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March 2016 Zero-sum repeated games: Counterexamples to the existence of the asymptotic value and the conjecture $\operatorname{maxmin}=\operatorname{lim}v_{n}$
Bruno Ziliotto
Ann. Probab. 44(2): 1107-1133 (March 2016). DOI: 10.1214/14-AOP997

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.

Citation

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Bruno Ziliotto. "Zero-sum repeated games: Counterexamples to the existence of the asymptotic value and the conjecture $\operatorname{maxmin}=\operatorname{lim}v_{n}$." Ann. Probab. 44 (2) 1107 - 1133, March 2016. https://doi.org/10.1214/14-AOP997

Information

Received: 1 December 2013; Revised: 1 December 2014; Published: March 2016
First available in Project Euclid: 14 March 2016

zbMATH: 1344.91006
MathSciNet: MR3474468
Digital Object Identifier: 10.1214/14-AOP997

Subjects:
Primary: 91A20
Secondary: 91A05, 91A15

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 2 • March 2016
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