Proper scoring rules, dominated forecasts, and coherence

The concept of coherent probabilities and conditional probabilities through a gambling argument and through a parallel argument based on a quadratic scoring rule was introduced by de Finetti (de Finetti, B. 1974. The Theory of Probability. John Wiley & Sons, New York). He showed that the two arguments lead to the same concept of coherence. When dealing with events only, there is a rich class of scoring rules that might be used in place of the quadratic scoring rule. We give conditions under which a general strictly proper scoring rule can replace the quadratic scoring rule while preserving the equivalence of de Finetti's two arguments. In proving our results, we present a strengthening of the usual minimax theorem. We also present generalizations of de Finetti's fundamental theorem of probability to deal with conditional probabilities. Key words: Brier score; finite additivity History: Received on November 14, 2008. Accepted on June 22, 2009, after 1 revision. Published online in Articles in Advance October 13, 2009. DOI 10.1287/deca.1090.0153
1. Introduction 1.1. Background There are two different elicitation methods that lead to subjective probability. The first is through fair betting odds. What is a price at which you would [...]