Comparative probability for conditional events: A new look through coherence

Abstract

We study, from the standpoint of coherence, comparative probabilities on an arbitrary family E of conditional events. Given a binary relation ≤·, coherence conditions on ≤· are related to de Finetti's coherent betting system: we consider their connections to the usual properties of comparative probability and to the possibility of numerical representations of ≤·. In this context, the numerical reference frame is that of de Finetti's coherent subjective conditional probability, which is not introduced (as in Kolmogoroff's approach) through a ratio between probability measures. Another relevant feature of our approach is that the family & need not have any particular algebraic structure, so that the ordering can be initially given for a few conditional events of interest and then possibly extended by a step-by-step procedure, preserving coherence. © 1993 Kluwer Academic Publishers.