Structure in the value function of zero-sum games of incomplete information

In this paper, we introduce plan-time sufficient statistics, representing probability distributions over joint sets of private information, for zero-sum games of incomplete information. We define a family of zero-sum Bayesian Games (zs-BGs), of which the members share all elements but the plan-time statistic. Using the fact that the statistic can be decomposed into a marginal and a conditional term, we prove that the value function of the family of zs-BGs exhibits concavity in marginal-space of the maximizing agent and convexity in marginal-space of the minimizing agent. We extend this result to sequential settings with a dynamic state, i.e., zero-sum Partially Observable Stochastic Games (zs-POSGs), in which the statistic is a probability distribution over joint action-observation histories. First, we show that the final stage of a zs-POSG corresponds to a family of zs-BGs. Then, we show by induction that the convexity and concavity properties can be extended to every time-step of the zs-POSG.