • By cultivating this uncertainty , the leader is able to make followers believe that punishment is possible more often than under conditions of complete information. • Even in those cases where punishment is “too costly ”, the leader may nevertheless engage in sanctioning behaviour in order to mislead followers into believing that her costs are lower than they actually are. Reputation and leadership (Calvert) • If the follower (Mr. t) supports the leader’s proposal the leader earns “credit” toward reelection or reappointment, a (where a> 1), while follower t receives whatever payoff he associates with achieving the group’s common objective (normalized to 0 without affecting the argument). • If the follower rebels, then the payoffs depend on what the leader does in response. a) If she acquiesces in the rebellion, allowing follower t to oppose her proposal , then she gets less than what she would have if the follower had obeyed (again normalized to 0) and the follower gets more (some positive level b < 1 rather than 0 if he did not rebel ). b) If the follower’s rebellion is punished, then the leaders bears a cost, -x and the follower gets b -1 , (< 0 as b<1) Reputation and leadership (Calvert) • When follower’s rebellion is punished follower t does not know the value of -x ; Rather , follower t has beliefs, namely that -x is costly ( equal to -1 ) with probability w, and costless (equal to 0) with probability 1 — w. • The leader on the contrary leader knows the value of -x in advance and she is always fully informed of the consequences of her actions. Reputation and leadership (Calvert) • The followers are capable of learning. Their belief , w, that punishment is costly changes as they observe the leader in action. • Thus , leaders not only have asymmetric information about the costliness of administering punishments , they also are able to influence subsequent beliefs of followers about this information . If, for instance, she is able to convince followers that punishments are mainly costless to her , then she is likely to induce more follower loyalty to her proposals (and thus not even have to use punishments). Reputation and leadership (Calvert) • The game is repeated twice. The leader faces the possible rebellion first of follower t1 and then of follower t2 . • First , if -xt actually is 0 (costless punishment ), then the leader should always punish rebellion. • Second , if -x2 = -1 (i.e., it is costly for the leader to punish the second follower ), then she should never punish t2; there would be no purpose served in doing so (since the game ends) and thus no point in bearing the cost of punishing. • What the leader should do if -x1 = -1? Calvert’s model shows that the leader’s response to rebellion by t1 when it is costly to punish depends, in very specific ways, on the beliefs followers hold ( and the way they update these beliefs) about the leader’s costs of punishment relative to the benefits they would secure from rebelling. • Calvert identifies very precisely when t1 should rebel, when a leader should punish that rebellion for certain , when a leader should “flip a coin” (or randomize in some other way) to determine whether to punish this particular transgression, and whether, conditional on all this, t2 should rebel (for certain or with some probability ). Reputation and leadership ( Calvert ) • Since a successful leader cannot be constantly monitoring and punishing followers, it is necessary to establish in the followers a habit of obedience, a rule of thumb that the leader’s wishes are to be followed. • Habits and rules of thumb do not occur in a world of perfect and costless information. Thus the presence of uncertainty, and the artful manipulation of it, are crucial for successful leadership. Somewhat counterintuitively, this holds true regardless of whether the leader is a ruthless dictator or a benevolent provider of collective action . Courts and Judges • Courts and Judges are an ovelooked topic . It is difficult to put forward sensible hypothesis about their «utility function » • They do not have to be reelected and rarely their compensation depends on their performance. • Courts as dispute resolvers, as coordinators, and as interpreters of rules . Dispute Resolution • The judge is responsible for managing fact -finding and judgment phases of dispute resolution . • A large part of the daily life of a judge involves the provision of an independent, experienced look at the facts, an assessment of whether the dispute involves a violation of a private agreement or a public law (or both), and finally a judgment namely a determination of which party (if either) is liable and, if so, what compensation is in order (to the private party victimized and, if judged a criminal activity, to the larger public). Coordination • Dispute resolution occurs after the fact — that is, after a dispute has taken place. In a manner of speaking, it represents a failure of the legal system, since one function of law and its judicial institutions is to discourage such disputes in the first instance. • Courts and judges are before -the -fact coordination mechanisms inasmuch as the anticipation of what happens once their services are called upon allows private parties to form rational expectations and thereby coordinate their actions in advance of possible disputes. • The court system is as important for what it doesn’t do as for what it does. The system of courts and law coordinates private behaviour by providing incentives and disincentives for specific actions. To the extent that these work, there are fewer disputes to resolve and thus less after – the -fact dispute resolution for courts and judges to engage in. • What makes the incentives and disincentives work is their power (are the rewards and penalties big or small?), their clarity, and the consistency with which judges administer them. Rule interpretation • Judges are not entirely free agents. In matching the facts of a specific case to judicial Principles and statutory guidelines, judges must engage in interpretive activity. • Interpreting the rules is the single most important activity in which higher Courts engage. This is because the court system is hierarchical in the sense that judgments by higher courts constrain the discretion of judges in lower courts. • At the highest levels, courts and judges engage not only in statutory interpretation, but constitutional interpretation as well . (when the “evaluation parameter is the Constitution) • Statutory and constitutional interpretations have a precedential authority over subsequent deliberation (and, in turn, are themselves influenced by earlier interpretations). Rule interpretation • The interpretive activity of judges and justices are themselves subject to review. • Statutory interpretation , even that conducted by the highest court is exposed to legislative review. If Congress is unhappy with a specific statutory interpretation then it may amend the legislation so as explicitly to reverse the court ruling. • If the court makes a constitutional ruling, Congress cannot then abrogate that ruling through new legislation but Congress can commence the process of Constitutional amendment , thereby effectively reversing judicial interpretations with which it disagrees. Posner’s type of « judicial behaviour » • Non profit analogy • Voting analogy • The spectator and game player analogies Non profit analogy • Judges’ salaries and other benefits are fixed independent of effort, quality of work, or any other performance based standard. They are somehow salaried managers of a nonprofit system. • Individual judges, like other nonprofit professionals, rationalize the energy and commitment they invest in their jobs by their desire for popularity, prestige, and reputation. They value these things partly for their own sake but they also value them because they conceive of these things as necessary attributes for career advancement, however remote the latter possibility might be. Voting analogy • Much of what a judge does may be conceived of as voting. Rendering judgment is, at least in part, the casting of a vote. Judges, however, are more “crucial” than other political actor as their participation takes place in a small -group setting in which the chances of being pivotal to an outcome are much more likely. • Since judges may not be certain exactly when they have a chance to make or break a majority as opposed to being a peripheral vote, they may have an incentive to try to get their decisions “right” in terms of their beliefs about the proper disposition of cases. • Judges “vote” not only on who should win a given case but, through the opinions they draft, why one side or the other should win. Judges vote not only with ballots but also with ideas — ideas about the facts, about judicial principles, about legal reasoning, and about moral values . It is through the opinions they draft, rather than the ballots they cast, that judges may influence a wider collection of interests, since these opinions serve to constrain lower -court judges in similar cases in the future Posner’s classification drawback • It is to rich and it tries to capture the judges in general • According to Shepsle it is better to focus on those judges who are “players” in the larger political game precisely because they do have ambition, both for their careers and their ideas about the law. Just as the models of elected politicians are most apt for representatives, senators, presidents, and those who want to become one of these. Legislators in Robes • We may think of judges as having policy preferences just like legislators. To the degree that higher courts do not merely resolve disputes between the participants in a case but, more significantly, shape the legal context in which millions of private citizens interact and in which thousands of public officials exercise power, these courts are critical in the formation and implementation of public policy. • It may be hypothesized that judges with policy preferences treat each case as though it might have impact, either directly on national politics or indirectly through its effects on how Constitutional or statutory law is subsequently interpreted. Legislators in Robes • Courts and judges are “players” in the policy game because of the separation of powers. • The legislative branch formulates policy (defined constitutionally and institutionally by a legislative process); the executive branch implements policy (according to well -defined administrative procedures, and subject to initial approval by the president or the legislative override of his veto); and that the courts, when asked, rule on the faithfulness of the legislated and executed policy either to the substance of the statute or to the constitution itself. • The courts, may strike down an administrative action either a) because it exceeds the authority granted in the relevant statute (statutory rationale) or b) because the statute itself exceeds the authority granted the legislature by the Constitution (constitutional rationale). • If a decisive coalition in the legislature is unhappy with this judicial action, it may either recraft the legislation (if the rationale for striking it down was statutory) or initiate a constitutional amendment (if the rationale for originally striking it down was constitutional). Legislators in Robes : a Model • Imagine that the policy options in any given setting may adequately be represented by a one -dimensional interval, [0, 100], over which all actors — legislative, executive, and judicial — have single -peaked preferences. Thus, the nine Supreme Court Xji • 100 Senators X Si and 435 representatives XHi have ideal points on this interval as well as the president’s executive agent by XA. (complexities associated with the presidential veto and veto overrides are ignored .) • Those with ideals near 0 may be thought of as on the extreme liberal end of opinion, while those with ideals near 100 are at the extreme conservative end. Legislators in Robes : a Model • As there is one dimension ( single peaked preferences) we can consider the median voter for H( ouse ) , S( enate ) and Supreme Court (J). We assume also the existence of the Status quo XQ • In this simple setting, then, we can see the impact of judicial oversight by focusing on XQ , XJ, xH , XS , and xA . These five “parameters” will determine the outcome of policy in any specific application. • We have to know the sequence in which behavior unfolds. Legislators in Robes : a Model Legislators in Robes : a Model XJ • XA implement XH not to be overuled by the Chambers. Such a policy is confirmed by XJ. XJ • XA implement XH not to be overuled by the Chambers. Such a policy is confirmed by XJ. Legislators in Robes : a Model XJ • No matter what the bureaucratic choice, the court can strike down the action and name its ideal, J= Xj , as the new policy. Since this lies in the legislative equilibrium set (bicameral core), the legislature will not react with new legislation. • If the bureaucratic agent doesn’t like being reversed, then she will anticipate the court’s move in advance and simply name A = Xj XJ Legislators in Robes : a Model XJ • Any action taken by the bureaucratic agent will be struck down by the court; it will declare the new policy as J = XS. This is the best the median justice could hope for, since any policy closer to Xj will trigger a legislative response. • a fully anticipatory response from the bureaucrat would have her declare A = XS, with the court allowing that policy to stand. • Even if the bureaucrat were, for some odd reason, to implement at the ideal policy of the median justice, the court would still strike it down and declare J = XS. If the court allowed this nominally more desirable policy to stand, then the legislature would get involved with corrective legislation, ultimately producing a policy less desirable to the court median than XS. Cabinet Government and Parliamentary Democracy • In the Parliamentary Democracy the government plays a crucial role as agenda setter. The large majority of parliamentary governments are coalition governments. • Coalition Theories are one of the most advanced section in the Positive Political Theory. • In the literature two important “spatial” theories consider the parties as policy seekers. 1)A cooperative game theory, proposed by Laver & Schofield that considers the government as a “policy platform” 2) A non cooperative game theory, proposed by Laver & Shepsle that considers the government as a “combination of ministries” Policy – seeking models in one dimension In these models the political actors are motivated also by the policy distance between the expected policies of the government coalitions and their policy platforms. de Swann : Cooperative game -unidimensional . The policy positions are ordered along one dimension. A political actor will prefer the winning coalition whose policy position is the nearest to its preferred policy position. In only one dimension any winning coalitions (in a majority voting game) must include the party where is located the median voter. This party is called the Core Party , it cannot be excluded by the winning coalition and it controls its formation. The coalitions in the Core (or the winning coalitions) can be more than one. According that de Swann the Core Party should prefer the coalition that minimize the difference in terms of seats among the actors on the left and on the right of the Core Party in the coalition or in other terms the Core Party should prefer balanced coalitions . Policy – seeking models L (45) C (15) R (40) Seats=100 L (55) CL (20) CR (10) R (15) L (25) CL (15) C (8) CR (5) R (47) Core Party a) b) c) a) According to de Swann L,C,R is better for C than C,R or C,L as |45 -40|<|0 -40|<|0 – 45| b) Of course the best one is L c) L,CL,C,CR,R is better for CR as |48 -47| < any other difference. Policy – seeking models L (45) C (15) R (40) Seats=100 L (55) CL (20) CR (10) R (15) L (25) CL (15) C (8) CR (5) R (47) a) b) c) Def. Pareto Set: the set of points in the policy space that: a) For any point not in the set there is in the set a point that is preferred by all political actors taken in consideration. b) Given a point in the set none else is considered better by all political actors c) For any winning coalition the Pareto set is given by the line connecting the political actors members of the coalition Policy – seeking models L (45) C (15) R (40) Seats=100 L (55) CL (20) CR (10) R (15) L (25) CL (15) C (8) CR (5) R (47) a) b) c) The Core Party is the party present in all Pareto Sets of all winning coalition. It always exists in a unidimensional world but.. Policy -seeking models in a bidimensional policy space (Laver & Schofield ) A (20) C (20) B (20) D (40) Considering to simplify the analysis, only the minimal winning coalitions, in this policy space no Party is “member” of all Pareto Sets of all coalition. There is always a majority that can defeat any party platform. A (20) C (20) B (20) D (40) In this situation there is a a party that is always included in all Pareto sets of all winning coalition. It is D. No majority can defeat the D’s political platform. A (20) C (20) B (20) D (40) However usually a centrally located party is a Core Party if it is quite big.Otherwise no core party exists. C is not a Core party as is not in the Pareto Set of the coalitions AD and DB. Even when a small party centrally located is a Core party such a equilibrium is structurally unstable. …. ARP:14 KVP:33 VVD:10 PvdA:33 CHU:10 Traditionalism Modernization Left Right A structurally stable core at the KVP position The election of June 1952 in Netherlands ARP:14 KVP:33 VVD:10 PvdA:33 CHU:10 Traditionalism Modernization Left Right A structurally stable core at the KVP position The election of June 1952 in Netherlands PvdA:33 KVP:33 CHU:10 VVD:10 Traditionalism Modernization Left Right A structurally unstable core at the ARP position ARP:14 PvdA:33 KVP:33 Traditionalism Modernization Left Right A structurally unstable core at the ARP position: after a small change in its policy position, ARP is not a Core Party any more as the Pareto set PdvA, KVP,VVD does not include it. ARP:14 CHU:10 VVD:10 PvdA:33 KVP:33 Traditionalism Modernization Left Right However even if it does not exist a Core Party, the area of the disequilibrium is delimited by the intersections of the median lines. The so called Cycle Set. Core+Cycle set= Heart ARP:14 CHU:10 VVD:10 median median median • Laver -Shepsle theory is a theory about government formation , is not a theory about “ platform ” bargaining . • Laver -Shepsle approach models a real decision making process , considers an initial status quo: it belongs to non cooperative game theory . Policy -seeking models in a bidimensional policy space (Laver -Shepsle) R R R R R R R R R R R R R P1 sel . Proposes x1 Proposes x2 Proposes xi Proposes xn P2 sel Pi sel Pn sel x1 Vetoed? x2 Vetoed? xi Vetoed? xn Vetoed? yes yes yes yes no no no no x1 installed? x2 installed? xi installed? xn installed? yes no no yes no yes no yes x1 new SQ x2 new SQ xi new SQ xn new SQ Cabinet Government and Parliamentary Democracy • In the Parliamentary democracy there is a division and specialization of labor at two different levels. 1) The legislature selects the executive in the first place; keeps it in place or replaces it with a different one; and considers various pieces of legislation, the most important of which is the annual budget 2) In a manner quite parallel to the arrangements involving committees in the U.S. Congress, each ministry of state has jurisdiction over specified dimensions of public policy, called a ministerial portfolio. In this domain the minister and his or her senior civil servants haveconsiderable discretion to interpret statutory authority and implement public policy. Cabinet Government and Parliamentary Democracy • Another way to think about this arrangement is as a chain of principal – agent relationships. 1) Parliament as principal delegates executive authority to a collective agent, the cabinet. 2) The cabinet as principal, in turn, delegates discretionary authority in various policy jurisdictions to its agents, namely particular cabinet ministers. To control its agents, and coordinate their activities, the cabinet employs various before -the -fact and after -the -fact mechanisms; One member of the cabinet plays the role of policeman and maestro — _the prime minister. Likewise, to control its agent, parliament exercises before -the -fact and after – the fact authority. Parliament votes a cabinet into office in the first place and, if it is unhappy with cabinet performance after the fact, may vote it out of office and replace it with an alternative cabinet. Government formation process • When either a new parliament convenes, an old government is defeated in a confidence procedure, or the old government resigns for whatever reason, the head of state appoints a leading politician, called a formateur , to try to assemble a new government. • The formateur negotiat es with other parties in assembling a distribution of ministerial portfolios to political parties — a government — that a parliamentary majority is prepared to support. • The parties receiving a ministerial. portfolio are said to be the government parties. Those not included in government are the opposition parties or external supporters. • Once in place, the new government governs. Each minister attends to day – to -day administration with senior civil servants in his or her ministry and, together with other ministers, meets regularly in cabinet to formulate and coordinate overall governmental policy. This will entail not only the implementation of existing policy but also the formulation of new policy. • The government will need to have firm control of parliamentary politics, setting parliament’s agenda and making sure it follows the line. There is a political tension between government and parliament. The government governs subject to keeping the confidence of parliament. Parliamentary government forms Government formation ( German example last 25 years ) • Main parties in the Bundestag: 1) Christian Democrats (CD) 2) Social Democrats (SD), 3) Free Democrats (FD), 4) the Greens (G) • No single party has a majority of seats • CD usually the biggest party. It can have a majority of seats with an alliance with any other party. • Any majority without CD requires all remaining parties coalesce . Government formation ( German example last 25 years ) • Thus, the set of possible majority coalitions ( minimal winning ) in the Bundestag during this period included 1) CD -SD 2) CD -FD 3) CD -G 4) SD -FD -G However a government (or cabinet) is not only a majority coalition. It is a distribution of ministerial portfolios among the parties (more accurately, among senior party politicians). Suppose there were only two key cabinet ministries in the government — Finance (F) and Foreign Affairs (FA). Government formation ( German example last 25 years ) • Two party motivations 1) Office seeking ( we assume for sake of argument that all parties care more about financial affairs ) 2) Policy seeking Government formation (Office seeking ) Government formation (Office seeking ) • There is a substantial amount of indifference in party preferences, • The only thing parties consider is whether they are in government Government formation (Policy seeking ) • The horizontal dimension falls into the jurisdiction of the Finance Ministry and the vertical dimension is the responsibility of the Foreign Affairs Ministry. • All parties in the Bundestag know that whichever party secures each of the ministries will push its ideal policy in that jurisdiction Government formation (Policy seeking ) • A party that obtains both portfolios, therefore will implement its ideal point. • A party that obtains one portfolio will implement the component of its ideal point that applies to the jurisdiction of that portfolio. • If the CDs and FDs form a multiparty majority government , with the CDs getting the Finance portfolio and the FDs the Foreign Affairs portfolio, then this government will implement the policies associated with the point identified by the intersection of the horizontal line through the FD ideal and the vertical line through the CD ideal , that is , the CD -FD government will implement the CD’s ideal economic policy and the FD’s ideal foreign policy . Government formation (Policy seeking ) Government formation (Policy seeking ) • While there are policies that Bundestag’s majority prefers, there is no government (alternative allocation of portfolios as reflected in the various lattice points) that it prefers. The government CD -FS has an empty winset lattice Government formation (Policy seeking ) • A very complicated scenario. CD is a minority government that maybe cannot substituted because CD is a strong party….