Elimination of Randomization in Certain Statistical Decision Procedures and Zero-Sum Two-Person Games

Abstract

The general existence of minimax strategies and other important properties proved in the theory of statistical decision functions (e.g., [3]) and the theory of games (e.g., [5]) depends upon the convexity of the space of decision functions and the convexity of the space of strategies. This convexity can be obtained by the use of randomized decision functions and mixed (randomized) strategies. In Section 2 of the present paper the authors state the extension (first announced in [1]) of a measure theoretical result known as Lyapunov's theorem [2]. This result is applied in Section 3 to the statistical decision problem where the number of distributions and decisions is finite. It is proved that when the distributions are continuous (more generally, "atomless," see footnote 7 below) randomization is unnecessary in the sense that every randomized decision function can be replaced by an equivalent nonrandomized decision function. Section 4 extends this result to the case when the decision space is compact. Section 5 extends the results of Section 3 to the sequential case. Sections 6 and 7 show, by counterexamples, that the results of Section 3 cannot be extended to the case of infinitely many distributions without new restrictions.4 Section 8 gives sufficient conditions for the elimination of randomization under maintenance of e-equivalence. Section 9 concludes with a restatement of the results in the language of the theory of gaines. 1. Introduction. We shall consider the following statistical decision problem: