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:
| Robert Nau | robert.nau@duke.edu |
| Robert Winkler | rwinkler@duke.edu |
| Victor Richmond Jose | vrj@duke.edu |