M-限制合作对策的最小二乘解

在经典合作对策中,最小二乘解是使得联盟分配值与联盟收益的期望偏差最小的分配方案,众多单值解可以看作它的特例.为了拓展最小二乘解的适用范围,本文公理化研究M-限制合作对策的最小二乘解,这类对策的联盟收益是否已知仅与联盟中局中人的个数有关.首先,基于经典合作对策的最小二乘解定义了M-限制合作对策的最小二乘解.然后,利用拉格朗日乘子法得到了该最小二乘解的具体表达式及其等价形式,并以此重新解释了最小二乘解的现实意义.最后,为了说明最小二乘解的公平合理性,根据该值与ESL值的关系提出了它的公理体系.第一种公理体系是有效性、对称性、线性、非本质对策性、公平对待性.基于该公理体系,替换部分公理可得到其他的公理体系,比如:公平对待性可替换为联盟单调性或者联盟占优单调性;对称性可替换为基数无异性.另外,如果线性弱化为可加性且非本质对策性强化为策略等价性,则也可以公理化刻画最小二乘解.

The least square value for M-restricted cooperative games

In classical cooperative game theory, the least square value provides a method, which minimizes the expected deviation between the payoffs and the worths of all coalitions, of allocating benefits to the players. This paper devotes to the extension of the theory of the least square value on the class of M-restricted cooperative games, a generalization of classical cooperative games, that is defined only for certain coalitions. Firstly, the least square value for M-restricted cooperative games is defined based on the least square value of classical cooperative games. Secondly, the explicit expression and its equivalence forms of the least square value are obtained by using the Lagrange multiplier approach, and then two realistic meanings of this value are reinterpreted. Finally, in order to show fairness and rationality of the least square value, several axiomatizations are proposed according to the relationship between this value and the ESL value. An axiomatization is efficiency, symmetry, linearity, the inessential game property, and the equal treatment property. Other axiomatizations can be obtained by replacing one or two of the axioms of the aforementioned one, such as the equal treatment property can be replaced with coalition monotonicity or coalition dominant monotonicity, symmetry can be replaced with the subsidy free cardinality property. In addition, if linearity is weaken to additivity and the inessential game property is strengthened to the strategic equivalence property, it is also verity to characterize the least square value.