Problem Set 1 1. Games in Strategic Form: Do questions 1-4 and ...

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L C RU 3,2 1,1 3,3M 5,3 6,6 2,3D 2,2 -1,5 5,4Is the solution a Nash equilibrium? Try solving the same game using IDDS; do you get adifferent prediction?(c). For prisoners’ dilemma, what are the rationalizable strategies? For battle of thesexes and matching pennies, show that all strategies for all players are rationalizable.(d). Show that no strictly dominated strategy is rationalizable, using your definition from i.Are weakly dominated strategies rationalizable? Show that dominant strategies are alwaysrationalizable. Are rationalizable strategies always dominant?(e). Show that in a pure-strategy Nash equilibrium, both players are using rationalizablestrategies.(a). A strategy s ∗ i is rationalizable if there exists a profile of opponent strategies s −ifor which u i (s ∗ i ,s −i) ≥ u i (s ′ i ,s −i) for any other s ′ i that i could pick.(b). Iterated deletion of unrationalizable strategies: (0) Pick a player. If that playerhas strategies that are not strictly/weakly rationalizable, remove it from the game.Do this for all players. (1) If any strategies were removed in step 0, repeat step 0.Otherwise, stop.The strategy U is not rationalizable for the row player, since it is not a best responseto anything. The strategy L is not rationalizable for the column player, since it is nota best response to anything.With U and L removed, R is no longer a best response to anything the row playermight use, so it is unrationalizable.With R removed, D is not rationalizable for the row player.This leaves (M,C) as the unique profile that survived iterated deletion of unrationalizablestrategies.IDDS fails to remove anything and does not provide a unique prediction.(c). The rationalizable strategy in prisoners’ dilemma is C, since it is a best responseto anything an opponent might play. In battle of the sexes, both strategies arerationalizable since F is a best response to F and B is a best response to B, so eachstrategy is a best response to something an opponent might play. In matching pennies,H is a best response for the row player to H and T is a best response to T, so bothof the row player’s strategies are rationalizable. For the column player, H is a bestresponse to T and T is a best response to H, so both of his strategies are rationalizable.(d). A strictly dominated strategy s 1 is one for which there exists another strategy,s 2 , satisfies u i (s 2 ,s −i ) > u i (s 1 ,s −i ) for all s −i . Consequently, s 2 provides a higherpayoff than s 1 for anything that i’s opponents might do, so s 1 is never a best-responseto anything, so it is not rationalizable.Weakly dominated strategies can be rationalizable, since it might be a best responseto some strategy an opponent might play, for example, inL Ru 2,∗ 5,∗d 2,∗ 1,∗8
the strategy d is a best response to L, so it is rationalizable, even though it is weaklydominated by u.Dominant strategies are always rationalizable. Strategy s ∗ i is a dominant strategyfor player i if u i (s ∗ i ,s −i) ≥ u i (s ′ i ,s −i) for every s −i and any s ′ i that player i could choose.But this implies that s ∗ i is a best responseto any strategy profile, so it is a rationalizablestrategy.Rationalizable strategies are not always dominant. The example above shows this,since u weakly dominates d.(e). InaNashequilibriums ∗ , foreveryplayer i, u i (s ∗ i ,s∗ −i ) ≥ u i(s ′ i ,s∗ −i )foranyothers ′ i that player i could choose. Then s∗ i is a best response to s∗ −i , so it is a rationalizablestrategy.5c. This question involves agents who care about how their payoff compares to that oftheir opponent.We start from a particular strategy profile (a i ,a j ) in a strategic form, where the playershave material payoffs u i (a i ,a j ), that capture the direct payoff to player i of a given strategyprofile (a i ,a j ). The kindness of player j to player i at (a i ,a j ) is given byk ji (a i ,a j ) = u i (a i ,a j )− u i(a i ,a j )+u j (a i ,a j )2This compares the material payoff of player i to the average of the two player’s materialpayoffs. If the average material payoff is larger than player i’s material payoff, k ji will benegative and player j is unkind to player i. If the average payoff is smaller than player i’smaterial payoff, k ji is positive and player j is being kind to player i. Note that if k ji > 0then k ij < 0, so that if one player is being kind, the other player must be unkind.Define the player’s kindness payoffs asU i (a i ,a j ) = u i (a i ,a j )+ck ji (a i ,a j )so that player i cares about his payoff per se, but also how kind his opponent is to him. Thevalue ccan bepositive or negative, and measures how strongly the “kindness”motive affectsthe players’ payoffs. A strategy profile a ∗ = (a ∗ i ,a∗ j ) is a pure-strategy Nash equilibrium ifis a Nash equilibrium in the strategic form game with payoffs (U i (a i ,a j ),U j (a i ,a j )).(a). For c = 3, rewrite the strategic form for Battle of the Sexes with payoffs of U i (a i ,a j )rather than u i (a i ,a j ), where the material payoffs are given byColumnL RRow U 2,1 0,0D 0,0 1,2Do the equilibriumpredictions change? Explain the role that cplays in changing theagents’motives. Now, for c = −1, rewrite the strategic form for Prisoners’ Dilemma with payoffs9
Page 1 and 2: Problem Set 11. Games in Strategic
Page 3 and 4: The Nash equilibrium is (A,A), sinc
Page 5 and 6: (1) CL RU 2,2,3 1,1,2B D 1,3,4 1,4,
Page 7: ight-side up and you drop the ball
Page 11: The usual profiles (U,L) and (D,R)

Problem Set 1 1. Games in Strategic Form: Do questions 1-4 and ...

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