Extensions of the Cav(u) Theorem for Repeated Games with Incomplete Information on One Side

• 成果类型: Article
• 署名作者: Gensbittel, Fabien
• 署名单位: Universite de Toulouse; Universite Toulouse 1 Capitole
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2014.0658
• 发表日期: 2015
• 页码: 80-104
• 关键词: partial-differential-equations; hamilton-jacobi equations; VISCOSITY SOLUTIONS; continuous-time; formulas

摘要

This work is devoted to extend several asymptotic results concerning repeated games with incomplete information on one side. The model we consider is a generalization of the classical model of Aumann and Maschler (Aumann et al. [Aumann RJ, Maschler M, Stearns RE (1995) Repeated Games with Incomplete Information (MIT Press, Cambridge, MA)]) to infinite action spaces and partial information. We prove an extension of the classical Cav(u) Theorem in this model for both the lower and upper value functions using two different methods: respectively a probabilistic method based on martingales and a functional one based on approximation schemes for viscosity solutions of Hamilton Jacobi equations similar to the dual differential approach of Laraki [Laraki R (2002) Repeated games with lack of information on one side: The dual differential approach. Math. Oper. Res. 27(2): 419-440]. Moreover, we show that solutions of these two asymptotic problems provide asymptotically optimal strategies for both players in any game of length n. All these results are based on a compact approach, which consists in identifying a continuous-time problem defined on the time interval [0, 1] representing the limit of a sequence of finitely repeated games, as the number of repetitions is going to infinity. Finally, our results imply the existence of the uniform value of the infinitely repeated game whenever the value of the non-revealing game exists.