The Maxmin Value of Repeated Games with Incomplete Information on One Side and Tail-Measurable Payoffs
Gil Bar Castellon Koltun, Ehud Lehrer, Eilon Solan
TL;DR
The paper extends Aumann-Maschler's framework to tail-measurable payoffs in two-player zero-sum repeated games with incomplete information on one side, proving that the maxmin value equals the concavification of the non-revealing game's value. It shows that the informed player can effectively reveal information at the outset, while the uninformed player uses a blockwise, martingale-based counterstrategy to bound outcomes, with the result proven via induction on the number of states and Jensen's inequality. Importantly, the authors provide counterexamples demonstrating that tail-measurable payoffs do not guarantee the existence of a value. The work connects to broader themes in information revelation, Blackwell-type games, and tail-objective payoffs, and raises open questions about explicit characterizations and extensions to other payoff classes.
Abstract
We study two-player zero-sum repeated games with incomplete information on one side, where the payoff function is tail measurable (and not necessarily the long-run average payoff). We show that the maxmin value equals the concavification of the value function of the non-revealing game. In addition, we provide an example demonstrating that, under tail-measurable payoffs, the value of the game may fail to exist.