基于生成层级的超图结构合作博弈分配规则设计和刻画研究

Cooperative games with hypergraph structure, shortly hypergraph games, have become one of new hot topics in cooperative games with limited cooperation, which can be used to solve the profit and/or cost allocation problems arising from cooperative situations with complex structures. The previous studies about the hypergraph games draw more attention on the current state of connectivity of the hypergraph. However, there is little deep study about the construction process of the connectivity. This project will start with the deconstruction of the connectivity of hypergraphs. By destroying the connectivity of a hypergraph, we build the spanning hierarchies. Then, by adopting the knowledge of graph theory, combination and mapping methods, we first design the allocation rules of hypergraph games, and then, provide the axiomatic and non-axiomatic characterizations of the allocation rules. Formally, firstly, defining the concepts of partial hierarchies, we propose the partial-ordering-based allocation rules by using the marginalistic allocation; Secondly, building the concepts of decomposed hierarchies, we introduce the decomposition-based allocation rules by applying the egalitarian distribution; Thirdly, we intend to achieve the axiomatic characterizations of the proposed allocation rules by analyzing the properties and uniquenesses of the two-types allocation rules. Finally, by exploring the relationships among the allocation rules, we aim to carry out the non-axiomatic characterizations of the allocation rules. The researches of this project play a major role in promoting the theory of cooperative games with limited structure and application to the related fields.

具有超图结构的合作博弈（简称超图博弈）可以解决现实生活中具有复杂结构联盟的利润分配和成本分摊问题，是受限合作博弈新兴的热点之一。以往对超图博弈的研究局限于超图连通的即时状态，但对其连通的构建过程却缺乏深入探索。本项目从超图的连通解构入手，通过破坏超图的连通性来构建生成层级，进而结合图论、组合和映射等方法，设计超图博弈的分配规则，并针对相应的分配规则给出其公理刻画和非公理刻画。具体地：建立超图结构的偏序层级，基于边际贡献理念，设计偏序分配规则；建立超图结构的分解层级，基于平等分配理念，设计分解分配规则；分析两类分配规则满足的性质公理和唯一性条件，完成分配规则的公理刻画；探索分配规则之间的内在联系，致力于分配规则的非公理刻画。本项目对推动具有限制结构合作博弈理论的发展以及相关领域的应用具有重要意义。

现实生活中的社交群、共同体以及协同组织往往具有复杂的联盟结构，其涉及到的合作效益分配或者成本分摊问题则可以通过具有超图结构合作博弈（简称超图博弈）进行研究。超图博弈中公平合理的分配规则以及科学的分配机制是促成参与者广泛合作以及联盟稳定的核心因素。本项目基于超图结构的连通性，探讨了超图博弈及相关问题的分配规则设计和刻画问题。项目的主要内容如下：首先，基于超图结构连通性和超图分解，构建层级生成树以表征参与者加入联盟次序和联盟形成过程，并以此设计超图博弈分配规则和完成公理刻画，阐述分配规则的合理性；其次，基于参与者缺席情况提出了超图博弈连通性的新假设，借此设计了超图博弈的自适分配规则，并利用参与者缺席对分配结果的影响提出“退群”性质公理完成自适分配规则的公理刻画；最后，基于边际贡献、比例分配和平等分配理念、过程化分配方法以及特殊参与者对联盟组织形式的影响，设计过程比例解、比例分离解、比例-平等分配等复合式分配规则并探索其合理性，揭示联盟形成的内在规律。上述部分理论可以用来分析和解决具有英国脱欧或美国“退群”等现象的分配问题、区域经济一体化贡献评估和考虑社交关系的破产清偿等经济管理问题，为决策者提供理论依据。相关理论成果及应用探索得到同行认可，部分成果发表在MMOR、ORL和AOR等运筹领域国际期刊。