Purication of Measure-Valued Maps

Given a measurable mapping f from a nonatomic Loeb probability space (T; T ; P ) to the space of Borel probability measures on a compact metric space A, we show the existence of a measurable mapping g from (T; T ; P ) to A itself such that f and g yield the same values for the integrals associated with a countable class of functions on T A. A corollary generalizes the classical result of Dvoretzky-Wald-Wolfowitz on puri…cation of measure-valued maps with respect to a …nite target space; the generalization holds when the domain is a nonatomic, vector-valued Loeb measure space and the target is a complete, separable metric space. A counterexample shows that the generalized result fails even for simple cases when the restriction of Loeb measures is removed. As an application, we obtain a strong puri…cation for every mixed strategy pro…le in …nite-player games with compact action spaces and di¤use and conditionally independent information. 1. Introduction In 1951, Dvoretzky, Wald and Wolfowitz used the Lyapunov theorem for vector measures to establish the following result in [9, Theorem 4] (also announced in [8, Theorem 1] and in [10, Theorem 2.1]). Theorem 1.1. Let A be a …nite set, (T; T ) a measurable space, and k; k = 1; ;m; …nite, nonatomic signed measures on (T; T ). Let f be a mapping from T to the spaceM(A) of probability measures on A such that for each a 2 A, f( )(fag) is T -measurable. Then there exists a T -measurable function g from T to A such that for each a 2 A, Z T f(t)(fag) k(dt) = k(ft 2 T : g(t) = ag): This theorem justi…es the elimination, i.e., puri…cation, of randomness in various settings. In games, for example, T represents the space of information available to the game’s players, and A represents the set of actions players may choose, given the available information t 2 T . 1 2 PETER LOEB AND YENENG SUN Each player’s objective is to maximize their own expected payo¤, which depends not only on that player’s choice of action but also on that of all the other players. (Our use of “their”is consistent with the increasing use of some form of they with singular, generic antecedents that has its origins in the fourteenth century; see [4].) For each player, a mapping from the space of information T to particular actions in A is called a pure strategy. If the mapping is not to A itself but to the spaceM(A) of probability measures on A, then that mapping is called a mixed strategy; here the player chooses a “lottery on A”. A Nash equilibrium is achieved when every player is satis…ed with their choice of strategy given the choices of all the other players. In quite general settings, such an equilibrium can be achieved when the players choose a mixed strategy. In the more restrictive settings where Theorem 1.1 or an extension applies, those strategies can then be puri…ed to obtain an equilibrium with the same expected payo¤ for all the players. In fact, Theorem 1.1 was applied by Dvoretzky, Wald andWolfowitz to the puri…cation of both statistical decision procedures (see [8, Theorems 5 and 6], [10, Theorems 3.1 and 3.2, Section 4, Theorems 5.1 and 5.2]), and of mixed strategies in two-person zero-sum games with …nite action sets (see [8, Theorems 2 and 3], [10, Section 9] on two-person zero-sum games). The relevance of Theorem 1.1 to the puri…cation problem in …nite games with …nite action spaces and incomplete and di¤use information was already suggested in [20, Footnote 3] and in [19, Section 5]. A uni…ed approach to puri…cation problems in …niteaction games using Theorem 1.1 is presented in [14]. Theorem 1.1 and the applications just noted are restricted to the case of a …nite action space A. We will remove that restriction by establishing a result valid for a compact metric space and even a complete separable metric space. Even when A is a closed, …nite interval in the real line, however, Example 2.7 below shows that there is no extension of Theorem 1.1 when T is the unit interval supplied with Lebesgue measure and another measure having a continuous density function. To obtain our extension, we require that T with its associated measures are nonatomic measure spaces of the kind introduced by the …rst author in [17], and now called “Loeb spaces”in the literature. Using such a space T , we will obtain a general extension of Theorem 1.1 and a corresponding application to games. In Section 2, we consider the puri…cation of measure-valued maps. Theorem 2.2 shows that for a measurable mapping f from a nonatomic Loeb probability space (T; T ; P ) to the space of Borel probability measures on a compact metric space A, one can …nd a measurable mapping g from (T; T ; P ) to A such that f and g yield the same values for the