On the distribution of pure strategy equilibria in finite games with vector payoffs -- Web of Science 5.0

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On the distribution of pure strategy equilibria in finite games with vector payoffs
Stanford W
MATHEMATICAL SOCIAL SCIENCES 33 (2): 115-127 APR 1997

Document type: Article

Language: English

Cited References: 16

Times Cited: 1

Abstract:
Two players, each with vector payoffs, no notion of substitution rates between coordinates, and no recourse to mixed strategies, play a finite matrix game. Thus preferences over outcomes are incomplete and a pure equilibrium (PE) is a pure strategy pair in which the choice of each player is maximal in his pure strategy set (no worse than anything else) given the other player's strategy. The coordinates of each payoff vector are real-valued random variables. For two-dimensional payoffs, we calculate the mean and variance of the number of PE points in the game and show that for any positive integer k, the probability of at least k PE points approaches one as pure strategy sets increase in size, and as a corollary, that this extends to n-dimensional payoffs for n greater than or equal to 3. Again for two-dimensional payoffs, in the limit as pure strategy sets expand, the distribution is normal with mean and identical variance increasing without bound. For symmetric bimatrix games with n-dimensional payoffs, the exact distribution of symmetric PE points is binomial with parameters depending on n and the size of pure strategy sets, and a similar central limit result applies. Thus, in the cases we study, the need to consider mixed strategy equilibria vanishes in the limit for large games when payoffs are vector valued. (C) 1997 Elsevier Science B.V.

Author Keywords:
central limit theorem, dependent random variables, finite games, Poisson approximations, pure strategy equilibria, vector payoffs

KeyWords Plus:
NASH EQUILIBRIA, POISSON

Addresses:
Stanford W, UNIV ILLINOIS,DEPT ECON MC144,601 S MORGAN ST,ROOM 2103,CHICAGO,IL 60607

Publisher:
ELSEVIER SCIENCE BV, AMSTERDAM

IDS Number:
WX965

ISSN:
0165-4896

Article 20 of 40

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