h1ansgamef07

# h1ansgamef07 - Department of Economics The Ohio State...

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Department of Economics The Ohio State University Econ 817—Game Theory Fall 2007 Prof. James Peck Homework #1 Answers 1. O-R, exercise 19.1. Answer: There are n players, and each player i has the action set, A i = { out } [0 , 1] . Each player prefers an action pro f le with more votes than any other player than one in which he/she ties for the most votes; prefers to tie than to be out; and prefers to be out rather than lose. With two players, the unique equilibrium is for both players to choose the median of the distribution, m = F 1 ( 1 2 ) . Th i si saNE ,becau sebo th f rms tie, and any deviation will cause that f rm to lose or be out. To see that this NE is unique, there cannot be a NE in which one of the players is out, because that player could guarantee at least a tie by choosing the right position. There cannot be a NE in which the players choose di f erent positions, because a player standing to lose could guarantee at least a tie, and a player standing to tie could move closer to the other player and thereby win. There cannot be a NE in which the players choose the same position other than the median, because they would stand to tie, but a player could deviate closer to the median and win. With three players, there cannot be a NE. If all three are out, then one player could choose a position and win. If two are out, then one of those players could guarantee at least a tie by choosing the right position. If one player is out, then the other two must be choosing the median voter position; in that case, the player that is out could choose a position close to the median voter position and receive almost half the votes, thereby winning. Finally, suppose all three voters choose positions. They must tie for f rst, because otherwise being out is preferred by a loser. If the players choose three distinct points, then one of the outside players could move closer to the middle and win. If two of the players choose the same position, then the other player could move closer to them and win.

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## This note was uploaded on 07/17/2008 for the course ECON 817 taught by Professor Peck during the Fall '07 term at Ohio State.

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h1ansgamef07 - Department of Economics The Ohio State...

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