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On the existence of equilibria in discontinuous games: three counterexamples

  • Guilherme Carmona

    ()

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 "!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 [6], also shows that nonquasiconcave games that are payoff secure and reciprocally upper semicontinuous may fail to have mixed strategy equilibria.

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File URL: http://hdl.handle.net/10.1007/s001820400187
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Article provided by Springer in its journal International Journal of Game Theory.

Volume (Year): 33 (2005)
Issue (Month): 2 (06)
Pages: 181-187

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Handle: RePEc:spr:jogath:v:33:y:2005:i:2:p:181-187
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  1. Baye, Michael R & Tian, Guoqiang & Zhou, Jianxin, 1993. "Characterizations of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs," Review of Economic Studies, Wiley Blackwell, vol. 60(4), pages 935-48, October.
  2. 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.
  3. Dasgupta, Partha & Maskin, Eric, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," Review of Economic Studies, Wiley Blackwell, vol. 53(1), pages 1-26, January.
  4. Simon, Leo K, 1987. "Games with Discontinuous Payoffs," Review of Economic Studies, Wiley Blackwell, vol. 54(4), pages 569-97, October.
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