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 ɛ-equilibria for all ɛ>0. We show by examples that there are:
1. quasiconcave, payoff secure games without pure strategy ɛ-equilibria for small enough ɛ>0 (and hence, without pure strategy Nash equilibria),
2. quasiconcave, reciprocally upper semicontinuous games without pure strategy ɛ-equilibria for small enough ɛ>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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Additional information
I wish to thank the editor, an associate editor and an anonymous referee for very helpful comments. I thank also John Huffstot for editorial assistance. Any remaining error is, of course, mine
Rights and permissions
About this article
Cite this article
Carmona, G. On the existence of equilibria in discontinuous games: three counterexamples. Int J Game Theory 33, 181–187 (2005). https://doi.org/10.1007/s001820400187
Issue Date:
DOI: https://doi.org/10.1007/s001820400187