Acyclic Gambling Games

33 Pages Posted: 30 May 2018

See all articles by Rida Laraki

Rida Laraki

Université Paris-Dauphine, PSL Research University

Jerome Renault

University of Toulouse 1 - Toulouse School of Economics (TSE)

Date Written: May 30, 2018


We consider 2-player zero-sum stochastic games where each player controls his own state variable living in a compact metric space. The terminology comes from gambling problems where the state of a player represents its wealth in a casino. Under natural assumptions (such as continuous running payoff and non expansive transitions), we consider for each discount factor the value vλ of the λ-discounted stochastic game and investigate its limit when λ goes to 0 (players are more and more patient). We show that under a new acyclicity condition, the limit exists and is characterized as the unique solution of a system of functional equations: the limit is the unique continuous excessive and depressive function such that each player, if his opponent does not move, can reach the zone when the current payoff is at least as good than the limit value, without degrading the limit value. The approach generalizes and provides a new viewpoint on the Mertens-Zamir system coming from the study of zero-sum repeated games with lack of information on both sides. A counterexample shows that under a slightly weaker notion of acyclicity, convergence of (vλ) may fail.

Keywords: Markov Decision Processes, Zero-Sum Stochastic Games, Asymptotic Value, Gambling Houses, Mertens-Zamir System, Splitting Games, Persuasion

Suggested Citation

Laraki, Rida and Renault, Jerome, Acyclic Gambling Games (May 30, 2018). Becker Friedman Institute for Research in Economics Working Paper No. 2018-34, Available at SSRN: or

Rida Laraki (Contact Author)

Université Paris-Dauphine, PSL Research University ( email )

Place du Maréchal de Lattre de Tassigny
Paris, 75016

Jerome Renault

University of Toulouse 1 - Toulouse School of Economics (TSE) ( email )

Place Anatole-France
Toulouse Cedex, F-31042

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