Recursive games: Uniform value, Tauberian theorem and the Mertens conjecture "Maxmin=lim v_n=lim v_λ"

2015/06/02 by Xiaoxi Li, Li, Xiaoxi, Xavier Venel +1
Computer Science · Decision Sciences · Economics, Econometrics and Finance · Mathematics · #Computability, Logic, AI Algorithms #Economic theories and models #FOS: Mathematics #Game Theory and Applications #Optimization and Control (math.OC) #Probability (math.PR) #math.OC #math.PR

paper · pdf · doi:10.48550/arxiv.1506.00949

32 pages

arxiv created 2015/06/02 · openalex publication_date 2015/06/02 · arxiv updated 2015/06/03 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

We study two-player zero-sum recursive games with a countable state space and finite action spaces at each state. When the family of n-stage values \v_n,n≥ 1\ is totally bounded for the uniform norm, we prove the existence of the uniform value. Together with a result in Rosenberg and Vieille (2000), we obtain a uniform Tauberian theorem for recursive games: (v_n) converges uniformly if and only if (v_λ) converges uniformly. We apply our main result to finite recursive games with signals (where players observe only signals on the state and on past actions). When the maximizer is more informed than the minimizer, we prove the Mertens conjecture Maxmin=lim_n→∞ v_n=lim_{λ→ 0}v_λ. Finally, we deduce the existence of the uniform value in finite recursive game with symmetric information.

Citations

• Existence of the uniform value in repeated games with a more informed controller
• A Tauberian theorem for nonexpansive operators and applications to zero-sum stochastic games

Related

• Uniform Value for Recursive Games with Compact Actions
• A Tauberian theorem for nonexpansive operators and applications to zero-sum stochastic games
• On Uniform Tauberian Theorems for Dynamic Games
• Mertens conjectures in absorbing games with incomplete information
• Tauberian theorems for general iterations of operators: applications to zero-sum stochastic games