On proper scoring rules and cumulative prospect theory

Scoring rules are traditional techniques to measure the association between a reported belief and an observed outcome. The condition that a scoring rule is proper means that an agent maximizes his expected score when he reports a belief that equals his true belief. The implicit assumption that the agent is risk neutral is, however, often unrealistic, at least when the underlying agent is a human. Modern decision theories based on rank-dependent utilities, such as cumulative prospect theory, have been shown to be more effective at describing how human beings make decisions under risk and uncertainty. Traditional proper scoring rules are, however, incompatible with cumulative prospect theory because they fail to satisfy a property called comonotonicity. In this paper, we provide novel insights on why comonotonicity is crucial to make proper scoring rules indeed proper when eliciting beliefs from cumulative prospect theory agents. After suggesting strategies to create comonotonic proper scoring rules, we propose calibration procedures to obtain an agent’s true belief by removing the influence of the agent’s value function and weighting functions from his reported belief, when beliefs are elicited by means of comonotonic proper scoring rules.