MIDTERM ANSWER KEY GAME THEORY, ECON 395

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1 MIDTERM ANSWER KEY GAME THEORY, ECON 95 SPRING, 006 PROFESSOR A. JOSEPH GUSE () There are positions available with wages w and w. Greta and Mary each simultaneously apply to one of them. If they apply to the same job they will have to split the offered wage in half between them. If they apply to different jobs they will get to keep the entire wage offered for their respective positions. For example, if Greta applies to position and Mary applies to position, Greta s payoff will be w, while Mary s will be w. However, if they both applied to position, then both of their payoffs would be w. (a) Draw a normal-form payoff matrix for this game. Mary Greta J J w w Job(J) w w w w Job(J) w w Where in each cell: u G u M (b) Describe any pure-strategy Nash equilibria, if they exist. If not, explain why they do not exist. ANSWER. Suppose we have condition Γ = {w < w and w < w }. In this case there are exactly PSNE represented by the strategy profiles (s G, s M ) = (J, J) and (s G, s M ) = (J, J). To see why these are NE consider the first equilibrium where Greta takes Job and Mary takes job. In this outcome Greta gets w and Mary gets w. If Greta deviates choosing J, she would split Mary s wage ending up with w instead. Since we are assuming the conditions in Γ, Greta obviously would rather not deviate. Similarly from Mary. The second equilibrium is justified by parallel arguments. See part (d), for a description of the equilibria in the cases where condition Γ does not hold. (c) Find and describe any mixed-strategy Nash equilibria. ANSWER. Again assume the conditions in Γ defined above hold. Let r be the probability that Greta will choose J. Since Greta has only two pure-strategies, r completely describes her strategy. Similarly let m be the probablility that Mary chooses

2 SPRING, 006 PROFESSOR A. JOSEPH GUSE J. Now let s find each players best response correspondences. First Greta s, r (m). To find r (m), we need to consider Greta s utility of playing r when Mary plays m. u G (r, m) = ( r)( m) w + ( r)mw + r( m)w + rm w differentiating this w.r.t r we get u G (r, m) r = ( m) w mw + ( m)w + m w = ( + m) w + w m w Notice how the derivative has no instances of r left in any of its terms. (This make employing our usual trick of setting the derivate equal to zero and finding some optimal r problematic). However the derivate can still tell us a lot. If it is positive, it means that Greta s expected utility is increasing in the probability that she choose J. If we find this to be the case for certain values of m, then it means that Greta should make r as high as possible, chooseing J with probability. A negative value would mean that she should choose J with probability, while it being equal to zero, would mean that she is indifferent. So let s ask when u G (r, m) r ( + m) w + w m w 0 w + w m( w + w ) w w m(w + w ) 0 w w (w + w ) m Hence by the argument made above, if m is strictly less than w w (w +w ), J is the only best response for Greta. If m is strictly greater than this amount, J is Greta s best response and if m equals w w (w +w ), then Greta may play a mixed strategy. Summarizing, we have 0 if m > w w (w +w ) r (m) = if m < w w (w +w ) [0, ] if m = w w (w +w ) An appeal to symmetry gives us a similar result for Mary s best response function

3 MIDTERM ANSWER KEY GAME THEORY, ECON 95 0 if r > w w (w +w ) m (r) = if r < w w (w +w ) [0, ] if r = w w (w +w ) Hence there is one mixed strategy equilibrium at (r, m) = ( w w (w +w ), w w (w +w )). Figure shows a picture of the two players BR correspondences. In it we see that they intersect at three points. The interior intersection represents the mixed strategy equilibrium. The two intersections at the corners represent the two PSNE described in the previous part. PSNE Mary s BR, m (r) Mixed-Strategy NE m =Prob(sM = J) w w (w +w ) 0 Greta s BR, r (m) PSNE w 0 w (w +w ) r =Prob(s G = J) Figure. Greta and Mary s Best Response Correspondences. (d) Does the number of Nash Equilibria in this game (pure and mixed) depends on the values of w and w. Explain. ANSWER. Yes, indeed. If condition Γ = {w < w and w < w } holds, then there are PSNE and Mixed Strategy NE as described in parts (b) and (c). There are 4 ways to violate Γ. We could have w = w, w > w, w = w, or w > w. I will discuss the first two cases. The last two cases are perfectly analagous to the first two. Case: w = w. Note what happens to the switching point in the players best response functions in this case. w w (w +w ) = 0. Hence the mixed strategy equilbrium in Figure becomes a PSNE in which both players choose J. Note that the other PSNE remain. So the tally in this case is PSNE and 0 Mixed. Case: w > w. Note what happens to the switching point in the players best response functions in this case. w w (w +w ) < 0. In other words such parameter

4 4 SPRING, 006 PROFESSOR A. JOSEPH GUSE values for w and w would push the Mixed-strategy eqm right out of existence. Moreover the PSNE involving the players taking different jobs disappear as well and we are left with just (J, J). So the tally in this case is PSNE and 0 Mixed.

5 MIDTERM ANSWER KEY GAME THEORY, ECON 95 5 () Three players {A, B, C}, must decide unanimously how to divide a dollar. The bargaining proceeds as follows. At time, t = 0, Player A proposes a division (x AA, x AB, x AC ) where each x ij 0 denotes how much of the dollar i is offering to give to j such that x AA +x AB +x AC. If players B and C both approve, the dollar is allocated according to A s proposal and the game ends. If not, at time t =, B proposes (x BA, x BB, x BC ) which is subject to the same summation and unanimous approval rules. If B s proposal fails, then at time t =, C proposes (x CA, x CB, x CC ) which again must be approved unanimously and sum to or less. If C s proposal fails, the game ends with all players receiving 0. (a) Are there any players with strictly dominated strategies in this game? If so, identify one such player and the at least on such strictly dominated strategy. ANSWER. No, there are no strictly dominated strategies. Most urges to answer otherwise to this question stem either from a imprecise notion of what a strategy is or from imprecise notion of what it mean for a strategy to be strictly dominated. Remember a pure strategy must specify which action a player will take whenever it might be that player s move - even if it seems very unlikely that the player will ever get to make the move. In this game a strategy for a player is a proposed allocation whenever it is that player s turn to make a proposal AND a plan for how to vote on every possible proposal the other players will make. Moreover the proposal and voting plan can be contigent on the history of observed action up to that point. For example player B s strategy could be something like, In the first round, Vote YES on A s proposal if A proposes at least 5 cents for me and at least 0 cents for player C or at least 50 cents for me no matter how much of the remain is split between C and A. In the second round propose 50 cents for A, 40 cents for me and 0 cents for C unless A proposed less than 5 cents for me in the first round, in which case propose 0 cents for A, 50 cents for me and 40 cents for C. In the last round for YES for anything unless the voting outcome in the nd round involved YES from A and NO from me and the proposal involves giving Player A exactly 4 cents. This strategy is not intended to make sense or be allowed in equilibrium, just to give an idea of a what a strategy can and may entail. A strategy can be very complicated and the space of strategies is very large. In order to be strictly dominated a stragey has to do strictly worse than some other strategy no matter what strategies the other players adopt. To see why there are no strictly dominated strategies for player A, consider a combination of Player B and C strategies where they always vote NO on any proposal which is not their own. Since unanimity is required for passage, this would mean that no matter what Player A did, no proposal would ever pass and all players (including A) would get 0. Therefore, against this combination of strategies any strategy A plays is a best response which means nothing is strictly dominated. Using the same argument,

6 6 SPRING, 006 PROFESSOR A. JOSEPH GUSE we come to the same conclusion about the existence of dominated strategies for player B, by considering a strategic environment in which A and C vote NO on everything. Similarly for player C. Hence no player has an strictly dominated strategies. (b) Assume that all the players share a common discount factor δ, describe the unique backwards induction solution to this game. Be sure to describe how what each player s proposal would be if they got a chance to propose and what the players expected utilities of playing the game are. ANSWER. To find the backward induction solution, we begin our analysis at the end of the game in the voting stage of the final round. In this round it is player C who proposes. Suppose player C proposed any proposal involving some positive shares for B and A. Voting YES on C s proposal, no matter how small their shares are is a weakly dominant behaviour in the last round. This is due to the fact that if players A and B do not agree to C s proposal they will receive 0. Player C, anticipating this, proposes to give players A and B very small shares - practically zero - keeping very nearly for herself. Therefore if the game continues to round, players should expect to receive (x CA, x CB, x CC ) = (0, 0, ). Now consider the second to last round in which player B is proposing. C will not accept anything less than δ from B, since she anticipates getting in the rd round. B will therefore propose (0, δ, δ). Note that B proposes giving 0 to A, since A anticipates 0 in the last round. Now consider the first round in which player A is proposing. Since C anticipates δ from B in the next round, A must give C at leat δ in this round. Since B anticipates δ from himself in the next round, A must give B at least δ( δ) in this round. This leaves at most δ for player A. Hence, A s proposal will be ( δ, δ δ, δ ). There are other equilibria in this subgame involving weakly dominated strategies. Consider for example player B only agreeing to C s proposal when it provides at least 0 cents to each of B and C. If A adopts the same voting strategy, this is an equilibrium. The key to seeing why this an equilibrium is to note that in the case the C proposes less than 0 cents for either A or B, neither A nor B would be a pivotal voter: Changing one s vote to YES (because you d rather get something less than 0 cents than nothing) won t change the outcome as long as the other voter sticks to her guns. This outline of play describes the only backward induction solution not involving weakly dominated strategies, but it is not the unique backward induction solution as the problem claims. As noted above, there are other equilbria in the final subgame besides the one presumed in this outline. Similarly an argument can made that other equilibria involving dominated strategies exist in subgames further up the game tree. For example, consider a strategy profile in which the two respondants (the players who are not proposing) in each round only vote YES as long as both respondants get at least. Proposers would then be forced to propose (,, ) in each round. This constitutes an equilibrium in every subgame. In cases where proposals include less than for a respondant, no voter is pivotal and therefore no voter has any (strict) incentive to change their vote to YES. In cases where both respondants are getting at least each, both respondants are pivotal, but have no incentive to change their votes to NO, since they will just postpone getting until the following round (or 0 if its the last round).

7 MIDTERM ANSWER KEY GAME THEORY, ECON 95 7 (c) Suppose that instead of a deterministic order of proposals, each player is selected to make a proposal in each round with probability. (That is, at each time t {0,, }, player A may propose with probability, player B with probability, and player A with probability.) How does this change the equilibrium? ANSWER. My answer focuses again on the subgame perfect (backward induction) solution not involving weakly dominated strategies. In the last round, the proposer can keep very nearly all the dollar to herself. In the second to last round, each player anticipates being the proposer in the last round with probability. Therefore the respondants in the second-to-last round require at least δ to approve and the second-to-last round prosposal is ( δ, δ, δ ) where the first share goes to the proposer (whoever that may be). In the first round, each respondant anticipates being the propose in the following round with probability and being one the responders with probability. Therefore the utility a responder gets from voting NO is always [ ( δ δ ) + ( )] δ which reduces to δ. But this is the same demand respondants have in the secondto-last round, which makes the first round proposal the same as the second-to-last: ( δ, δ, δ ). When the game starts (before the first proposer selection is made) all players expect a payoff equal to exactly. This can be confirmed by multiplying times the payoff a player will get in each role or by noting that since the whole dollar is distributed in equilibrium and each player has a equal chance of getting any of the shares, the expected payoff must be for each player. For any anticpated shares (x, y, z) each going to player i with probability, i s expected payoff must be x + y + z = (x + y + z) =.