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ISIPTA'09 -
SIXTH INTERNATIONAL SYMPOSIUM ON

IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS

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Durham University, Department of Mathematical Sciences

Durham, United Kingdom

Tuesday 14 to Saturday 18 July 2009

## ELECTRONIC PROCEEDINGS

## Robert Nau, Victor Richmond Jose, Robert Winkler

# Duality Between Maximization of Expected Utility and Minimization of Relative Entropy When Probabilities are Imprecise

### Abstract

In this paper we model the problem faced by a risk-averse 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

** Keywords. ** decision theory, decision analysis, relative entropy, utility theory, imprecise probabilities, portfolio optimization

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** Authors addresses: **

**Robert Nau**

Fuqua School of Business

Duke University

Durham, NC 27708-0120

USA

**Victor Richmond Jose**

Box 90120

Duke University

Durham, NC

27708-0120

**Robert Winkler**

Fuqua School of Business

Duke University

Durham, NC 27708

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