208PS04 - Economics 208 Problems Vincent Crawford,...

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Economics 208 Problems Winter 2004 Vincent Crawford, Department of Economics, UCSD 1-2 (counts as two questions). Consider the Battle of the Sexes game. Assume, here and below, that the structure is common knowledge, that both players are self-interested, and that there are no observable differences between the players or their roles in the game. In each of the variations described below, say whether you would expect the players to be able to coordinate on one of the efficient pure-strategy equilibria, and what strategies you would expect the players to use, on average. Briefly but clearly explain your answers. Fights Ballet Fights 1 3 0 0 Ballet 0 0 3 1 Battle of the Sexes (a) The original simultaneous-move game is a complete model of the players' situation. (b) The game is modified so that Row chooses her/his strategy first and Column gets to observe her/his choice before choosing her/his own strategy. (c) Row chooses her/his strategy first and Column does NOT get to observe her/his choice before choosing her/his own strategy. (d) Row chooses her/his strategy first, Column observes her/his choice before choosing her/his own strategy, but Row then gets to observe Column’s choice and costlessly revise her/his own choice, and this decision ends the game (so that Column cannot revise her/his choice). (e) The original simultaneous-move game is a complete model of the players' situation, except that Row (only) can make a non-binding suggestion about the strategies players should use before they choose them. (f) The original simultaneous-move game is a complete model of the players' situation, except that both players can make simultaneous, non-binding suggestions about the strategies players should use before they choose them. (g) The original simultaneous-move game is a complete model of the players' situation, except that players can make sequential, non-binding suggestions about the strategies players should use before they choose them, say with Row making the first suggestion.
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3. Consider the following two-person guessing game. Each player has her/his own target, lower limit, and upper limit. These are possibly different across players, and they influence players’ payoffs as follows. Players make simultaneous guesses, which are required to be within their limits. Each player then earns 1000 points minus the distance between her/his guess and the product of her/his target times the other's guess. (a-d) Find the Nash equilibrium or equilibria for the following targets and limits: a) Lower Limit Target Upper Limit Player 1 200 0.7 600 Player 2 400 1.5 700 b) Lower Limit Target Upper Limit Player 1 300 0.7 500 Player 2 400 1.3 900 c) Lower Limit Target Upper Limit Player 1 400 0.5 900 Player 2 300 0.7 900 d) Lower Limit Target Upper Limit Player 1 300 1.3 500 Player 2 200 1.5 900 (e) State and prove a general result that determines the equilibrium as a function of the targets and limits for these guessing games. (f) Would you expect intelligent people randomly paired from students who have not
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208PS04 - Economics 208 Problems Vincent Crawford,...

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