On the structure of the set of Nash equilibria of weakly nondegenerate bimatrix games

In two-person games where each player has a finite number of pure strategies, the set of Nash equilibria is a finite set when a certain nondegeneracy condition is satisfied. Recent investigations have shown that for n×n games, the cardinality of this finite set is bounded from above by a function φ(n) with 2ⁿ–1 ≤ φ(n) ≤ (27/4)^n/2–1, where n is the maximal number of pure strategies of any player. In the present paper, we generalize this result to a class of games which may not satisfy the nondegeneracy condition. The set of Nash equilibria may be infinite; it is shown that it consists of no more than φ(n) arc-connected components.