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ch07_solutions_solved edit.doc

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Ettore Damiano

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Solutions to Chapter 7 ExercisesSOLVED EXERCISESS1 False A players equilibrium mixture is devised in order to keep her opponent indifferent among all of her the opponents possible mixed strategies thus a players equilibrium mixture yields the opponent the same expected payoff against each of the players pure strategies Note that the statement will be true for zerosum games because when your opponent is indifferent in such a game it must also be true that you are indifferent as wellS2 aThe game most resembles an assurance game because the two Nash equiibria occur when both players play the same move In an assurance game both players prefer to make the same move but there is also a preferred Nash equilibrium with higher payoffs for both players In this game Risky Risky is the preferred equilibrium because it has higher payoffs but there is a chance that the players will play the worse Nash equilibrium with lower payoffs Even worse the players might not play an equilibrium at all Without convergence of expectations these results can occur and this is characteristic of an assurance gamebThe two purestrategy Nash equilibria for this game are Risky Risky and Safe SafeS3aThere is no purestrategy Nash equilibrium here hence the search for an equilibrium in mixed strategies Rows pmix probability p on Up must keep Column indifferent and so must satisfy 16p201p6p401p this yields p23067 and 1p033 Similarly Columns qmix probability q on Left must keep Row indifferent and so must satisfy q4 1q2q31q the correct q here is 05bRows expected payoff is 25 Columns expected payoff1733cJoint payoffs are larger when Row plays Down but the highest possible payoff to Row occurs when Row plays Up Thus in order to have a chance of getting 4 Row must play Up occasionally If the players could reach an agreement always to play Down and Right both would get higher expected payoffs than in the mixedstrategy equilibrium This might be possible with repetition of the game or if guidelines for social conduct were such that players gravitated toward Solutions to Chapter 7 Solved Exercises1 of 6
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