The joint continuity of the expected payoff functions

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Abstract

Glicksberg [Glicksberg, I.L., 1952. A further generalization of the Kakutani fixed point theorem, with applications to Nash equilibrium points. In: Proceedings of the American Mathematical Society 3, pp. 170–174] generalized the Kakutani fixed point theorem to the setting of locally convex spaces and used it to prove that every k-person strategic game with action sets convex compact subsets of locally convex spaces and continuous payoff functions has a Nash equilibrium. He subsequently used this result to establish the following fundamental theorem of game theory: Every k-person strategic game with action sets metrizable compact topological spaces and continuous payoff functions has a mixed strategies equilibrium. However, in his proof of the latter result, Glicksberg did not show that the expected payoff functions were jointly continuous, something that was required for the existence of a mixed strategies equilibrium.

The joint continuity of the expected payoff functions is not obvious at all. A general proof was presented by Glycopantis and Muir [D. Glycopantis, A. Muir, 2000. Continuity of the payoff functions, Economic Theory 16, 239–244]. An alternative proof was obtained by Zarichnyi [M. Zarichnyi, 2004. Continuity of the payoff function revisited, Economics Bulletin 3, 1–4]. A proof can also be obtained by using Billingsley [P. Billingsley, 1968. Convergence of Probability Measures, Wiley, New York, Theorem 3.2, p. 21] or by invoking a result of Balder [E.J. Balder, 1988. Generalized equilibrium results for games with incomplete information, Mathematics of Operations Research 13, 265–276, Theorem 3.1, p. 273]. In this note an alternative proof of the joint continuity of the expected payoff functions is given that is based on measure theoretic techniques.

Introduction

One of the fundamental and most useful theorems of non-cooperative game theory is the one that asserts that every strategic game with a finite number of players has a mixed strategies equilibrium. This result appeared in the literature in several variations and in the works of many authors; see for instance Debreu (1952) and Nash (1950). Its most general formulation is the one obtained by Glicksberg (1952) that can be stated as follows: Every strategic game with a finite number of players having action sets convex compact subsets of Hausdorff locally convex spaces and continuous payoff functions has a mixed strategies equilibrium.

A modern proof of this result is the one given by Glicksberg (1952) and is based on his generalization of the Kakutani fixed point theorem to the locally convex framework; see Theorem 2.1. In turn, Glicksberg’s proof of the generalization of Kakutani’s fixed point theorem to the locally convex setting is basically an adaptation of the proof of an elegant fixed point theorem obtained by Bohnenblust and Karlin (1950). There is, however, one assumption in the proof that Glicksberg did not verify. Namely, that of the joint continuity of the expected payoff functions. In most modern advanced books in game theory (see for instance Friedman, 1986; Fudenberg and Tirole, 1996; Myerson, 1991; Osborne and Rubinstein, 1994) this point is overlooked and the existing proofs of this basic result in the textbook literature of game theory are incomplete.

Glycopantis and Muir (2000) proved the continuity of the payoff function working, as in Glicksberg, with compact Hausdorff spaces of pure strategies,1 employing the fundamental Stone–Weierstrass theorem. Zarichnyi (2004) provides, under the same general conditions, an alternative proof by introducing into mathematical economics the properties of Milyutin maps.

The objective of this note is to present another proof of the joint continuity of the expected payoff functions when the action sets are metrizable compact spaces based on functional analytic and measure theoretic methods. The main result of this note is Lemma 3.4. The reason for providing a detailed proof adjusted to such conditions, stems from the fact that these are the usual requirements imposed on pure strategies (actions) of the agents.2 The paper also contains two examples that seem to be missing from the literature. The first one (Example 2.2) shows that in the classical result (Theorem 2.1) regarding the existence of Nash equilibrium the separate quasi-concavity of the payoff functions cannot be dropped. The second example (Example 3.2) demonstrates that even when the action sets of a strategic game are compact and convex subsets of finite dimensional Euclidean spaces and the payoff functions are continuous and separately quasi-concave, the game besides (necessarily) having pure strategies equilibria it can also have mixed strategies equilibria.

To help the reader understand the problem, the material has been spread out into three small sections. In Section 2 we simply state the basic existence of Nash equilibrium theorem in the locally convex setting (Theorem 2.1). In Section 3 we state the existence of mixed strategies equilibrium (Theorem 3.3) and establish Lemma 3.4, the key result needed to prove the joint continuity of the expected payoff functions. In the appendix section we have collected the major results from functional analysis and measure theory that are needed for this note. For terminology and notation we follow Aliprantis and Border (1999).

Section snippets

The existence of Nash equilibrium: the pure strategies case

We discuss here very briefly the classical existence theorem of Nash equilibria for strategic games. So, we consider a k-player strategic game. The action set of player i is Ai. The set of all action profiles is the Cartesian product A=i=1kAi. We shall write a=(a1,,ak) for the arbitrary action profile. As usual, if aA, then for each player i we letai=(a1,,ai1,ai+1,,ak),the actions of all players with respect to a except player’s i, and Ai=jiAj. Following the standard notation, for

The existence of Nash equilibrium: the mixed strategies case

We now remove the convexity assumption from the action sets. But we shall assume that each Ai is a metrizable compact topological space. This guarantees that each C(Ai), the Banach space of all continuous real-valued functions defined on Ai, is separable and its norm dual is the Banach space ca(Ai), the Borel signed measures of bounded total variation defined on the Borel σ-algebra BAi. The weak topology σ(ca(Ai),C(Ai)) will be denoted by w and will be called the w-topology of ca(Ai).

Acknowledgements

This research is supported in part by the NSF grants EIA-0075506, SES-0128039, DMS-0437210, and ACI-0325846. We thank Eric Balder for pointing out to us references Balder (1988); Billingsley (1968); Allan Muir and Nicholas Yannelis for their constructive comments and an anonymous referee for several useful suggestions.

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