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.

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关于适当的计分规则和累积前景理论

计分规则是衡量报告的信念与观察到的结果之间关联的传统技术。计分规则是适当的条件意味着，当一个代理人报告一个等于其真实信念的信念时，该代理人将最大化其预期得分。但是，至少在底层主体是人的情况下，隐含的假设是，主体是风险中性的，通常是不现实的。基于等级依赖效用的现代决策理论，例如累积前景理论，已被证明在描述人类如何在风险和不确定性下做出决策方面更为有效。但是，传统的适当计分规则与累积前景理论不兼容，因为它们不能满足称为共调性的性质。在本文中，我们提供了新颖的见解，说明当从累积前景理论主体中得出信念时，共调性对于使正确的评分规则确实正确至关重要的重要性。在提出创建单调的正确评分规则的策略建议后，我们提出了校准程序，以通过用单调的正确评分规则引起信念时，从其报告的信念中消除代理的价值函数和权重函数的影响，从而获得代理的真实信念。