Abstract: A cooperative game with transferable utility (a TU-game) is a pair (N,v), N being the finite set of players and v :2^N→R with v(φ)=0, the characteristic function of the game, that is a real valued map that assigns to each coalition S$\subseteq $N the worth v(S) that its members can obtain by cooperating. The worth v(S) represents the economic possibilities of the coalition S if it is formed. A central issue is to find a method to distribute the benefits of cooperation among these players. A (single-valued) solution for TU-games is a function that assigns to every TU-game a vector with the same dimension as the size of the player set, where each component of the vector represents the payoff assigned to the corresponding player. The Shapley value (Shapley, 1953) probably is the most eminent single-valued solution concept for this type of games. In 1977, Myerson assigned to every communication situation (N,v,L) the Shapley value of network-restricted game (N,v^L), and the value is also called the Myerson value. The Myerson value is the unique allocation rule that satisfies component efficiency and fairness. Haeringer (1999) further extended the Myerson value to the weighted graph game and proposed the weighted Myerson value.
The consistency axiom was first introduced by Hart and Mas-Colell in 1989 and was used to characterize the Shapley value. Consistency requires that when a portion of the participants leaves the coalition with the corresponding payoff, the payments to the remaining participants in the coalition remain unchanged. Since then, the consistency axiom has been widely used in the axiomatic characterization of values. Winter (1992) applied consistency to coalition structure games and characterized the famous Owen value by using consistency and four other properties. Inspired by Hart and Mas-Colell, Dragan (1996) constructed a reduced game for the Banzhaf value and accordingly proposed the corresponding consistency axiom, and then gave the axiomatic characterization of the Banzhaf value by using consistency and standardness. On the other hand, Albizuri and Zarzuelo (2009)extended consistency to hypergraph games and proposed the CS consistency axiom, then uniquely characterized the Myerson value on hypergraph games. In general, when using the consistency axiom to characterize the value, it is necessary to use the potential function as a tool to complete the proof that the value satisfies the consistency. However, in this paper, the use of this tool is avoided.
First of all, we define the weighted reduced graph game and the weighted reduced graph, and then we propose the w-consistency. The w-consistency means when reducing the weighted graph game to a coalition, the weighted reduced graph game consists of the corresponding weighted reduced game and the weighted reduced graph. The gain of each participant in the coalition in the weighted reduced graph game is exactly equal to the gain of that participant in the original assignment graph game. Then, we give an important lemma which shows the relationship between the Harsanyi dividends of each coalition in the restricted game of weighted reduced graph and restricted game of original weighted graph. Based on this lemma, we can easily prove that the weighted Myerson value satisfies the w-consistency. In addition, we propose the w-standardness. For a two-participant weighted graph game, w-standardness requires that if they are not connected, each participant receives the utility generated by himself. If they are connected, each participant first receives the utility generated by himself and then the remaining utility is distributed in the ratio of its own weight to the total weight. We give another lemma which shows that if an allocation satisfies the w-consistency and the w-standardness, it also satisfies the component efficiency. Finally, we give the axiomatic characterization of the weighted Myerson value by the w-consistency and the w-standardness.

Key words: graph game; weighted Myerson value; reduced game; consistency

摘要： 在具有图结构的合作对策中,Myerson值是最重要的分支有效解,它是Shapley值在图对策上的推广。HAERINGER进一步将Myerson值推广到赋权图对策上,提出了赋权Myerson值。在合作对策中,一致性在值的公理化刻画中被普遍使用,它要求当一部分参与者带着应得的支付离开联盟后,联盟内剩余参与者的支付保持不变。一般地,利用一致性公理刻画合作对策的值时,需要借助潜能函数做工具才能完成值满足一致性的证明。然而,本文在提出赋权图对策上的缩减对策和缩减图后,避开了潜能函数的概念,直接建立了在赋权Myerson值下每个联盟在缩减图限制对策和原图限制对策下红利之间的关系式,以此实现值满足一致性的证明,并由此利用权意义下的一致性和标准性给出了赋权Myerson值的公理化刻画。

关键词: 图对策, 赋权Myerson值, 缩减对策, 一致性