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.