Acyclic Gambling Games



Laraki, Rida ORCID: 0000-0002-4898-2424 and Renault, Jerome
(2020) Acyclic Gambling Games. MATHEMATICS OF OPERATIONS RESEARCH, 45 (4). pp. 1237-1257.

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Abstract

<jats:p> We consider two-player, zero-sum stochastic games in which each player controls the player’s own state variable living in a compact metric space. The terminology comes from gambling problems in which the state of a player represents its wealth in a casino. Under standard assumptions (e.g., continuous running payoff and nonexpansive transitions), we consider for each discount factor the value v<jats:sub>λ</jats:sub> of the λ-discounted stochastic game and investigate its limit when λ goes to zero. 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 the player’s opponent does not move, can reach the zone when the current payoff is at least as good as 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 (v<jats:sub>λ</jats:sub>) may fail. </jats:p>

Item Type: Article
Uncontrolled Keywords: zero-sum stochastic games, Markov decision process, asymptotic and uniform value, gambling theory, Mertens-Zamir system, splitting games
Depositing User: Symplectic Admin
Date Deposited: 22 Jul 2019 08:21
Last Modified: 03 Apr 2023 21:48
DOI: 10.1287/moor.2019.1030
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3050153