## The Annals of Applied Probability

### Uniform value in recursive games

#### Abstract

We address the problem of existence of the uniform value in recursive games. We give two existence results: (i) the uniform value is shown to exist if the state space is countable, the action sets are finite and if, for some $a>0$, there are finitely many states in which the limsup value is less than $a$; (ii) for games with nonnegative payoff function, it is sufficient that the action set of player 2 is finite. The finiteness assumption can be further weakened.

#### Article information

Source
Ann. Appl. Probab., Volume 12, Number 4 (2002), 1185-1201.

Dates
First available in Project Euclid: 12 November 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1037125859

Digital Object Identifier
doi:10.1214/aoap/1037125859

Mathematical Reviews number (MathSciNet)
MR1936589

Zentralblatt MATH identifier
1040.91018

Subjects
Primary: 91A15: Stochastic games
Secondary: 91A05: 2-person games

#### Citation

Solan, Eilon; Vieille, Nicolas. Uniform value in recursive games. Ann. Appl. Probab. 12 (2002), no. 4, 1185--1201. doi:10.1214/aoap/1037125859. https://projecteuclid.org/euclid.aoap/1037125859

#### References

• BEWLEY, T. and KOHLBERG, E. (1976). The asy mptotic theory of stochastic games. Math. Oper. Res. 1 197-208.
• EVERETT, H. (1957). Recursive games. In Contributions to the Theory of Games (H. W. Kuhn and A. W. Tucker, eds.) 3 47-78. Princeton Univ. Press.
• LEHRER, E. and SORIN, S. (1992). A uniform Tauberian theorem in dy namic programming. Math. Oper. Res. 17 303-307.
• MAITRA, A. and SUDDERTH, W. (1993). Borel stochastic games with limsup pay off. Ann. Probab. 21 861-885.
• MERTENS, J. F. and NEy MAN, A. (1981). Stochastic games. Internat. J. Game Theory 10 53-66.
• MERTENS, J. F., SORIN S. and ZAMIR, S. (1994). Repeated games. CORE Discussion Papers 9420, 9421, 9422. Louvain-la-Neuve.
• NOWAK, A. S. (1984a). On zero-sum stochastic games with general state space, I. Probab. Math. Statist. 4 13-32.
• NOWAK, A. S. (1984b). On zero-sum stochastic games with general state space, II. Probab. Math. Statist. 4 143-152.
• NOWAK, A. S. (1985). Universally measurable strategies in zero-sum stochastic games. Ann. Probab. 13 269-287.
• ROSENBERG, D., SOLAN, E. and VIEILLE, N. (2001). Stopping games with randomized strategies. Probab. Theory Related Fields 119 433-451.
• ROSENBERG, D. and VIEILLE, N. (2000). The maxmin of recursive games with incomplete information on one side. Math. Oper. Res. 25 23-35.
• SECCHI, P. (1997). Stationary strategies for recursive games. Math. Oper. Res. 22 494-512.
• THUIJSMAN, F. and VRIEZE, K. (1992). A note on recursive games. Game Theory and Economic Applications. Lecture Notes in Econom. and Math. Sy stems 389 133-145. Springer, Berlin.
• EVANSTON, ILLINOIS 60208 AND SCHOOL OF MATHEMATICAL SCIENCES TEL AVIV UNIVERSITY TEL AVIV 69978 ISRAEL E-MAIL: e-solan@northwestern.edu DÉPARTEMENT ECONOMIE ET FINANCE HEC SCHOOL OF MANAGEMENT 78 351 JOUY EN JOSAS FRANCE E-MAIL: vieille@hec.fr