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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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    5. 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.
    6. 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.
    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. 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.
    9. 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.
    10. 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.
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