敏感最优性与Blackwell最优性阈值的更紧界

中文摘要 AI 辅助

在完美信息的双人随机博弈中，Blackwell最优性与敏感最优性概念将经典的平均收益最优性和折扣最优性准则推广到更具远见的偏好。我们给出了Blackwell阈值$\alpha_{\sf bw}$和$d$-敏感阈值$\alpha_{\sf d}$的界，这些阈值定义为最小的折扣因子，超过该折扣因子时，折扣最优策略分别与Blackwell最优策略和$d$-敏感最优策略一致。我们改进的界通过聚焦于多项式的“约化”族而优于先前工作，并且关键的是，我们的界在控制Blackwell阈值的最小多项式的次数和高度方面是紧的。我们将基于Mahler和Cauchy界的经典根分离应用于我们的约化族，以推导文献中关于$\alpha_{\sf bw}$的最强上下界，并且我们是首个在多链随机设置中获得$\alpha_{\sf d}$的界的工作。

英文摘要

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.

发表机构

• INRIA（法国国家信息与自动化研究所）
• CMAP
• Ecole Polytechnique（巴黎综合理工学院）
• ISOM Department（ISOM系）
• HEC Paris（巴黎高等商学院）
• CIFASIS-CONICET（CIFASIS-阿根廷国家科学与技术研究理事会）

机构由 AI 辅助整理，请以论文原文为准。