NDLI: Probabilities of Pure Nash Equilibria in Matrix Games when the Payoff Entries of One Player Are Randomly Selected

Content Provider	Springer Nature Link
Author	Peterson, J. M. Simaan, M. A.
Copyright Year	2007
Abstract	The Nash equilibrium in pure strategies represents an important solution concept in nonzero sum matrix games. Existence of Nash equilibria in games with known and with randomly selected payoff entries have been studied extensively. In many real games, however, a player may know his own payoff entries but not the payoff entries of the other player. In this paper, we consider nonzero sum matrix games where the payoff entries of one player are known, but the payoff entries of the other player are assumed to be randomly selected. We are interested in determining the probabilities of existence of pure Nash equilibria in such games. We characterize these probabilities by first determining the finite space of ordinal matrix games that corresponds to the infinite space of matrix games with random entries for only one player. We then partition this space into mutually exclusive spaces that correspond to games with no Nash equilibria and with r Nash equilibria. In order to effectively compute the sizes of these spaces, we introduce the concept of top-rated preferences minimal ordinal games. We then present a theorem which provides a mechanism for computing the number of games in each of these mutually exclusive spaces, which then can be used to determine the probabilities. Finally, we summarize the results by deriving the probabilities of existence of unique, nonunique, and no Nash equilibria, and we present an illustrative example.
Starting Page	401
Ending Page	410
Page Count	10
File Format	PDF
ISSN	00223239
Journal	Journal of Optimization Theory and Applications
Volume Number	137
Issue Number	2
e-ISSN	15732878
Language	English
Publisher	Springer US
Publisher Date	2008-01-03
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Matrix games Pure Nash equilibria Ordinal games Random payoffs Operations Research/Decision Theory Engineering Applications of Mathematics Theory of Computation Optimization Calculus of Variations and Optimal Control; Optimization
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Control and Optimization Management Science and Operations Research

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