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Comparative probability for conditional events: A new look through coherence. (English) Zbl 0785.90006

Summary: We study, from the standpoint of coherence, comparative probabilities on an arbitrary family \(\mathcal E\) of conditional events. Given a binary relation \(\prec\cdot\), coherence conditions on \(\prec\cdot\) 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 \(\prec\cdot\). 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 \(\mathcal E\) 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.

MSC:

91B08 Individual preferences
91B16 Utility theory
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