On the existence of equilibria in discontinuous games: three counterexamples

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 non-quasiconcave games that are payoff secure and reciprocally upper semicontinuous may fail to have mixed strategy equilibria.