Durham University, Department of Mathematical Sciences
Durham, United Kingdom
Tuesday 14 to Saturday 18 July 2009


Robert Nau, Victor Richmond Jose, Robert Winkler

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


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

Victor Richmond Jose
Box 90120
Duke University
Durham, NC

Robert Winkler
Fuqua School of Business
Duke University
Durham, NC 27708

E-mail addresses:

Robert Nau
Victor Richmond Jose
Robert Winkler

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