Finite Linear Qualitative Probability

Abstract

Many years ago, Bruno de Finetti asked whether certain axioms for a comparative probability ordering relation ≻ on the subsets of ann-element set that are necessary for the existence of an order-preserving probability measure are also sufficient. This was answered in the negative forn⩾5 by Kraft, Pratt, and Seidenberg [Annals of Mathematical Statistics30(1959), 408–419]. The present paper extends their analysis of comparative probabilities that satisfy de Finetti's axioms but lack order-preserving measures when it is assumed also that ≻ is a linear order. We refer to a linear order on the subsets of {1, 2, …, n} that satisfies de Finetti's axioms as an LQP (linear qualitative probability), and say that it is an NLQP (nonrepresentable LQP) if it has no order-preserving probability measure. The paper characterizes all NLQPs forn=5, shows that everyn⩾5 has an NLQP that violates the simplest extension of de Finetti's additivity axiom yet has an order-preserving measure on the family of subsets of every (n−1)-element set, and proves that asn→∞ there is no upper bound on the number of ≻ comparisons needed to verify the nonexistence of an order-preserving measure. Special examples for smallnillustrate other facets of NLQPs, including the necessity of considering multiplicities of an ≻ comparison in testing whether an LQP has an order-preserving measure.