[论文解读] Desirability and the birth of incomplete preferences

Marco Zaffalon, Enrique Miranda|arXiv (Cornell University)|Jun 1, 2015

Decision-Making and Behavioral Economics参考文献 20被引用 13

一句话总结

本文建立了基于混合独立性的不完全偏好与 imprecise probability 中可取性理论之间的等价性，表明可取性为决策理论提供了更自然且更一般的理论基础。研究证明，仅通过最差行为、弱化的阿基米德公理，且无需正则性假设或完整的概率评估，即可构建不完全偏好，即使在不可数空间中亦成立。

ABSTRACT

We establish an equivalence between two seemingly different theories: one is the traditional axiomatisation of incomplete preferences on horse lotteries based on the mixture independence axiom; the other is the theory of desirable gambles developed in the context of imprecise probability. The equivalence allows us to revisit incomplete preferences from the viewpoint of desirability and through the derived notion of coherent lower previsions. On this basis, we obtain new results and insights: in particular, we show that the theory of incomplete preferences can be developed assuming only the existence of a worst act---no best act is needed---, and that a weakened Archimedean axiom suffices too; this axiom allows us also to address some controversy about the regularity assumption (that probabilities should be positive---they need not), which enables us also to deal with uncountable possibility spaces; we show that it is always possible to extend in a minimal way a preference relation to one with a worst act, and yet the resulting relation is never Archimedean, except in a trivial case; we show that the traditional notion of state independence coincides with the notion called strong independence in imprecise probability---this leads us to give much a weaker definition of state independence than the traditional one; we rework and uniform the notions of complete preferences, beliefs, values; we argue that Archimedeanity does not capture all the problems that can be modelled with sets of expected utilities and we provide a new notion that does precisely that. Perhaps most importantly, we argue throughout that desirability is a powerful and natural setting to model, and work with, incomplete preferences, even in case of non-Archimedean problems. This leads us to suggest that desirability, rather than preference, should be the primitive notion at the basis of decision-theoretic axiomatisations.

研究动机与目标

• 建立基于混合独立性的传统不完全偏好理论与 imprecise probability 中可取性框架之间的正式等价性。
• 证明不完全偏好可仅通过最差行为进行公理化，从而无需假设存在最好行为或正则性假设。
• 利用可取性框架重新表述并统一关键概念，如状态独立性、完全偏好和信念。
• 证明相干下预估（下期望）在建模非阿基米德问题方面，比传统期望值集合更具有效性。
• 主张可取性应作为决策理论基础中的原始概念，而非偏好。

提出的方法

• 将可取性理论扩展至向量值赌局，以建模决策问题中的行为与偏好。
• 利用相干可取性集合的自然扩展，推导出作为下期望泛函的相干下预估。
• 应用完全严格可取性的概念，以刻画集合的相干性及其对正线性组合的封闭性。
• 采用弱化后的阿基米德公理，允许非正则概率和不可数的可能性空间。
• 使用状态层面可取性与条件赌局的概念，以建模状态依赖偏好并推导相干性条件。
• 证明任何不完全偏好关系均可最小化地扩展为包含最差行为的关系，并表明该扩展在非平凡情况下永远不满足阿基米德性。

实验结果

研究问题

• RQ1 不完全偏好能否在不假设存在最好行为或概率正则性的前提下进行公理化？
• RQ2 基于混合独立性的不完全偏好理论与 imprecise probability 中的可取性框架之间是否存在根本性等价？
• RQ3 能否利用可取性原则，以更弱、更一般的形式重新定义传统的状态独立性概念？
• RQ4 阿基米德公理是否能完全涵盖由期望值集合可建模的问题范围，还是存在局限性？
• RQ5 可取性能否作为比偏好更基础的概念，用于决策理论的公理化？

主要发现

• 任何不完全偏好关系的最小扩展至包含最差行为的关系始终存在，且仅在平凡情况下为阿基米德性。
• 不完全偏好理论可仅通过最差行为的存在来发展，无需假设最好行为或概率正则性。
• 弱化后的阿基米德公理足以确保相干性，并允许不可数的可能性空间与非正则概率。
• 传统的状态独立性概念等价于 imprecise probability 中的强独立性，从而导致其定义比以往认为的要弱得多。
• 由可取性导出的相干下预估在建模非阿基米德问题方面，比期望值集合提供了更一般、更灵活的框架。
• 研究表明，可取性作为决策理论公理化的基础，比偏好更具自然性和更强的表达力，尤其在处理不完全或不精确信息时。