Expected utility, rank-dependent utility, and prospect theory all use generalized weighted averages of utilities for evaluating prospects. We have used tradeoff consistency conditions to provide measurements and behavioral foundations for such models. Alternative conditions, based on a bisymmetry condition, have been used in the literature to obtain behavioral foundations. These conditions use certainty equivalents of prospects, so that a richness assumption must be added that certainty equivalents always exist. To avoid details concerning null events, we will assume that S is finite and that all states are nonnull. The latter is implied by strong monotonicity in the following assumption.

Structural Assumption E.1. Structural Assumption 1.2.1 (decision under uncertainty) holds with S finite. Further, ≽ is a monotonic and strongly monotonic weak order, and for each prospect a certainty equivalent exists. □

Although the following multisymmetry condition is a static preference condition, it is best explained by thought experiments using multistage uncertainty. Consider Figure E.1, where we use backward induction (Appendix C) to evaluate the prospects. The indifference sign ∼ indicates that backward induction generates the same certainty equivalent for both two-multistage prospects.