On the Existence of Equilibria in Discontinuous Games: Three Counterexamples
New University of Lisbon
September 29, 2003
FEUNL Working Paper No. 438
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 nonquasiconcave games that are payoff secure and reciprocally upper semicontinuous may fail to have mixed strategy equilibria.
Number of Pages in PDF File: 11working papers series
Date posted: February 20, 2006
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