Definable Zero-Sum Stochastic Games

• 成果类型: Article
• 署名作者: Bolte, Jerome; Gaubert, Stephane; Vigeral, Guillaume
• 署名单位: Universite de Toulouse; Universite Toulouse 1 Capitole; Inria; Institut Polytechnique de Paris; Ecole Polytechnique; Universite PSL; Universite Paris-Dauphine
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2014.0666
• 发表日期: 2015
• 页码: 171-191
• 关键词: Incomplete information; nonexpansive-mappings; Absorbing games; one side; optimization; EQUATIONS; sets

摘要

Definable zero-sum stochastic games involve a finite number of states and action sets, and reward and transition functions, that are definable in an o-minimal structure. Prominent examples of such games are finite, semi-algebraic, or globally subanalytic stochastic games. We prove that the Shapley operator of any definable stochastic game with separable transition and reward functions is definable in the same structure. Definability in the same structure does not hold systematically: we provide a counterexample of a stochastic game with semi-algebraic data yielding a non-semi-algebraic but globally subanalytic Shapley operator. Our definability results on Shapley operators are used to prove that any separable definable game has a uniform value; in the case of polynomially bounded structures, we also provide convergence rates. Using an approximation procedure, we actually establish that general zero-sum games with separable definable transition functions have a uniform value. These results highlight the key role played by the tame structure of transition functions. As particular cases of our main results, we obtain that stochastic games with polynomial transitions, definable games with finite actions on one side, and definable games with perfect information or switching controls have a uniform value. Applications to nonlinear maps arising in risk sensitive control and Perron-Frobenius theory are also given.