# Duality between maximization of expected utility and minimization of relative entropy when probabilities are imprecise

Published

Conference Paper

In this paper we model the problem faced by a riskaverse decision maker with a precise subjective probability distribution who bets against a risk-neutral opponent or invests in a financial market where the beliefs of the opponent or the representative agent in the market are described by a convex set of imprecise probabilities. The problem of finding the portfolio of bets or investments that maximizes the decision maker's expected utility is shown to be the dual of the problem of finding the distribution within the set that minimizes a measure of divergence, i.e., relative entropy, with respect to the decision maker's distribution. In particular, when the decision maker's utility function is drawn from the commonly used exponential/ logarithmic/power family, the solutions of two generic utility maximization problems are shown to correspond exactly to the minimization of divergences drawn from two commonly-used parametric families that both generalize the Kullback-Leibler divergence. We also introduce a new parameterization of the exponential/ logarithmic/power utility functions that allows the power parameter to vary continuously over all real numbers and which is a natural and convenient parameterization for modeling utility gains relative to a non-zero status quo wealth position.

### Duke Authors

### Cited Authors

- Nau, RF; Jose, VRR; Winkler, RL

### Published Date

- December 1, 2009

### Published In

- Isipta 2009 Proceedings of the 6th International Symposium on Imprecise Probability: Theories and Applications

### Start / End Page

- 337 - 346

### Citation Source

- Scopus