Abstract: Acooperative game with transferable utility (TU-game) is a pair (N,v)consisting of a set N of players and a characteristic function v assigning a worth v(S) to each coalition S$\subseteq$N. The central question to be answered in cooperative game theory is how the value v(N) should be divided among the players N. The Shapley value (Shapley, 1953) provides a principled way to do this. However, it is assumed that all players can communicate with each other and also that all possible coalitions are feasible in the classical TU-game. Myerson(1977) modified this assumption by introducing restrictions in the communication among players through a graph. In this new setting, some of the coalitions become infeasible. Myerson, then, defined the graph-restricted game, and he proposed the Shapley value of the graph-restricted game as an allocation rule, now called the Myerson value, for TU-games restricted by graphs. Moreover, he presented an axiomatic characterization of the defined value using two properties: component efficiency and either fairness or balanced contributions. The component efficiency states that the worth of a component of the graph is distributed among its members. Thus generally the Myerson value is not efficient. However, in practice, the graph structure may not affect the formation of the grand coalition, but it will affect the bargaining power of players because of their different positions in the network. This prompts people to consider the effective extension of Myerson value. Several effective extensions of the Myerson value have been put forward in the literature.
Li and Shan(2020) introduced an efficient extension of the Myerson value, called the efficient quotient Myerson value, as a two-step value. The efficient quotient Myerson value proposed here distributes the worth of the grand coalition in two steps. Firstly, players within one component act collectively to bargain with other components, all components play the quotient game and obtain a payoff prescribed by the Shapley value. The surplus of the difference between the obtained payoff and the worth of the component is distributed equally among component members. An axiomatic characterization of the value for graph games is established. They show that the efficient quotient Myerson value is the unique allocation rule that satisfies quotient component efficiency, fair distribution of surplus within component and coherence with the Myerson value for connected graphs. Quotient component efficiency states that the sum of the payoffs obtained by the members within a component is equal to the Shapley value obtained by this component in the quotient game. The fair distribution of surplus within component property requires that any two members in the same component have the same payoff changes in the sub-game restricted on the component. The coherence with the Myerson value for connected graphs property suggests that each playerreceives the Myerson value in the connected graph games.
In this paper we introduce a new property, called quasi-fairness of surplus with quotient. This property means that breaking an link will exert the same influence on the quasi-payoffs of the two players associated with the link, that is to say, the quasi-payoff changes of the two players are the same, where the quasi-payoff of a player is the difference between his payoff in the original distribution rule and the amount of surplus assigned to the player. We show that the efficient quotient Myerson value can be axiomatically characterized by efficiency, quasi-fairness of surplus with quotient and coherence with the Myerson value for connected graphs. In addition, we compare this value with other allocation rules through an application example.

Key words: TU-game, graph game, myerson value, efficient quotient Myerson value

摘要： 在具有图结构的合作对策中,Myerson值(Myerson,1977)是一个著名的分配规则,它可以由分支有效性和公平性或者平衡贡献性所唯一确定。在实际中,图结构可能并不影响大联盟的形成,只是由于参与者在网络中所处的位置不同,对其讨价还价能力会产生影响。换句话说,图结构会对分配格局产生影响,但对大联盟的形成没有影响。这促使人们开始考虑Myerson值的有效推广问题。文献中已经提出了Myerson的几种有效推广形式。2020年,Li和Shan提出了有效商Myerson值并给出了公理化刻画,它是Myerson值一种新的有效推广形式。本文首先引入了准商盈余公平性这一性质, 然后结合有效性和Myerson值黏性给出了有效商Myerson值的新公理化刻画。其次, 通过应用案例,将该值和其他值做了比较分析。

关键词: TU-对策, 图对策, Myerson值, 有效商Myerson值