SSB Preferences: Nonseparable Utilities or Nonseparable Beliefs

It is around 1980 that new era for decision making under risk/uncertainty began to uncover numerous alternative representations which generalize the traditional (subjective) expected utility maximization. The initial major contributors include Chew and MacCrimmon (1979), Chew (1983), Fishburn (1982), Kahneman and Tversky (1979), Machina (1982), Quiggin (1981), and Schmeidler (1988) (first appeared in 1981 as a discussion paper). One of Fishburn's works in this area is the discovery of an axiomatic structure of SSB (skew-symmetric bilinear) preferences in decision making under risk, and its numerical representation, dubbed SSB utility (see Fishburn, 1982). Since then, he published a series of papers which study SSB preferences and their numerical representations in various contexts in decision making under risk/uncertainty (see a survey, Fishburn, 1988b). This paper further explores representational aspects of SSB preferences particularly in decision making under uncertainty and discusses their necessary and sufficient axiomatizations. Three representational forms will be examined. One of them is known as an SSA (skew-symmetric additive) representation first explored by Fishburn, 1984a. The other two are new in the literature, one of which seems to be a more natural application of SSB utility to decision making under uncertainty than SSA representation. A characteristic feature of the first two representations is nonseparability of utilities for decision outcomes. The last one is a generalization of subjective expected utility (SEU) which replaces subjective probabilities with non-separable representation of comparative beliefs first discovered by Fishburn (1983a and b).