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A direct proof of the existence of pure strategy equilibria in large generalized games with atomic players

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  • Alvaro Riascos V.
  • Juan Pablo Torres-Martínez

Abstract

Consider a game with a continuum of players where only a finite number of them are atomic. Objective functions and admissible strategies may depend on the actions chosen by atomic players and on aggregate information about the actions chosen by non-atomic players. Only atomic players are required to have convex sets of admissible strategies and quasi-concave objective functions. In this context, we prove the existence of pure strategy Nash equilibria, a result that extends Rath (1992, Theorem 2) to generalized games and gives a direct proof of a special case of Balder (1999, Theorem 2.1). Our proof has the merit of being simple, based only on standard fixed point arguments and finite dimensional real analysis.

Suggested Citation

  • Alvaro Riascos V. & Juan Pablo Torres-Martínez, 2010. "A direct proof of the existence of pure strategy equilibria in large generalized games with atomic players," Working Papers wp311, University of Chile, Department of Economics.
    • Handle: RePEc:udc:wpaper:wp311
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    References listed on IDEAS

    as
    1. Balder, Erik J., 1999. "On the existence of Cournot-Nash equilibria in continuum games," Journal of Mathematical Economics, Elsevier, vol. 32(2), pages 207-223, October.
    2. Aumann, Robert J., 1976. "An elementary proof that integration preserves uppersemicontinuity," Journal of Mathematical Economics, Elsevier, vol. 3(1), pages 15-18, March.
    3. Rath, Kali P, 1992. "A Direct Proof of the Existence of Pure Strategy Equilibria in Games with a Continuum of Players," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(3), pages 427-433, July.
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    Cited by:

    1. Rubén Poblete-Cazenave & Juan Torres-Martínez, 2013. "Equilibrium with limited-recourse collateralized loans," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 53(1), pages 181-211, May.

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    More about this item

    Keywords

    Generalized games; Non-convexities; Pure-strategy Nash equilibrium.;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium

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