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Poor convexity and Nash equilibria in games

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  • Tadeusz Radzik

Abstract

This paper considers two-person non-zero-sum games on the unit square with payoff functions having a new property called poor convexity. This property describes “something between” the classical convexity and quasi-convexity. It is proved that various types of such games have Nash equilibria with a very simple structure, consisting of the players’ mixed strategies with at most two-element supports. Since poor convexity is a basic notion in the paper, also a theory of poorly convex functions is also developed. Copyright The Author(s) 2014

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  • Tadeusz Radzik, 2014. "Poor convexity and Nash equilibria in games," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(1), pages 169-192, February.
    • Handle: RePEc:spr:jogath:v:43:y:2014:i:1:p:169-192
    DOI: 10.1007/s00182-013-0379-5
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    References listed on IDEAS

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    1. Carmona, Guilherme, 2009. "An existence result for discontinuous games," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1333-1340, May.
    2. McClendon, J F, 1986. "Existence of Solutions of Games with Some Non-convexity," International Journal of Game Theory, Springer;Game Theory Society, vol. 15(3), pages 155-162.
    3. Wojciech Połowczuk & Piotr Więcek & Tadeusz Radzik, 2007. "On the existence of almost-pure-strategy Nash equilibrium in n-person finite games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(1), pages 141-152, February.
    4. Radzik, Tadeusz, 1993. "Nash Equilibria of Discontinuous Non-Zero-Sum Two-Person Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(4), pages 429-437.
    5. Philippe Bich, 2009. "Existence of pure Nash equilibria in discontinuous and non quasiconcave games," Documents de travail du Centre d'Economie de la Sorbonne 09061, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    6. Philippe Bich, 2009. "Existence of pure Nash equilibria in discontinuous and non quasiconcave games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00750953, HAL.
    7. Philippe Bich, 2009. "Existence of pure Nash equilibria in discontinuous and non quasiconcave games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00426402, HAL.
    8. Tadeusz Radzik, 2000. "Characterization of optimal strategies in matrix games with convexity properties," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(2), pages 211-227.
    9. Philippe Bich, 2009. "Existence of pure Nash equilibria in discontinuous and non quasiconcave games," Post-Print halshs-00426402, HAL.
    10. Radzik, Tadeusz, 1991. "Pure-strategy [epsiv]-Nash equilibrium in two-person non-zero-sum games," Games and Economic Behavior, Elsevier, vol. 3(3), pages 356-367, August.
    11. Philippe Bich, 2009. "Existence of pure Nash equilibria in discontinuous and non quasiconcave games," Post-Print hal-00750953, HAL.
    12. Carmona, Guilherme, 2010. "Polytopes and the existence of approximate equilibria in discontinuous games," Games and Economic Behavior, Elsevier, vol. 68(1), pages 381-388, January.
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