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Games with continuous payoff functions and the problem of measurability

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  • Christian Ewerhart

Abstract

This paper examines the definition and continuity of expected payoffs in compact games with continuous payoff functions. There are three main results. First, we confirm that Glicksberg’s (1952) original definition of expected payoffs as an iterated integral is mathematically sound under general conditions. Second, we show that the now more common definition as a single integral is both rigorous and equivalent to the original when strategy spaces are either Hausdorff or second countable. Third, we offer an alternative proof of the continuity of expected payoffs without imposing the Hausdorff separation axiom. Together, these results lead to a strengthening of Glicksberg’s theorem on equilibrium existence in compact Hausdorff games with continuous payoff functions.

Suggested Citation

  • Christian Ewerhart, 2025. "Games with continuous payoff functions and the problem of measurability," ECON - Working Papers 467, Department of Economics - University of Zurich.
    • Handle: RePEc:zur:econwp:467
    as

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    File URL: https://www.zora.uzh.ch/id/eprint/276544/1/econwp467.pdf
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    References listed on IDEAS

    as
    1. Partha Dasgupta & Eric Maskin, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 53(1), pages 1-26.
    2. Aliprantis, Charalambos D. & Glycopantis, Dionysius & Puzzello, Daniela, 2006. "The joint continuity of the expected payoff functions," Journal of Mathematical Economics, Elsevier, vol. 42(2), pages 121-130, April.
    3. Partha Dasgupta & Eric Maskin, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, II: Applications," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 53(1), pages 27-41.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Compact games; expected payoffs; weak* topology; measurability; continuity;
    All these keywords.

    JEL classification:

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

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