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ISIPTA'07 -
FIFTH INTERNATIONAL SYMPOSIUM ON

IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS

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Charles University, Faculty of Mathematicsand Physics

Prague, Czech Republic

16-19 July 2007

## ELECTRONIC PROCEEDINGS

## Robert Nau, Robert Winkler, Victor Richmond Jose

# Entropy minimization and imprecise probabilities

### Abstract

Suppose that a risk-averse expected utility maximizer with a precise probability distribution p bets optimally against a risk neutral opponent (or equivalently an incomplete market for contingent claims) whose beliefs are described by a convex set Q of probability distributions. The utility-maximization problem turns out to be precisely the dual of the problem of finding the distribution q in Q that minimizes a generalized divergence with respect to p. A special case is the one in which the decision maker has logarithmic utility, in which case the divergence is just the Kullback-Leibler divergence, but we present a closed-form solution for the entire family of linear-risk-tolerance (a.k.a. HARA) utility functions and show that this corresponds to a particular parametric family of generalized divergences, which is derived from an entropy measure originally proposed by Arimoto and which is also related to the pseudospherical scoring rule originally proposed by I.J. Good.

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** Authors addresses: **

Robert Nau

Fuqua School of Business

Duke University

Durham, NC 27708-0120

USA

Robert Winkler

Fuqua School of Business

Duke University

Durham, NC 27708

Victor Richmond Jose

Box 90120

Duke University

Durham, NC

27708-0120

** E-mail addresses: **

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