The Expected Number of Nash Equilibria of a Normal Form Game

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 R -super-S. 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 T i . 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 T i . 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. Copyright The Econometric Society 2005.