时间：2026-02-06 分类：行政管理
随着全球经济一体化程度加深和国际关系日益密切，双赢合作已成为当今核心趋势。合作博弈作为研究合作问题的有力工具，主要探讨如何在参与者之间分配合作产生的收益。Shapley值作为合作博弈中最重要的解之一，具有重要的研究意义和价值。本文主要从可加性、均衡贡献性、边际性、公平性、简约一致性、相关一致性以及一些特殊参与者性质等角度，介绍Shapley值公理化的相关研究成果，最后从未来研究视角进行简要总结。
关键词
合作博弈;Shapley值;公理化
论文《效用可转移合作博弈的Shapley值公理化研究进展综述》发表在《运筹学学报(中英文)》，版权归《运筹学学报(中英文)》所有。本文来自网络平台，仅供参考。
1 引言
在社会经济发展中，合作有助于创造更多价值，而合作收益的公平分配是合作过程中亟需解决的关键问题。合作博弈理论起源于1881年Edgeworth的研究，1944年von Neumann和Morgenstern的著作奠定了其理论基础。效用可转移合作博弈是合作博弈的重要类型，指参与者之间的效用可以相互转移。
1953年，Shapley首次提出了合作博弈的Shapley值，该值通过计算参与者对所有联盟的边际贡献的加权平均来分配收益，其定义为：对于任意合作博弈(< N, v> in G^{N})及参与者(i in N)，(Sh_{i}(N, v)=sum_{S subseteq N, S i i} frac{(n-s) !(s-1) !}{n !}[v(S)-v(S ackslash {i} )])(其中(n=|N|)，(s=|S|))。
Shapley值自提出以来，因其合理的收益分配逻辑，成为合作博弈领域的核心研究对象。公理化研究是刻画Shapley值本质特征的重要方式，即通过一系列公理(性质)唯一确定Shapley值。本文将从可加性、均衡贡献性、边际性、公平性、简约一致性、相关一致性和特殊参与者性质等多个维度，系统梳理Shapley值公理化的研究进展。
2 Shapley值公理化的主要研究方向
2.1 基于可加性的公理化研究
Shapley本人在1953年的开创性研究中，证明了Shapley值是合作博弈空间上唯一同时满足负载性、可加性和匿名性的解。此后，Shubik进一步拓展，提出Shapley值是唯一满足有效性、可加性、对称性和哑元参与者性的解。
2011年，Einy和Haimanko引入“收益单调性”公理，证明Shapley值是唯一满足收益单调性、可加性、对称性和零元参与者性的解。2019年，Casajus提出“符号对称性”公理(对称参与者收益符号相同)，将对称性弱化为符号对称性，仍唯一确定Shapley值。2024年，Chen等提出“穿越不变性”公理，结合可加性和哑元参与者性，完成了Shapley值的新公理化刻画。
2.2 基于均衡贡献性的公理化研究
Myerson于1980年提出“均衡贡献性”公理，即任意两个参与者之间，一方离开大联盟对另一方收益的影响与另一方离开对其的影响相等。证明了Shapley值是唯一满足有效性和均衡贡献性的解。
Kamijo和Kongo对均衡贡献性进行拓展，2010年提出“均衡圈贡献性”，2018年提出“间接均衡贡献性”，分别结合剔除零元参与者性、弱剔除零元参与者性与有效性，均唯一确定Shapley值，丰富了均衡贡献性公理体系。
2.3 基于边际性的公理化研究
Young于1985年提出“边际性”公理：若参与者在两个博弈中对所有联盟的边际贡献相同，则其收益相同;同时提出“强单调性”公理：若边际贡献均更大，则收益更大。证明Shapley值是唯一满足有效性、对称性和边际性(或强单调性)的解。
Chun引入“联盟策略不变性”和“公平排序性”公理，结合有效性和零博弈性，完成Shapley值的公理化。2021年，Casajus进一步提出“二阶边际性”和“二阶强单调性”，考虑参与者间的二阶交互效应，结合有效性和对称性，实现了更精细的公理化刻画。2023年，Shan等加入“收益损失性”和“非本质博弈性”，基于边际性公理形成新的公理化体系。
2.4 基于公平性的公理化研究
2002年，van den Brink提出“公平性”公理：若两个参与者在某博弈中是对称的，则在该博弈与任意博弈的叠加中，两人收益增量相等。证明Shapley值是唯一满足有效性、零元参与者性和公平性的解。
Casajus对公平性进行深化，2011年提出“微分边际性”公理，2017年提出“弱微分边际性”公理(弱化了收益增量相等的要求，仅要求符号一致)，均证明在结合有效性和零元参与者性时，唯一确定Shapley值。2014年，Casajus剔除有效性和可加性，仅通过收益损失性、哑元参与者性和公平性(或微分边际性)便唯一确定Shapley值，简化了公理化条件。
2.5 基于简约一致性的公理化研究
简约一致性刻画解在子博弈中的稳定性，核心思想是：若将大联盟中部分参与者的收益固定，剩余参与者构成的子博弈中，各参与者的收益应与原博弈一致。
Hart和Mas-Colell于1989年提出“自简约一致性”公理，证明Shapley值是唯一满足自简约一致性和二人标准博弈性的解，或唯一满足自简约一致性、有效性、对称性和平移协变性的解。Calleja和Llerena将自简约一致性弱化为“弱自简约一致性”，结合弱公平性和哑元参与者性，仍唯一确定Shapley值。Oishi提出“平移协议一致性”公理，构建了新的简约一致性公理化框架。
2.6 基于相关一致性的公理化研究
相关一致性关注解在博弈变换中的稳定性，即解在初始博弈和经特定规则变换后的博弈中保持不变。
Hamiache于2001年提出“Ham iache博弈变换”和“相关一致性”公理，证明当变换参数(0<lambda<frac{2}{n})时，Shapley值是唯一满足该相关一致性的解。Xu等对博弈变换进行改进，提出“对偶相似相关一致性”，结合连续性和非本质博弈性，完成Shapley值的公理化，拓展了相关一致性的适用范围。
2.7 基于特殊参与者性质的公理化研究
特殊参与者性质聚焦于不同类型参与者(如无关参与者、非联合生产参与者等)对解的影响。
Manuel等提出“敌对参与者性”和“无关参与者零贡献性”，证明Shapley值是唯一满足有效性、这两个公理的解。Besner于2022年提出“非联合生产参与者性”和“可分离博弈性”，2023年进一步提出“抵制博弈均衡性”，分别结合有效性，唯一确定Shapley值。2024年，Shan等引入“相互依赖参与者微分边际性”，丰富了特殊参与者性质的公理化体系。
3 总结与展望
Shapley值的公理化研究已形成多维度、多层次的体系，从早期的可加性、对称性等基础公理，到后续的均衡贡献性、边际性、一致性等拓展公理，再到对各类公理的弱化与组合，逐步揭示了Shapley值“公平分配合作收益”的核心本质。这些研究不仅深化了对Shapley值的理论认知，也为其在经济、管理、政治等领域的应用提供了坚实的理论支撑。
未来研究可从以下方向推进：一是拓展至更复杂的博弈类型(如效用不可转移博弈、模糊合作博弈等);二是结合实际应用场景(如供应链合作、公共资源分配等)，提出更具针对性的公理;三是探索公理化与算法实现的结合，提升Shapley值的实际应用效率;四是进一步弱化或组合现有公理，构建更简洁、更具包容性的公理化体系，推动合作博弈理论的持续发展。
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