TY - JOUR

T1 - Concave/convex weighting and utility functions for risk

T2 - A new light on classical theorems

AU - Wakker, Peter P.

AU - Yang, Jingni

N1 - JEL classification: D81, C60
Publisher Copyright:
© 2021 The Author(s)

PY - 2021/9/1

Y1 - 2021/9/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85111920844&partnerID=8YFLogxK

U2 - 10.1016/j.insmatheco.2021.07.002

DO - 10.1016/j.insmatheco.2021.07.002

M3 - Article

AN - SCOPUS:85111920844

SN - 0167-6687

VL - 100

SP - 429

EP - 435

JO - Insurance: Mathematics and Economics

JF - Insurance: Mathematics and Economics

ER -