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Existence of Equilibria in Discontinuous and Nonconvex Games

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  • Rabia Nessah
  • Guoqiang Tian
Abstract
This paper investigates the existence of pure strategy, dominant-strategy, and mixed strategy Nash equilibria in discontinuous and nonconvex games. We introduce a new notion of very weak continuity, called weak transfer continuity, which holds in a large class of discontinuous economic games and is easy to check. We show that it, together with the compactness of strategy space and the quasiconcavity of payoff functions, permits the existence of pure strategy Nash equilibria. Our equilibrium existence result neither implies nor is implied by the existing results in the literature such as those in Baye et al. [1993] and Reny [1999]. We provide sufficient conditions for weak transfer continuity by introducing notions of weak transfer upper continuity and weak transfer lower continuity. These conditions are satisfied in many economic games and are often quite simple to check. We also introduce the notion of weak dominant transfer upper continuity, and use it to study the existence of dominant strategy equilibria. We then generalize these results and those in Baye et al. [1993] and Reny [1999] without assuming any form of quasi-concavity of payoff functions or convexity of strategy spaces.
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  • Rabia Nessah & Guoqiang Tian, 2009. "Existence of Equilibria in Discontinuous and Nonconvex Games," Levine's Working Paper Archive 814577000000000206, David K. Levine.
    • Handle: RePEc:cla:levarc:814577000000000206
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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.
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    Cited by:

    1. Roberto Ghiselli Ricci, 2021. "A note on a Tarski type fixed-point theorem," International Journal of Game Theory, Springer;Game Theory Society, vol. 50(3), pages 751-758, September.
    2. Nessah, Rabia, 2011. "Generalized weak transfer continuity and the Nash equilibrium," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 659-662.
    3. Tian, Guoqiang, 2015. "On the existence of equilibria in games with arbitrary strategy spaces and preferences," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 9-16.
    4. Horst, Ulrich & Moreno-Bromberg, Santiago, 2010. "Efficiency and equilibria in games of optimal derivative design," SFB 649 Discussion Papers 2010-035, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    5. repec:hum:wpaper:sfb649dp2010-035 is not listed on IDEAS

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