Abstract: Classical cooperative games assume that any coalition can be formed. However, in real cooperative situations, the assumption is not the truth. In many cooperative cases, there may not be a direct cooperative relation between two players, but through a third player as a middleman, an indirect relationship is established between them. Different from classical cooperative games, this class of games is called restricted cooperative games. A coalition structure game is one branch of restricted cooperative games, corresponding to Shapley value for classical cooperative games. Owen established Owen value for coalition structure games. Winter extended Owen value to the NTU game. In the coalition structure game, the relation structures between players in coalition cannot be expressed explicitly. Considering the impact of different relationship structures on coalition payoffs, Myerson proposed communication games. Corresponding to Shapley value for classical cooperative games, he also proposed a new allocation rule, which is called Myerson value. According to Myerson’s communication games, for players set {1,2,3}, if the connected ways among the three players are {12,13} and {12,23} respectively, then for players set {1,3}, their payoffs under the two cooperation structures will show differences. But for players set {1,2,3}, because they are connected, their payoffs under the two cooperation structures are considered equal, both of which are equal to the payoffs of the grand coalition {1,2,3}. However, in many situations, the payoffs of players {1,2,3} in different connected ways are different, and the communication game cannot indicate this difference.
Considering the payoff difference caused by the different cooperative ways among players, Jackson and Wolinsky modified the Myerson’s model of the communication game and built the network game model, which can distinguish the difference in connected ways between players. Jackson then proposed a new allocation rule for the network game—the link-based flexible network allocation rule, which is an allocation rule with respect to non-directed graph. Generally, non-directed graph represents a symmetry relationship between players. But for asymmetry relationships among players, the explanatory power of Jackson’s network game models has limitations. Earlier in 1990s, GILLES et al., DERKS and GILLES, van den BRINK and GILLES, van den BRINK, and GILLES and OWEN etc. begun to concentrate on the difference in payoffs generated by asymmetry relations among players and described this structure by directed graph. Based on previous conclusions for the directed graph game,SLIKKER et al. made further research on this issue. Especially, they discussed the existence ofallocation rules which satisfies Component Efficiency and the Hierarchical Payoff Property.
In cooperative games, the basic cooperation relation derives from the interaction between two players, which can be expressed with three types: the initiative-passive, the passive-initiative and the interactive. Obviously, the payoffs of coalitions with different relation structures showing differences are a natural thing. For a complex group of players, according to different regulations and rules, players will form different mutual relations. Naturally, the payoffs of group, the allocation of payoffs, the satisfaction degree of allocation, should take on different characterization. Compared to set function, relationship function has advantages in depicting the difference in payoffs created by different relation structures, so it should be a more powerful tool in depicting cooperative games. To a degree, the directed network game has described the difference in payoffs resulting from different directed relations, but more often than not, it only discusses specific directed networks. Compared to the directed network game, the cooperative game with relation function is a generalized model of games, of which classical cooperative games, non-directed graph game and directed network game etc. are all the special cases.
In this paper, focused on the relation structures among players, cooperative games with relation function are established. As the extension of the allocation rule for classical cooperative games, the Shapley value for games with relation function is proposed, its relative properties are proved, and the stability of relation structures are discussed. Further, based on the concept of stability for games with relation function and PROMETHEE method, an approach to ranking different relation structures is proposed, which is verified by a numerical case as well.

Key words: cooperative game, relation function, Shapley allocation, PROMETHEE method

摘要： 联盟内部结构不同会导致收益不同。MYERSON R,JACKSON M和SLIKKER A建立了不同的合作博弈模型,如交流博弈、网络博弈、有向网络博弈等,用来描述联盟内局中人之间不同关系结构对联盟收益的影响。在合作博弈中,联盟的超值源自于局中人之间相互关系的建立,而有序对是两个局中人之间关系的最基本表达。本文定义了关系函数,与经典合作博弈中的集函数相比,关系函数能够从更微观角度反映合作博弈的内在特征。经典合作博弈、交流博弈和网络博弈都可以看作是基于关系函数的博弈特例。作为经典合作博弈Shapley值的推广,提出了基于关系函数博弈的Shapley分配规则,并证明了相关性质。此外,对于如何评价合作博弈不同关系结构的稳定性,基于PROMETHEE方法提出评价不同关系结构稳定性的方法,并通过数值算例进行了验证。

关键词: 合作博弈, 关系函数, Shapley分配, PROMETHEE方法