Rationality Institon behaviour exam ( needs Political science major) I have health problems and I also have another exam I dont have time to study for it
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 ? Rationality Institon behaviour exam ( needs Political science major) I have health problems and I also have another exam I dont have time to study for it 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".