Random binary choices that satisfy stochastic betweenness

Experimental evidence suggests that the process of choosing between lotteries (risky prospects) is stochastic and is better described through choice probabilities than preference relations. Binary choice probabilities admit a Fechner representation if there exists a utility function u such that the probability of choosing a over b is a non-decreasing function of the utility difference u(a)−u(b). The representation is strict if u(a)≥u(b) precisely when the decision-maker is at least as likely to choose a from {a,b} as to choose b. Blavatskyy (2008) obtained necessary and sufficient conditions for a strict Fechner representation in which u has the expected utility form. One of these is the Common Consequence Independence (CCI) axiom (ibid., Axiom 4), which is a stochastic analogue of the mixture independence condition on preferences. Blavatskyy also conjectured that by weakening CCI to a condition we call Stochastic Betweenness–a stochastic analogue of the betweenness condition on preferences (Chew, 1983)–one obtains necessary and sufficient conditions for a strict Fechner representation in which u has the implicit expected utility form (Dekel, 1986). We show that Blavatskyy’s conjecture is false, and provide a valid set of necessary and sufficient conditions for the desired representation.