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Existence of Solution of Minimax Inequalities, Equilibria in Games and Fixed Points Without Convexity and Compactness Assumptions

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Abstract

This paper characterizes the existence of equilibria in minimax inequalities without assuming any form of quasiconcavity 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 non-quasiconcave payoff functions and nonconvex and/or noncompact strategy spaces.

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Notes

  1. Banach Fixed Point Theorem: Let (K,d) be a complete metric space and let f:K→K be a d-contraction (d∈[0,1[). Then, f has a unique fixed point.

  2. Brouwer–Schauder–Tychonoff Fixed Point Theorem: Let K be a nonempty, compact and convex subset of a locally convex Hausdorff space, and let f:K→K be a continuous function. Then the set of fixed points of f is compact and nonempty.

  3. Halpern–Bergman Fixed Point Theorem: Let K be a nonempty, compact and convex subset of a locally convex Hausdorff space X, and let C:K⇉X be an inward pointing upper semicontinuous mapping with nonempty, closed and convex values. Then C has a fixed point.

  4. Kakutani–Fan–Glicksberg Fixed Point Theorem: Let K be a subset nonempty, compact and convex of a locally convex Hausdorff space, and let C:K⇉K have closed graph and nonempty and convex values. Then the set of fixed points of C is nonempty and compact.

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Acknowledgements

The authors would like to thank the two anonymous referees for insightful comments which have substantially improved the quality of the paper.

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Authors and Affiliations

  1. IESEG School of Management, CNRS-LEM (UMR 8179), 3 rue de la Digue, 59000, Lille, France

    Rabia Nessah

  2. Department of Economics, Texas A&M University, College Station, TX, 77843, USA

    Guoqiang Tian

Authors
  1. Rabia Nessah
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  2. Guoqiang Tian
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Correspondence to Rabia Nessah.

Additional information

Communicated by Suliman Saleh Al-Homidan.

Financial support from the National Natural Science Foundation of China (NSFC-70773073) and the Program to Enhance Scholarly and Creative Activities at Texas A&M University as well as from Cheung Kong Scholars Program at the Ministry of Education of China is gratefully acknowledged.

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Nessah, R., Tian, G. Existence of Solution of Minimax Inequalities, Equilibria in Games and Fixed Points Without Convexity and Compactness Assumptions. J Optim Theory Appl 157, 75–95 (2013). https://doi.org/10.1007/s10957-012-0176-5

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