## 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

#### 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
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

#### 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

#### References

• Aumann, R. J. and Maschler, M. B. (1995). Repeated Games with Incomplete Information. MIT Press, Cambridge, MA.
• Bewley, T. and Kohlberg, E. (1976). The asymptotic theory of stochastic games. Math. Oper. Res. 1 197–208.
• Forges, F. (1982). Infinitely repeated games of incomplete information: Symmetric case with random signals. Internat. J. Game Theory 11 203–213.
• Gensbittel, F., Oliu-Barton, M. and Venel, X. (2014). Existence of the uniform value in repeated games with a more informed controller. Journal of Dynamics and Games 1 411–445.
• Kohlberg, E. and Zamir, S. (1974). Repeated games of incomplete information: The symmetric case. Ann. Statist. 2 1040–1041.
• Mertens, J.-F. (1987). Repeated games. In Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986) 1528–1577. Amer. Math. Soc., Providence, RI.
• Mertens, J.-F. and Neyman, A. (1981). Stochastic games. Internat. J. Game Theory 10 53–66.
• Mertens, J. F., Sorin, S. and Zamir, S. (1994). Repeated Games. CORE DP 9420-22.
• Mertens, J.-F. and Zamir, S. (1971). The value of two-person zero-sum repeated games with lack of information on both sides. Internat. J. Game Theory 1 39–64.
• Mertens, J.-F. and Zamir, S. (1985). Formulation of Bayesian analysis for games with incomplete information. Internat. J. Game Theory 14 1–29.
• Neyman, A. (2008). Existence of optimal strategies in Markov games with incomplete information. Internat. J. Game Theory 37 581–596.
• Philippou, A. N., Georghiou, C. and Philippou, G. N. (1983). A generalized geometric distribution and some of its properties. Statist. Probab. Lett. 1 171–175.
• Renault, J. (2006). The value of Markov chain games with lack of information on one side. Math. Oper. Res. 31 490–512.
• Renault, J. (2012). The value of repeated games with an informed controller. Math. Oper. Res. 37 154–179.
• Rosenberg, D. (2000). Zero sum absorbing games with incomplete information on one side: Asymptotic analysis. SIAM J. Control Optim. 39 208–225.
• Rosenberg, D., Solan, E. and Vieille, N. (2002). Blackwell optimality in Markov decision processes with partial observation. Ann. Statist. 30 1178–1193.
• Rosenberg, D., Solan, E. and Vieille, N. (2003). The maxmin value of stochastic games with imperfect monitoring. Internat. J. Game Theory 32 133–150.
• Rosenberg, D., Solan, E. and Vieille, N. (2004). Stochastic games with a single controller and incomplete information. SIAM J. Control Optim. 43 86–110.
• Rosenberg, D. and Vieille, N. (2000). The maxmin of recursive games with incomplete information on one side. Math. Oper. Res. 25 23–35.
• Shapley, L. S. (1953). Stochastic games. Proc. Natl. Acad. Sci. USA 39 1095–1100.
• Sorin, S. (1984). “Big match” with lack of information on one side. I. Internat. J. Game Theory 13 201–255.
• Sorin, S. (1985). “Big match” with lack of information on one side. II. Internat. J. Game Theory 14 173–204.
• Sorin, S. (2002). A First Course on Zero-Sum Repeated Games. Mathématiques & Applications (Berlin) 37. Springer, Berlin.
• Venel, X. (2015). Commutative stochastic games. Math. Oper. Res. 40 403–428.
• Vigeral, G. (2013). A zero-zum stochastic game with compact action sets and no asymptotic value. Dyn. Games Appl. 3 172–186.