Thresholds for sensitive optimality and Blackwell optimality in stochastic games

TL;DR

This paper studies thresholds in stochastic games that determine when strategies are optimal under different criteria, providing new bounds that improve understanding of the complexity involved in computing these strategies.

Contribution

It introduces the first bounds on the $d$-sensitive threshold beyond the mean-payoff case and improves bounds for the Blackwell threshold using advanced algebraic techniques.

Findings

New bounds on the $d$-sensitive threshold $oldsymbol{ ext{alpha}_d}$.

Improved bounds on the Blackwell threshold $oldsymbol{ ext{alpha}_{Bw}}$.

Enhanced understanding of the complexity in computing optimal strategies.

Abstract

We investigate refinements of the mean-payoff criterion in two-player zero-sum perfect-information stochastic games. A strategy is Blackwell optimal if it is optimal in the discounted game for all discount factors sufficiently close to $1$. The notion of $d$-sensitive optimality interpolates between mean-payoff optimality (corresponding to the case $d=-1$) and Blackwell optimality ($d=+\infty$). The Blackwell threshold $\alpha_{\sf Bw} \in [0,1[$ is the discount factor above which all optimal strategies in the discounted game are guaranteed to be Blackwell optimal. The $d$-sensitive threshold $\alpha_{\sf d} \in [0,1[$ is defined analogously. Bounding $\alpha_{\sf Bw}$ and $\alpha_{\sf d}$ are fundamental problems in algorithmic game theory, since these thresholds control the complexity for computing Blackwell and $d$-sensitive optimal strategies, by reduction to discounted games which can be solved in $O\left((1-\alpha)^{-1}\right)$ iterations. We provide the first bounds on the $d$-sensitive threshold $\alpha_{\sf d}$ beyond the case $d=-1$, and we establish improved bounds for the Blackwell threshold $\alpha_{\sf Bw}$. This is achieved by leveraging separation bounds on algebraic numbers, relying on Lagrange bounds and more advanced techniques based on Mahler measures and multiplicity theorems.