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Scoring Rules, Generalized Entropy, and Utility Maximization

The Fuqua School of Business, Duke University, Durham, North Carolina 27708
The Fuqua School of Business, Duke University, Durham, North Carolina 27708
The Fuqua School of Business, Duke University, Durham, North Carolina 27708
vrj{at}duke.edu
rnau{at}duke.edu
rwinkler{at}duke.edu

Information measures arise in many disciplines, including forecasting (where scoring rules are used to provide incentives for probability estimation), signal processing (where information gain is measured in physical units of relative entropy), decision analysis (where new information can lead to improved decisions), and finance (where investors optimize portfolios based on their private information and risk preferences). In this paper, we generalize the two most commonly used parametric families of scoring rules and demonstrate their relation to well-known generalized entropies and utility functions, shedding new light on the characteristics of alternative scoring rules as well as duality relationships between utility maximization and entropy minimization. In particular, we show that weighted forms of the pseudospherical and power scoring rules correspond exactly to measures of relative entropy (divergence) with convenient properties, and they also correspond exactly to the solutions of expected utility maximization problems in which a risk-averse decision maker whose utility function belongs to the linear-risk-tolerance family interacts with a risk-neutral betting opponent or a complete market for contingent claims in either a one-period or a two-period setting. When the market is incomplete, the corresponding problems of maximizing linear-risk-tolerance utility with the risk-tolerance coefficient β are the duals of the problems of minimizing the pseudospherical or power divergence of order β between the decision maker's subjective probability distribution and the set of risk-neutral distributions that support asset prices.

Subject classifications: decision analysis; theory; probability; entropy; utility/preference; theory; finance; portfolio.
History: Received September 2006; revision received March 2007; accepted August 2007.