Games with continuous payoff functions and the problem of measurability

Christian Ewerhart

No 467, ECON - Working Papers from Department of Economics - University of Zurich

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.

Keywords: Compact games; expected payoffs; weak* topology; measurability; continuity (search for similar items in EconPapers)
JEL-codes: C72 (search for similar items in EconPapers)
Date: 2025-03
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Persistent link: https://EconPapers.repec.org/RePEc:zur:econwp:467

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