Partially Observable Risk-Sensitive Markov Decision Processes

Abstract

We consider the problem of minimizing a certainty equivalent of the total or discounted cost over a finite and an infinite time horizon that is generated by a partially observable Markov decision process (POMDP). 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 simple example, namely, a risk-sensitive Bayesian house selling problem, is considered to illustrate our results.