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In expected utility theory, risk attitudes are modeled entirely in terms of utility. In the rank-dependent theories, a new dimension is added: chance attitude, modeled in terms of nonadditive measures or nonlinear probability transformations that are independent of utility. Most empirical studies of chance attitude assume probabilities given and adopt parametric fitting for estimating the probability transformation. Only a few qualitative conditions have been proposed or tested as yet, usually quasi-concavity or quasi-convexity in the case of given probabilities. This paper presents a general method of studying qualitative properties of chance attitude such as optimism, pessimism, and the "inverse-S shape" pattern, both for risk and for uncertainty. These qualitative properties can be characterized by permitting appropriate, relatively simple, violations of the sure-thing principle. In particular, this paper solves a hitherto open problem: the preference axiomatization of convex ("pessimistic" or "uncertainty averse") nonadditive measures under uncertainty. The axioms of this paper preserve the central feature of rank-dependent theories, i.e. the separation of chance attitude and utility.
Peter P. Wakker (Sun,) studied this question.