On the Dubins and Savage Characterization of Optimal Strategies

An elegant characterization of optimal strategies for gambling problems was given by Dubins and Savage in the finitely additive setting of their book How to Gamble If You Must. An exposition of their ideas is given here in a measurable, countably additive framework. With the additional measurability assumptions, it becomes possible to treat a larger class of payoff functions. Also, necessary and sufficient conditions are given for a strategy to be i-optimal, a problem not considered by Dubins and Savage. 1. Definitions and preliminaries. This section establishes the framework for the sequel and reports certain technical measurability results needed there. Most of the notation and definitions below are borrowed or adapted from [3]. The term Borel set is used here to mean a Borel subset of a complete separable metric space. Let X be a Borel set. Denote by XW(X) the Borel subsets of X and by 9(X) the set of all countably additive probability measures on X(X). If 9(X) is given the usual weak topology, then it has the structure of a Borel set and the Borel subsets of 9(X) may be described as the smallest a-field of subsets which makes r r(A) a measurable function from 9~'(X) to the Borel line for each A in <@(X). (A thorough discussion of the weak topology is in Chapter II of [6] and the Borel structure of 9(X) is explored