We study whether we can weaken the conditions given in Reny  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 , also shows that non-quasiconcave games that are payoff secure and reciprocally upper semicontinuous may fail to have mixed strategy equilibria.
- Discontinuous games
- Nash equilibrium
- Payoff security
- Reciprocal upper semicontinuity