# Rationality Insertion behavior exam

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 ?