Working Papers in Economics and Statistics 2013-18
We show that concepts introduced by Aumann more than thirty years ago throw a new light on purification in games with extremely dispersed private information. We show that one can embed payoff-irrelevant randomization devices in the private information of players and use these randomization devices to implement mixed strategies as deterministic functions of the private information. This approach gives rise to very short, elementary, and intuitive proofs for a number of purification results that previously required sophisticated methods from functional analysis or nonstandard analysis. We use our methods to prove a general purification theorem for games with private information in which a player's payoffs can depend in arbitrary ways on events in the private information of other players and in which we allow for shared information in a general way.