On the Existence of Equilibria in Discontinuous Games: Three Counterexamples

Abstract:     
We study whether we can weaken the conditions given in Reny [4] and still obtain existence of pure strategy Nash equilibria in quasiconcave normal form games, or, at least, existence of pure strategy ε -equilibria for all ε > 0. We show by examples that there are:

1. quasiconcave, payoff secure games without pure strategy ε -equilibria for small enough ε > 0 (and hence, without pure strategy Nash equilibria),

2. quasiconcave, reciprocally upper semicontinuous games without pure strategy ε -equilibria for small enough ε > 0, and

3. payoff secure games whose mixed extension is not payoff secure.

The last example, due to Sion and Wolfe [6], also shows that nonquasiconcave games that are payoff secure and reciprocally upper semicontinuous may fail to have mixed strategy equilibria.

Number of Pages in PDF File: 11

working papers series