This chapter presents rank-dependent utility for uncertainty as a natural generalization of the same theory for risk. Much of the material in this chapter can be obtained from Chapters 7 and 8 (on rank dependence under risk) by using a word processor to search for “probability p” and by then replacing it with “event E” everywhere. Similarly, much of this chapter can be obtained from Chapter 4 (on EU under uncertainty) by searching for “outcome event E” and replacing it with “ranked event ER,” and “subjective probabilities of events” with “decision weights of ranked events.” Most of this chapter should, accordingly, not be surprising. I hope that the readers will take this absence of a surprise as a surprise in a didactic sense. A good understanding of the material of §2.1–§2.3, relating risk to uncertainty, will facilitate the study of this chapter.

The literature on rank dependence for uncertainty has commonly used a “comonotonicity” concept that, however, is not very tractable for applications. Hence, our analysis will use ranks instead, in the same way as we did for risk. Comonotonicity is analyzed in Appendix 10.12.

Probabilistic sophistication

The assumption of expected utility maximization has traditionally been divided into two assumptions. The first one entails that all uncertainties can be quantified in terms of probabilities, so that event-contingent prospects can be replaced by probability-contingent prospects (Cohen, Jaffray, & Said 1987, Introduction; Savage 1954 Theorem 5.2.2).