非合作-合作两型博弈的Shapley值纯策略纳什均衡解求解方法

在合作中又有竞争的"经济全球化"时代背景下，经济实体之间越来越多地体现出竞争与合作交织的特点，既有策略的选择，同时也有利益的分配或者成本的分摊，即竞争与合作相互联系。为此，Brandenburger和Stuart提出了非合作-合作两型博弈模型为这类博弈提供了有效的工具。目前非合作-合作两型博弈研究较少，且Brandenburger和Stuart提出的非合作-合作两型博弈存在一些不足：合作博弈用核心求解可能为空或者不唯一。Shapley值是一种重要的合作博弈单值解，满足匿名性、有效性、可加性和虚拟性，表达形式简单且唯一，对一些成本分摊问题和利益分配问题，给决策者提供了一个公平满意的分配方案。因此本文研究将Shapley值作为合作博弈的解时非合作-合作两型博弈解存在的条件。为了分析本文提出的基于Shapley值的非合作-合作两型博弈的新理论框架，首先给出了其特征函数满足的联盟无外部性条件。在满足此条件下，我们进一步证明了非合作-合作两型博弈解存在的条件及性质。结合数值实例比较分析合作博弈用核心和Shapley值求解非合作-合作两型博弈解的优缺点。研究表明：当用Shapley值求解合作博弈解，降低了非合作-合作两型博弈解存在条件。因此，本文的研究不仅弥补了Brandenburger和Stuart提出的非合作-合作两型博弈中合作博弈的核心为空或者不唯一的情况，而且为非合作-合作两型博弈的解提供新的理论框架，从而为既有竞争又有合作的博弈问题提供新的求解方法，因此，本文的研究具有一定的理论价值和应用价值。

A Solution Method for Shapley-based Equilibrium Strategies of Biform Games

Under the background of the "economic globalization" with both competition and cooperation, economic entities are increasingly reflecting the characteristics of competition and cooperation. They have not only the choice of strategies, but also the distribution of benefits or the allocation of costs. That is, competition and cooperation are interrelated. Therefore, Brandenburger and Stuart proposed a Biform game model to provide an effective tool for this game. At present, there are a few Biform game researches, and there are some shortcomings in the Biform game proposed by Brandenburger and Stuart:the core solution of cooperative games may be empty or not unique. Shapley value is an important single-valued solution of cooperative game, satisfying anonymity, validity, additivity, virtuality, and the expression form is simple and unique. It provides players with a fair and satisfactory allocation scheme for some cost allocation problems and benefit allocation problems. Therefore, when Shapley value is used as the solution of cooperative game, the conditions are studied for the existence of Biform game solutions. In order to analyze the new theoretical framework of the Biform game based on Shapley value proposed in this paper, the condition is first given that the characteristic function satisfies the no externalities of coalition (Shorthand for CNE, it means any player changes strategy will be not affect the return value of the coalition that it does not participate in). Under the satisfaction of this condition, the existence and nature of the Biform game solution are proven. The advantages and disadvantages of using core and Shapley value to solve Biform game solutions are compared and analyzed with numerical examples. The research shows that when the cooperative game solution is solved by the Shapley value, the existence conditions of the Biform game solution are reduced. Therefore, the research in this paper not only makes up for the Biform game proposed by Brandenburger and Stuart, the core of the cooperative game is empty or not unique, and provides a new theoretical framework for the solution of the Biform game. Therefore, it provides a new solution method for the game problem of both competition and cooperation. Therefore, the research of this paper has certain theoretical value and application value.