具有结盟限制的合作对策的加权Shapley值

The weighted Shapley value for games with restricted cooperation

现实生活中,联盟的形成通常是有约束的,而且可行联盟的结构是无规律的,此类问题可以用具有结盟限制的合作对策模型来描述.本文研究了结盟有限制而且局中人具有权重的合作对策的收益分配方案,并基于收益分配方案进行了联盟稳定性分析.因为联盟红利可以描述为联盟形成获得的额外收益,因此本文将联盟红利以局中人权重为比例进行分配,定义了加权Shapley值.然后,用两种公理体系,即分支有效性与比例公平性,有效性与加权平衡贡献性,刻画了该加权Shapley值,并讨论了各公理之间的关系.同时,基于加权Shapley值对联盟稳定性进行了分析并建立了数学模型,得到了一些有趣结论,并发现加权Shapley值满足非本质联盟可去性、无关支撑性和关联性.最后,以指派对策为例,表明具有结盟限制的合作对策模型的实用性以及其加权Shapley值的有效性.

This paper analyzes the weighted Shapley value for cooperative games in which partial coop- eration is based on a set system, which is the set of feasible coalitions that can be formed by players in a game, and the structure of feasible coalitions is completely free. In this context, we define the weighted Shapley value which distributes the Harsanyi dividends proportional to the weights of players. We provide two axiomatic characterizations of the weighted Shapley value： one by means of component efficiency and proportional fairness, and the other with efficiency and weighted balanced contributions. Moreover, the stability of feasible coalitions is analyzed, and some properties of the weighted Shapley value are obtained. Finally, we give two examples about assignment games showing that our method is feasible and effective.