# Political Analysis

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