Payoff-Relevant States in Dynamic Games with Infinite Action Spaces

Maskin and Tirole have defined payoff-relevant states in discrete time dynamic games with observable actions in terms of a partition of the set of histories. Their proof that this partition is unique cannot be applied, when action spaces are infinite or when players are unable to condition on calendar time. This note provides a unified proof of existence and uniqueness for these cases. The method of proof is useful for problems other than the one treated here. To illustrate this, a well known characterization of common knowledge is generalized.