Proper scoring rules, dominated forecasts, and coherence. Decision Analysis 6(4

De Finetti introduced the concept of coherent previsions and conditional pre-visions through a gambling argument and through a parallel argument based on a quadratic scoring rule. He shows that the two arguments lead to the same concept of coherence. When dealing with events only, there is a rich class of scoring rules which 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 prevision to deal with conditional previsions. 1. Introduction. Chapter 3 of Theory of Probability (de Finetti, 1974, vol. 1) pro-vides two criteria of coherence for previsions (or probabilistic forecasts) over a set of bounded random variables. The first criterion is formulated under the assumption that pre-visions serve as fair prices for buying and/or selling contracts of the form c(X − P (X))