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

TL;DR

This paper establishes Tauberian theorems for operator iterations and applies them to analyze convergence and strategy construction in zero-sum stochastic games with weighted payoffs, especially as players become more patient.

Contribution

It introduces new Tauberian theorems for general operator iterations and applies them to derive conditions for convergence and strategy design in weighted stochastic games.

Findings

Asymptotic value existence implies convergence of weighted game values.

Constructs asymptotically optimal strategies based on discounted game solutions.

Provides conditions for convergence in finite state and action stochastic games.

Abstract

This paper proves several Tauberian theorems for general iterations of operators, and provides two applications to zero-sum stochastic games where the total payoff is a weighted sum of the stage payoffs. The first application is to provide conditions under which the existence of the asymptotic value implies the convergence of the values of the weighted game, as players get more and more patient. The second application concerns stochastic games with finite state space and action sets. This paper builds a simple class of asymptotically optimal strategies in the weighted game, that at each stage play optimally in a discounted game with a discount factor corresponding to the relative weight of the current stage.