# Stochastic Comparative Statics in Markov Decision Processes

## TL;DR

This paper introduces stochastic comparative statics in Markov decision processes, analyzing how the distribution of future optimal decisions responds to parameter changes in multi-period stochastic optimization.

## Contribution

It develops a framework for stochastic comparative statics, providing new results on how optimal decisions vary with parameters in Markov decision processes.

## Key findings

- Optimal decisions change predictably with payoff and transition parameters.
- Results apply to economic models like investment and pricing.
- Provides insights into stationary distribution comparisons.

## Abstract

In multi-period stochastic optimization problems, the future optimal decision is a random variable whose distribution depends on the parameters of the optimization problem. We analyze how the expected value of this random variable changes as a function of the dynamic optimization parameters in the context of Markov decision processes. We call this analysis \emph{stochastic comparative statics}. We derive both \emph{comparative statics} results and \emph{stochastic comparative statics} results showing how the current and future optimal decisions change in response to changes in the single-period payoff function, the discount factor, the initial state of the system, and the transition probability function. We apply our results to various models from the economics and operations research literature, including investment theory, dynamic pricing models, controlled random walks, and comparisons of stationary distributions.