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Existence of Equilibrium in Minimax Inequalities, Saddle Points, Fixed Points, and Games without Convexity Sets

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Author Info
Rabia Nessah () (IÉSEG School of Management, LEM-CNRS (UMR 8179))
Guoqiang Tian (Texas A&M University, USA)

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Abstract

This paper characterizes the existence of equilibria in minimax inequalities without assuming any form of quasi-concavity of functions and convexity or compactness of choice sets. A new condition, called “local dominatedness property”, is shown to be necessary and further, under some mild continuity condition, sufficient for the existence of equilibrium. We then apply the basic result obtained in the paper to generalize the existing theorems on the existence of saddle points, fixed points, and coincidence points without convexity or compactness assumptions. As an application, we also characterize the existence of pure strategy Nash equilibrium in games with discontinuous and nonquasiconcave payoff functions and nonconvex and/or noncompact strategy spaces.

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File URL: http://my.ieseg.fr/bienvenue/DownloadDoc.asp?Fich=519821983_2008-ECO-15_Nessah_Guoqiang.pdf
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Publisher Info
Paper provided by IESEG School of Management in its series Working Papers with number 2008-ECO-15.

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Length: 24 pages
Date of creation: Nov 2008
Date of revision:
Handle: RePEc:ies:wpaper:e200815

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Related research
Keywords: minimax inequality; saddle points; fixed points; coincidence points; discontinuity; non-quasiconcavity; non-convexity; and non-compactness.;

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  1. Dasgupta, Partha & Maskin, Eric, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," Review of Economic Studies, Blackwell Publishing, vol. 53(1), pages 1-26, January. [Downloadable!] (restricted)
  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.
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