Coherent Preferences

Abstract

De Finetti has defined coherent previsions and coherent probabilities, and others have described concepts of coherent actions or coherent decisions. Here we consider a related concept of coherent preferences. Willingness to accept one side of a bet is an example of a preference. A set of preferences is called incoherent if reversal of some subset yields a uniform increase in utility, as with a sure win for a collection of bets. In both probability and statistical models (where preferences are conditional on data) separating hyperplane theorems show that coherence implies existence of a probability measure from which the preferences could have been inferred. Relationships to confidence intervals and to decision theory are indicated. No single definition of coherence is given which covers all cases of interest. The various cases distinguish between probability and statistical models and between finite and infinite spaces. No satisfactory theory is given for continuous statistical models.