The Annals of Probability

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

Bruno Ziliotto

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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.

Article information

Source
Ann. Probab., Volume 44, Number 2 (2016), 1107-1133.

Dates
Received: December 2013
Revised: December 2014
First available in Project Euclid: 14 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1457960392

Digital Object Identifier
doi:10.1214/14-AOP997

Mathematical Reviews number (MathSciNet)
MR3474468

Zentralblatt MATH identifier
1344.91006

Subjects
Primary: 91A20: Multistage and repeated games
Secondary: 91A05: 2-person games 91A15: Stochastic games

Keywords
Repeated games asymptotic value public signals symmetric information stochastic games

Citation

Ziliotto, Bruno. Zero-sum repeated games: Counterexamples to the existence of the asymptotic value and the conjecture $\operatorname{maxmin}=\operatorname{lim}v_{n}$. Ann. Probab. 44 (2016), no. 2, 1107--1133. doi:10.1214/14-AOP997. https://projecteuclid.org/euclid.aop/1457960392


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