Comparison of Information Structures for Zero-Sum Games and a Partial Converse to Blackwell Ordering in Standard Borel Spaces
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In statistical decision theory involving a single decision-maker, one says that an information structure is better than another one if for any cost function involving a hidden state variable and an action variable which is restricted to be only a function of some measurement, the solution value under the former is not worse than the value under the latter. For finite probability spaces, Blackwell's celebrated theorem on comparison of information structures leads to a complete characterization on when one information structure is better than another. For stochastic games with incomplete information, due to the presence of competition among decision makers, in general such an ordering is not possible since additional information can lead to equilibria perturbations with positive or negative values to a player. However, for zero-sum games in a finite probability space, Pęski introduced a complete characterization of ordering of information structures. In this paper, we obtain an infinite dimensional (standard Borel) generalization of Pęski's result. A corollary of our analysis is that more information cannot hurt a decision maker taking part in a zero-sum game in standard Borel spaces. During our analysis, we establish two novel supporting results: (i) a partial converse to Blackwell's ordering of information structures in the standard Borel space setup and (ii) a refined existence result for equilibria in zero-sum games with incomplete information when compared with the prior literature.
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