This paper analyzes concave and convex utility and probability distortion functions for decision under risk (law-invariant functionals). We characterize concave utility for virtually all existing models, and concave/convex probability distortion functions for rank-dependent utility and prospect theory in complete generality, through an appealing and well-known condition (convexity of preference, i.e., quasiconcavity of the functional). Unlike preceding results, we do not need to presuppose any continuity, let be differentiability.

An example of a new light shed on classical results: whereas, in general, convexity/concavity with respect to probability mixing is mathematically distinct from convexity/concavity with respect to outcome mixing, in Yaari's dual theory (i.e., Wang's premium principle) these conditions are not only dual, as was well-known, but also logically equivalent, which had not been known before.

本文分析了风险下决策的凹凸效用和概率失真函数（律不变函数）。我们通过一个吸引人的和众所周知的条件（偏好的凸性，即函数的准凹性）来表征几乎所有现有模型的凹效用，以及完全通用的秩相关效用和前景理论的凹/凸概率失真函数。与前面的结果不同，我们不需要预设任何连续性，假设是可微性。

一个新的例子揭示了经典结果：虽然一般来说，关于概率混合的凸/凹在数学上不同于关于结果混合的凸/凹，在 Yaari 的对偶理论（即王的溢价原则）中，这些条件不仅是众所周知的对偶，而且在逻辑上是等价的，这在以前是未知的。