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Acyclic Gambling Games

Laraki, Rida and Renault, Jérôme (2019) Acyclic Gambling Games. Mathematics of Operations Research. (In Press)

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Abstract

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 standard assumptions (e.g. continuous running payoff and nonexpansive 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. 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 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 (vλ) may fail.

Item Type: Article
Language: English
Date: 2019
Refereed: Yes
Subjects: B- ECONOMIE ET FINANCE
Divisions: TSE-R (Toulouse)
Site: UT1
Date Deposited: 25 Jul 2019 11:54
Last Modified: 25 Jul 2019 11:54
OAI ID: oai:tse-fr.eu:123247
URI: http://publications.ut-capitole.fr/id/eprint/32646

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