A duality between utility transforms and probability distortions

TL;DR

This paper reveals a fundamental mathematical duality between utility transforms and probability distortions, unifying key decision theories under a common framework without extra assumptions.

Contribution

It establishes that probability distortions and utility transforms are characterized by mutual commutation, providing a unified axiomatic foundation for decision theories under risk.

Findings

Probability distortions are characterized by commutation with utility transforms.

Utility transforms are characterized by commutation with probability distortions.

Rank-dependent utility transforms can be characterized by set commutation under monotonicity.

Abstract

In this paper, we establish a mathematical duality between utility transforms and probability distortions. These transforms play a central role in decision under risk by forming the foundation for the classic theories of expected utility, dual utility, and rank-dependent utility. Our main results establish that probability distortions are characterized by commutation with utility transforms, and utility transforms are characterized by commutation with probability distortions. These results require no additional conditions, and hence each class can be axiomatized with only one property. Moreover, under monotonicity, rank-dependent utility transforms can be characterized by set commutation with either utility transforms or probability distortions.