Totally convex preferences for gambles. (Q1867794)

The author considers a set of gambles with a binary strict preference relation. Fishburn showed that there is a nontransitive convex representation if the relation has an open and totally convex structure. The author gives a necessary and sufficient condition that there is a transitive convex representation of the preference relation (Theorem 3.1). Relations to Dekel's implied representation are shown. A second theorem (Theorem 4.1) gives a necessary and sufficient condition for the existence of a separable SSB representation. A final theorem (Theorem 4.2) gives a necessary and sufficient condition for a nonnegative component of a separable SSB representation. Relations to Chew's weighted linear utility are shown.