On the Existence of Equilibria in Discontinuous Games: Three Counterexamples
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 "!equilibria for all " > 0. We show by examples that there are: 1. quasiconcave, payoff secure games without pure strategy "!equilibria for small enough " > 0 (and hence, without pure strategy Nash equilibria), 2. quasiconcave, reciprocally upper semicontinuous games without pure strategy "!equilibria for small enough " > 0, and 3. payoff secure games whose mixed extension is not payoff secure. The last example, due to Sion and Wolfe , also shows that nonquasiconcave games that are payoff secure and reciprocally upper semicontinuous may fail to have mixed strategy equilibria.
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- Michael R. Baye & Guoqiang Tian & Jianxin Zhou, 1993. "Characterizations of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs," Review of Economic Studies, Oxford University Press, vol. 60(4), pages 935-948.
- Partha Dasgupta & Eric Maskin, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," Review of Economic Studies, Oxford University Press, vol. 53(1), pages 1-26.
- Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, vol. 67(5), pages 1029-1056, September.
- Leo K. Simon, 1987. "Games with Discontinuous Payoffs," Review of Economic Studies, Oxford University Press, vol. 54(4), pages 569-597.