Reducible Markov Decision Processes and Stochastic Games

Abstract

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 paper, we introduce reducible MDPs whose exact solution can be obtained by solving a simpler MDP, termed the coordinate MDP. The value function and an optimal policy of a reducible MDP are linear functions of those of the coordinate MDP. The coordinate MDP does not involve the multi-dimensional endogenous state. Thus, we achieve dimension reduction on the reducible MDP by solving the coordinate 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 for formulating problems as reducible models and illustrate ways to transform a model into the reducible framework. We demonstrate the applicability and modeling flexibility of reducible models in a wide range of contexts including capacity and inventory management and duopoly competition.

Keywords: Markov decision processes, stochastic games, dimension reduction, applications