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

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