Markov decision processes with observation costs: framework and computation with a penalty scheme

TL;DR

This paper develops a framework for Markov decision processes with observation costs, formulating the problem as quasi-variational inequalities and proposing penalty methods for accurate solutions, with applications demonstrated through numerical experiments.

Contribution

It introduces a novel approach to MDPs with observation costs using QVIs and penalty methods, extending existing models to include observation timing and uncertainty.

Findings

Comparison principle ensures solution uniqueness.

Penalty methods achieve high-accuracy solutions.

Numerical experiments validate the framework's effectiveness.

Abstract

We consider Markov decision processes where the state of the chain is only given at chosen observation times and of a cost. Optimal strategies involve the optimisation of observation times as well as the subsequent action values. We consider the finite horizon and discounted infinite horizon problems, as well as an extension with parameter uncertainty. By including the time elapsed from observations as part of the augmented Markov system, the value function satisfies a system of quasi-variational inequalities (QVIs). Such a class of QVIs can be seen as an extension to the interconnected obstacle problem. We prove a comparison principle for this class of QVIs, which implies uniqueness of solutions to our proposed problem. Penalty methods are then utilised to obtain arbitrarily accurate solutions. Finally, we perform numerical experiments on three applications which illustrate our framework.