A generalization of comparative probability on finite sets

Let ? be a binary relation on a finite algebra $\text{A}$ of events A, B,…, where A ? B is interpreted as “A is more probable than B.” Conventional subjective probability is concerned with the existence of a probability measure P on $\text{A}$ that agrees with ? in the sense that A ? B ? P(A) > P(B). Because evidence suggests that some people's comparative probability judgments do not admit an agreeing probability measure, this paper explores a more flexible scheme for representing ? numerically. The new representation has A ? B ? p(A, B) > 0, where p is a monotonic and normalized skew-symmetric function on $\text{A}$ × $\text{A}$ that replaces P's additivity by a conditional additivity property. Conditional additivity says that $\text{p(A ? B, C) + p(?, C) = p(A, C) + p(B, C)}$ whenever A and B are disjoint. The paper examines consequences of this representation, presents examples of ? that it accommodates but which violate the conventional representation, formulates axioms for ? on $\text{A}$ that are necessary and sufficient for the representation, and discusses specializations in which p in separable in its arguments.