Ellsberg Revisited: A New Look at Comparative Probability

Abstract

Suppose you are given the opportunity of guessing whether it will snow or not in Chicago next Christmas. If you guess correctly, you win 1,000; if not, you win nothing. Which event, snow or no snow, would you bet on? It is widely accepted among decision theorists that your answer reveals which of the two events you deem more probable. Furthermore, if your choices over a field of events obey certain postulates of coherency and consistency, then there is a probability measure $P$ on the field that reflects your choices: you regard $A$ as more probable than $B$ if, and only if, $P(A) > P(B)$. Numerous experiments have shown that people often violate those postulates, so they lack full descriptive validity. Moreover, because of systematic and persistent violations of one of the postulates--an independence axiom-the theory has been questioned on its normative adequacy as a guide to well-reasoned judgments and choices. The purpose of the present paper is to examine a weaker set of postulates that avoids the independence axiom as well as the usual assumption of fully transitive preferences. Despite this weakening, the assumptions imply that there is a unique normalized functional $\rho$ on pairs of events that preserves choices in the sense that $A$ is more probable than $B$ if, and only if, $\rho(A, B) > 0$. The functional $\rho$ has several nice mathematical properties, including "conditional additivity," that reflect vestiges of numerical probability, and it is related to the conventional measure $P$ by $\rho(A, B) = P(A) - P(B)$ when the omitted independence axiom is coupled to the other postulates.