随机选择的条件概率层次结构
A Conditional Probability Hierarchy for Stochastic Choice
- Lingnan College, Laboratory of Mezzoeconomics and Regional Industrial Coordinated Development, Shenzhen Institute of Economics of Lingnan College, Sun Yat-sen University（岭南学院，中观经济与区域产业协调发展实验室，岭南学院深圳经济研究院，中山大学）
- Stern School of Business, Tandon School of Engineering, NYU Shanghai, New York University（斯特恩商学院，坦顿工程学院，纽约大学上海分校，纽约大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究引入点条件概率空间构建四级随机选择规则族层次结构，涵盖无限选择集，还将其与随机选择公理联系起来，建立全序与PCPSs关系，证明PCPSs可更简洁表示选择。
AI中文摘要：
我们引入点条件概率空间（PCPSs）作为随机选择的基本构建块。这个概念可追溯到雷尼（1955年），他提出条件概率空间（CPSs）作为概率论的基础。卢斯（1959年）指出了CPSs与随机选择之间的联系，塞雷亚 - 维奥廖等人（2021年）进一步发展了这种联系。一个PCPS是一个CPS，其每个分量概率测度都集中在一个单元素选择上。我们构建了一个基于PCPS的四级随机选择规则族层次结构。一级由PCPSs组成，二级由PCPSs的“条件一致”混合组成，三级包括PCPSs的所有概率混合，四级由PCPSs的所有有符号混合组成。我们在一般测度理论层面构建层次结构，涵盖无限选择集。我们还将层次结构的每一级与随机选择的著名公理联系起来，即随机显示偏好弱公理、无关替代品独立性和无荷兰赌。我们建立了全序与PCPSs之间的关系，并证明了PCPSs在某种意义上可以是一种更简洁的选择表示。
英文摘要：
We introduce point conditional probability spaces (PCPSs) as primitive building blocks for stochastic choice. This concept goes back to Rényi (1955), who proposed conditional probability spaces (CPSs) as a basis for probability theory. Luce (1959) noted the connection between CPSs and stochastic choice, and Cerreia-Vioglio et al. (2021) have developed the connection further. A PCPS is a CPS each of whose component probability measures concentrates on a singleton selection. We build a four-level PCPS-based hierarchy of families of stochastic choice rules. Level 1 consists of PCPSs, Level 2 is made up of finitely additive "conditionally consistent" mixtures of PCPSs, Level 3 comprises all finitely additive probabilistic mixtures of PCPSs, and Level 4 consists of all finitely additive signed probabilistic mixtures of PCPSs. We construct our hierarchy at a general measure-theoretic level that encompasses infinite choice sets. We also connect each level of our hierarchy to well-known axioms for stochastic choice, namely, the Weak Axiom of Stochastic Revealed Preference, Independence of Irrelevant Alternatives, and no Dutch book. We establish the relationship between total orders and PCPSs and demonstrate a sense in which PCPSs can be a more parsimonious representation of choice.
