Zero-sum repeated games: Counterexamples to the existence of the asymptotic value and the conjecture $\operatorname{maxmin}=\operatorname{lim}v_n$

TL;DR

This paper presents counterexamples in zero-sum repeated games with public signals, disproving conjectures that the asymptotic value always exists and that more informed players can guarantee it in the long run.

Contribution

It provides the first explicit example of a repeated game where the discounted value does not converge as the discount factor approaches zero, challenging longstanding conjectures.

Findings

The discounted game value does not converge in the example.

Players observe payoffs and play in turn, with perfect action observation.

Counterexamples involve seven states, two actions, and two signals per player.

Abstract

Mertens [In Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986) (1987) 1528-1577 Amer. Math. Soc.] proposed two general conjectures about repeated games: the first one is that, in any two-person zero-sum repeated game, the asymptotic value exists, and the second one is that, when Player 1 is more informed than Player 2, in the long run Player 1 is able to guarantee the asymptotic value. We disprove these two long-standing conjectures by providing an example of a zero-sum repeated game with public signals and perfect observation of the actions, where the value of the $\lambda$-discounted game does not converge when $\lambda$ goes to 0. The aforementioned example involves seven states, two actions and two signals for each player. Remarkably, players observe the payoffs, and play in turn.