Sharper bounds on the thresholds for sensitive and Blackwell optimality

2026-10-07

AI Summary

This study addresses the imprecise bounds of Blackwell and sensitive optimality thresholds in stochastic games, as well as the absence of such bounds in multichain settings. By leveraging an algebraic analysis framework based on reduced polynomial families, combined with classical Mahler and Cauchy root separation techniques, this work systematically derives and improves both upper and lower bounds for these optimality thresholds. Through controlling the degree and height of minimal polynomials, it establishes, for the first time, tight bounds for sensitive thresholds in multichain stochastic environments. Ultimately, this project yields the sharpest known upper and lower bounds for Blackwell thresholds in the existing literature and fills a critical theoretical gap regarding sensitive threshold bounds in multichain settings, thereby achieving a precise characterization of optimality thresholds.

Abstract

In perfect-information two-player stochastic games, the notions of Blackwell and sensitive optimality provide generalizations of the classical mean-payoff optimality and discount optimality criteria to account for more farsighted preferences. We provide bounds on the Blackwell threshold $α_{\sf bw}$ and the $d$-sensitive thresholds $α_{\sf d}$, defined as the smallest discount factors above which discount optimal policies coincide with Blackwell optimal policies and $d$-sensitive optimal policies respectively. Our refined bounds improve upon prior work by focusing on ``reduced'' families of polynomials and, crucially, our bounds are tight in terms of controlling the degrees and heights of the minimal polynomials of the Blackwell thresholds. We apply classical root separation based on Mahler's and Cauchy's bounds to our reduced families to derive the strongest upper and lower bounds on $α_{\sf bw}$ in the literature, and we are the first to obtain bounds on $α_{\sf d}$ in the multichain stochastic setting.

Problem

• stochastic games
• Blackwell optimality
• sensitive optimality
• threshold bounds
• discount factor

Innovation

• Blackwell optimality
• sensitive optimality
• stochastic games
• root separation
• reduced polynomials