Search

We consider how an unconditional, finite-valued, finitely additive probability P on a countable set may localize its non-conglomerability (non-disintegrability). Non-conglomerability, a characteristic of merely finitely additive probability, occurs when the unconditional probability of an event P(E) lies outside the closed interval of conditional probability values, [infhe pi P(E|h), suphe pp(E|h)], taken from a countable partition p = hj:j=1,...}. The problem we address is how to identify events and partitions where a finite-valued, finitely additive probability fails to satisfy conglomerability. We focus on the extreme case of 2-valued finitely additive probabilities that are not countably additive. These are, equivalently, non-principal ultrafilters. Evidently, the challenge we face is that given a countable partition, at most one of its elements has positive probability under P. Thus, we must find ways of regulating the coherent conditional probabilities, given null events, that cohere with the unconditional probability P. Our analysis of P proceeds by the use of combinatorial properties of the associated non-principal ultrafilter UP. We show that when ultrafilter UP is not minimal in the Rudin-Keisler partial order of b(w)\w, we may locate a partition in which P fails to satisfy the conglomerability principle by examining (at most) countably many partitions. This result is then applied to finitely additive probabilities that assume only finitely many values. By contrast, if ultrafilter UP is Rudin-Keisler minimal, then P is simultaneously conglomerable in each finite collection of partitions, though not simultaneously conglomerable in all partitions.

Let κ be an uncountable cardinal. Using the theory of conditional probability associated with de Finetti (1974) and Dubins (1975), subject to several structural assumptions for creating sufficiently many measurable sets, and assuming that κ is not a weakly inaccessible cardinal, we show that each probability that is not κ-­additive has conditional probabilities that fail to be conglomerable in a partition of cardinality no greater than κ. This generalizes our (1984) result, where we established that each finite but not countably additive probability has conditional probabilities that fail to be conglomerable in some countable partition.

We give necessary and sufficient conditions for a scoring rule to be proper (or strictly proper) for a quantile if utility is linear, and the distribution is unrestricted. We also give results when the set of distributions is limited, for example, to distributions that have first moments.