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The Expected Number of Nash Equilibria of a Normal Form Game

Andrew McLennan

Abstract

Fix finite pure strategy sets S1,…,Sn, and let S=S1×⋯×Sn. In our model of a random game the agents' payoffs are statistically independent, with each agent's payoff uniformly distributed on the unit sphere in RS. For given nonempty T1⊂S1,…,Tn⊂Sn we give a computationally implementable formula for the mean number of Nash equilibria in which each agent i's mixed strategy has support Ti. The formula is the product of two expressions. The first is the expected number of totally mixed equilibria for the truncated game obtained by eliminating pure strategies outside the sets Ti. The second may be construed as the "probability" that such an equilibrium remains an equilibrium when the strategies in the sets Si∖Ti become available.