关键词: 图对策, 赋权Myerson值, 缩减对策, 一致性

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