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^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.