Partially Observable Risk-Sensitive Markov Decision Processes

TL;DR

This paper addresses risk-sensitive decision-making in partially observable Markov decision processes by embedding the problem into an observable MDP with an extended state space, providing conditions for optimal policies and simplifying the exponential utility case.

Contribution

It introduces a novel approach to solve risk-sensitive POMDPs without change of measure techniques by extending the state space with joint distributions.

Findings

Optimal policies exist under certain conditions.

The exponential utility case simplifies the problem significantly.

Numerical example illustrates the impact of certainty equivalent parameters.

Abstract

We consider the problem of minimizing a certainty equivalent of the total or discounted cost over a finite and an infinite time horizon which is generated by a Partially Observable Markov Decision Process (POMDP). The certainty equivalent is defined by $U^{-1}(EU(Y))$ where $U$ is an increasing function. In contrast to a risk-neutral decision maker this optimization criterion takes the variability of the cost into account. It contains as a special case the classical risk-sensitive optimization criterion with an exponential utility. We show that this optimization problem can be solved by embedding the problem into a completely observable Markov Decision Process with extended state space and give conditions under which an optimal policy exists. The state space has to be extended by the joint conditional distribution of current unobserved state and accumulated cost. In case of an exponential utility, the problem simplifies considerably and we rediscover what in previous literature has been named information state. However, since we do not use any change of measure techniques here, our approach is simpler. A small numerical example, namely the classical repeated casino game with unknown success probability is considered to illustrate the influence of the certainty equivalent and its parameters.