Pure strategies in games with private information

Abstract

Pure strategy equilibria of finite player games with informational constraints have been discussed under the assumptions of finite actions, and of independence and diffuseness of information. We present a mathematical framework, based on the notion of a distribution of a correspondence, that enables us to handle the case of countably infinite actions. In this context, we extend the Radner-Rosenthal theorems on the purification of a mixed-strategy equilibrium, and present a direct proof, as well as a generalized version of Schmeidler's large games theorem, on the existence of a pure strategy equilibrium. Our mathematical results pertain to the set of distributions induced by the measurable selections of a correspondence with a countable range, and rely on the Bollobás-Varopoulos extension of the marriage lemma.

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