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Existence of justifiable equilibrium

Author

Listed:
  • Flesch, Janos

    (QE / Mathematical economics and game the)

  • Vermeulen, Dries

    (QE / Operations research)

  • Zseleva, Anna

    (higher school of economics, saint petersburg)

Abstract

We present a general existence result for a type of equilibrium in normal-form games. We consider nonzero-sum normal-form games with an arbitrary number of players and arbitrary action spaces. We impose merely one condition: the payoff function of each player is bounded. We allow players to use finitely additive probability measures as mixed strategies. Since we do not assume any measurability conditions, for a given strategy profile the expected payoff is generally not uniquely defined, and integration theory only provides an upper bound, the upper integral, and a lower bound, the lower integral. A strategy profile is called a justifiable equilibrium if each player evaluates this profile by the upper integral, and each player evaluates all his possible deviations by the lower integral. We show that a justifiable equilibrium always exists. Our equilibrium concept and existence result are motivated by Vasquez (2017), who defines a conceptually related equilibrium notion, and shows its existence under the conditions of finitely many players, separable metric action spaces and bounded Borel measurable payoff functions. Our proof borrows several ideas from Vasquez (2017), but is more direct as it does not make use of countably additive representations of finitely additive measures by Yosida and Hewitt (1952).

Suggested Citation

  • Flesch, Janos & Vermeulen, Dries & Zseleva, Anna, 2018. "Existence of justifiable equilibrium," Research Memorandum 016, Maastricht University, Graduate School of Business and Economics (GSBE).
  • Handle: RePEc:unm:umagsb:2018016
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    File URL: https://cris.maastrichtuniversity.nl/portal/files/26161786/content
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    References listed on IDEAS

    as
    1. A. Maitra & W. Sudderth, 1998. "Finitely additive stochastic games with Borel measurable payoffs," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(2), pages 257-267.
    2. Ehud Lehrer, 2009. "A new integral for capacities," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 39(1), pages 157-176, April.
    3. Stinchcombe, Maxwell B., 2005. "Nash equilibrium and generalized integration for infinite normal form games," Games and Economic Behavior, Elsevier, vol. 50(2), pages 332-365, February.
    4. Flesch, János & Vermeulen, Dries & Zseleva, Anna, 2017. "Zero-sum games with charges," Games and Economic Behavior, Elsevier, vol. 102(C), pages 666-686.
    5. Capraro, Valerio & Scarsini, Marco, 2013. "Existence of equilibria in countable games: An algebraic approach," Games and Economic Behavior, Elsevier, vol. 79(C), pages 163-180.
    6. Harris, Christopher J. & Stinchcombe, Maxwell B. & Zame, William R., 2005. "Nearly compact and continuous normal form games: characterizations and equilibrium existence," Games and Economic Behavior, Elsevier, vol. 50(2), pages 208-224, February.
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    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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