AI 中文总结
本文研究渗流马尔可夫决策过程，证明一致值与0-最优策略存在性，并引入维度提升性质以近似渗流临界阈值。
AI 中文摘要
我们研究渗流马尔可夫决策过程（PMDPs），其中决策者在d维整数格上反复移动一个令牌，转移是确定性的，而边被赋予随机收益。收益在决策过程开始前被揭示，决策者的目标是在固定时间范围内最大化平均累积收益。我们证明了当时间范围趋于无穷时，一致值的存在性以及0-最优策略的存在性。在伯努利收益的特殊情形下，我们建立了一致值的连续性结果。我们还给出了几个一维例子，以说明最优策略可能非常复杂。然后，我们引入了一个维度提升性质，该性质允许PMDPs被低维的PMDPs近似，并可用于近似渗流理论中的临界概率阈值。
英文摘要
We study Percolation Markov Decision Processes (PMDPs), in which the decision maker repeatedly moves a token through the d-dimensional integer lattice with deterministic transitions and random payoffs assigned to the edges. Payoffs are revealed before the beginning of the decision process and the decision maker aims to maximize the average accumulated payoff over a fixed horizon. We establish the existence of the uniform value (as the horizon tends to infinity) and 0-optimal strategies. In the particular case of Bernoulli payoffs, we establish continuity results for the uniform value. We also present several one-dimensional examples to illustrate that optimal strategies may be very complex. Then, we introduce a dimensional lifting property that allows PMDPs to be approximated by PMDPs of lower dimension and could be used to approximate critical probability thresholds in percolation theory.
