The maximal number of regular totally mixed Nash equilibria

Let S=Pi(i=1)(n) S-i be the strategy space for a finite n-person game. Let (s(10),...,s(n0)) epsilon S be any strategy n-tuple, and let T-i=S-i-{s(i0)}, i=1,...,n. We show that the maximum number of regular totally mixed Nash equilibria of a game with strategy sets S-i is the number of partitions P={P-1,...,P-n} of boolean OR(r) T-i such that, for each i, \P-i\=\T-t\ and P-i boolean AND T-i=0. The bound is tight, as we give a method for constructing a game with the maximum number of equilibria. (C) 1997 Academic Press.