On the maximal number of Nash equilibria in an n x n bimatrix game

In Quint and Shubik (1994) it was shown that for each odd number y less than or equal to 2(n) - 1, there is an nxn bimatrix game with exactly y Nash equilibria, and it was conjectured that the number 2(n) - 1 is an upper bound for the number of Nash equilibria of an arbitrary nondegenerate n x n bimatrix game. In this paper, we give an upper bound derived from the theory of convex polytopes. This bound is not necessarily tight; we show that in the case n = 4, the Quint-Shubik bound applies in the sense that there are no more than 2(4) - 1 = 15 equilibria. (C) 1997 Academic Press.