ch07_solutions_solved edit

ch07_solutions_solved edit - Solutions to Chapter 7...

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Solutions to Chapter 7 Exercises SOLVED EXERCISES S1. False. A player’s equilibrium mixture is devised in order to keep her opponent indifferent among all of her (the opponent’s) possible mixed strategies; thus, a player’s equilibrium mixture yields the opponent the same expected payoff against each of the player’s pure strategies. Note that the statement will be true for zero-sum games, because when your opponent is indifferent in such a game, it must also be true that you are indifferent as well. S2 (a) The 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 game. (b) The two pure-strategy Nash equilibria for this game are (Risky, Risky) and (Safe, Safe). S3. (a) There is no pure-strategy Nash equilibrium here, hence the search for an equilibrium in mixed strategies. Row’s p-mix (probability p on Up) must keep Column indifferent and so must satisfy 16p + 20(1 – p) = 6p + 40(1 – p); this yields p = 2/3 = 0.67 and (1 – p) = 0.33. Similarly, Column’s q-mix (probability q on Left) must keep Row indifferent and so must satisfy q + 4 (1 – q) = 2q + 3(1 – q); the correct q here is 0.5. (b) Row’s expected payoff is 2.5. Column’s expected payoff = 17.33. (c) Joint 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 mixed-strategy 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 Exercises 1 of 6
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the outcome that maximized total payoff. S4. The two pure-strategy Nash equilibria are (Don’t Help, Help) and (Help, Don’t Help).
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This note was uploaded on 08/08/2011 for the course ECON 171 taught by Professor Charness,g during the Spring '08 term at UCSB.

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ch07_solutions_solved edit - Solutions to Chapter 7...

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