The Dvoretzky-Wald-Wolfowitz theorem and purification in atomless finite-action games

Abstract

In 1951, Dvoretzky, Wald and Wolfowitz (henceforth DWW) showed that corresponding to any mixed strategy into a finite action space, there exists a pure-strategy with an identical integral with respect to a finite set of atomless measures. DWW used their theorem for purification: the elimination of randomness in statistical decision procedures and in zero-sum two-person games. In this short essay, we apply a consequence of their theorem to a finite-action setting of finite games with incomplete and private information, as well as to that of large games. In addition to simplified proofs and conceptual clarifications, the unification of results offered here re-emphasizes the close connection between statistical decision theory and the theory of games.