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.
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A first draft of this paper was completed when Khan and Rath were visiting the Institute for Mathematical Sciences at the National University of Singapore in August 2003; they thank the Institute for supporting their visit. A preliminary version was presented at the Midwest Economic Theory Conference held at Indiana University, Bloomington in October 2003; the authors are grateful to Eric Balder, Robert Becker, William Thomson and Myrna Wooders for questions and helpful suggestions.
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Khan, M.A., Rath, K.P. & Sun, Y. The Dvoretzky-Wald-Wolfowitz theorem and purification in atomless finite-action games. Int J Game Theory 34, 91–104 (2006). https://doi.org/10.1007/s00182-005-0004-3
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DOI: https://doi.org/10.1007/s00182-005-0004-3