Recursive games: Uniform value, Tauberian theorem and the Mertens conjecture "$Maxmin=\lim v_n=\lim v_\lambda$"

TL;DR

This paper establishes conditions for the existence of the uniform value in recursive games, proves a Tauberian theorem linking different types of convergence, and confirms the Mertens conjecture in certain informational settings.

Contribution

It introduces a uniform Tauberian theorem for recursive games and proves the Mertens conjecture when the maximizer has more information than the minimizer.

Findings

Existence of the uniform value under total boundedness of $v_n$

Uniform convergence of $(v_n)$ and $(v_ extlambda)$ are equivalent

Validation of the Mertens conjecture in games with asymmetric information

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 $\{v_n,n\geq 1\}$ 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 $(v_\lambda)$ 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 $Maxmin=\lim_{n\to\infty} v_n=\lim_{\lambda\to 0}v_\lambda$. Finally, we deduce the existence of the uniform value in finite recursive game with symmetric information.