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Reducible Markov Decision Processes and Stochastic Games

机译：还原马尔可夫决策过程和随机游戏

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摘要

Markov decision processes (MDPs) provide a powerful framework for analyzing dynamic decision making. However, their applications are significantly hindered by the difficulty of obtaining solutions. In this study, we introduce reducible MDPs whose exact solutions can be obtained by solving simpler MDPs, termed the coordinate MDPs. The value function and an optimal policy of a reducible MDP are linear functions of those of the associated coordinate MDP. Because the coordinate MDP does not involve the multi-dimensional endogenous state, we achieve dimension reduction on a reducible MDP. Extending the MDP framework to multiple players, we introduce reducible stochastic games. We show that these games reduce to simpler coordinate games that do not involve the multi-dimensional endogenous state. We specify sufficient conditions for the existence of a pure-strategy Markov perfect equilibrium in reducible stochastic games and derive closed-form expressions for the players' equilibrium values. The reducible framework encompasses a variety of linear and nonlinear models and offers substantial simplification in analysis and computation. We provide guidelines and illustrative examples on formulating problems as reducible models. We demonstrate the applicability and modeling flexibility of reducible models in a wide range of contexts including capacity and inventory management and duopoly competition.
机译：马尔可夫决策过程（MDPS）为分析动态决策提供了强大的框架。但是，它们的应用是通过获得解决方案的难度来显着阻碍。在这项研究中，我们引入了可通过求解更简单的MDP来获得精确解决方案的可还原MDP，称为坐标MDP。 Repucible MDP的值函数和最佳策略是关联坐标MDP的线性函数。因为坐标MDP不涉及多维内源状态，所以我们在可还原MDP上实现尺寸减少。将MDP框架扩展到多个玩家，我们推出可减少的随机游戏。我们表明这些游戏减少了更简单的坐标游戏，不涉及多维内源状态。我们为纯策略Markov的存在条件提供了足够的条件，在可还原随机游戏中获得完美均衡，并导出用于球员的平衡值的闭合表达。可还原框架包括各种线性和非线性模型，并在分析和计算方面提供了大量的简化。我们提供关于将问题制定为可还原型号的指南和说明性示例。我们展示了在各种背景下还原模型的适用性和建模灵活性，包括能力和库存管理和Duoly竞争。

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