Independence on Relative Probability Spaces and Consistent Assessments in Game Trees

Abstract

Relative probabilities compare the likelihoods of any pair of events, even those with probability zero. Definitions of weak and strong independence of random variables on finite relative probability spaces are introduced. The former is defined directly, while the latter is defined in terms of approximations by ordinary probabilites. Our main result is a characterization of strong independence in terms of weak independence and exchangeability. This result is applied to game theory to obtain a natural interpretation ofconsistent assessment, an essential yet controversial ingredient in the definition of sequential equilibrium.Journal of Economic LiteratureClassification Numbers: C60, C72.