A Tauberian Theorem for Nonexpansive Operators and Applications to Zero-Sum Stochastic Games

Abstract

We prove a Tauberian theorem for nonexpansive operators, and apply it to the model of zero-sum stochastic game. Under mild assumptions, we prove that the value of the λ-discounted game v λ converges uniformly when λ goes to 0 if and only if the value of the n-stage game v n converges uniformly when n goes to infinity. This generalizes the Tauberian theorem of Lehrer and Sorin [6] to the two-player zero-sum case. We also provide the first example of a stochastic game with public signals on the state and perfect observation of actions, with finite state space, signal sets and action sets, in which for some initial state k 1 known by both players, (v λ (k 1 )) and (v n (k 1 )) converge to distinct limits.

Communicating zero-sum product stochastic games

“…If v n and v λ converge as n goes to infinity and λ goes to 0, and the limits are equal, then the game is said to have an asymptotic value. The Tauberian theorem of Ziliotto (2016a) applies in this setting and the asymptotic value exists if (v λ ) λ ∈(0, 1] converges as λ goes to 0.…”

An accretive operator approach to ergodic zero-sum stochastic games

“…Shapley operators. From their analysis, it is possible to infer asymptotic properties of the games (see e.g., Rosenberg and Sorin [RS01], Neymann [Ney03], Sorin [Sor04], Ziliotto [Zil16a]). In this paper, following this so-called "operator approach", we focus on the optimality equation (known as average case optimality equation, Shapley equation or ergodicity equation) T (u) = λe + u, where T : R n → R n is the Shapley operator of a game with n states, and e denotes the unit vector of R n , i.e., the vector whose coordinates are all equal to 1.…”

Tauberian theorems for general iterations of operators: Applications to zero-sum stochastic games

“…Many papers prove convergence of (v n ), (v λ ) and (v θ ) to the same limit in specific Assumption 1. There exists C > 0 such that for all α, β ∈ [0, 1], for all λ, λ ′ ∈ [0, 1], for all f, g ∈ X, The following proposition stems from the proof of Ziliotto [22,Theorem 1.2] (for further details, see Section 2). Proposition 1.…”

“…A natural question is to ask whether there is a link between the convergence of (v n ), (v λ ) and (v θ ) * . Ziliotto [22] has proved that in a very general stochastic game model, with possibly infinitely many states and actions, (v n ) converges uniformly if and only if (v λ ) converges uniformly † . This paper aims at generalizing such a result to a more general family of values (v θ ).…”

“…The contribution of this paper is twofold. In the same stochastic game model as in [22], it provides conditions under which the uniform convergence of (v n ) or (v λ ) implies the uniform convergence of (v θ ). Second, in the case where the state space and the action sets are finite, it proves that the following discounted strategy is asymptotically optimal in the θ-weighted game, as θ 1 goes to 0: at each stage m ≥ 1, play optimally in the discounted game with discount factor θ m /( m ′ ≥m θ m ′ ) ‡ .…”