具有超图合作结构的赋权Position值
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摘要
具有超图交流结构的可转移效用合作对策,也称为超图对策,它由一个三元组(N,v,H)所组成,其中(N,H)是一个可转移效用对策(简称TU-对策),而(N,H)是一个超图(超网络)。在超图对策中,除Myerson值(Myerson)外,Position值(Meessen)是另一个重要的分配规则。该模型要求把超图结构中每条超边Shapley的值平均分配给它所包含的点,而不考虑每个点的交流能力或合作水平。本文引入超图结构中点的度值来度量每条超边中每个点的交流能力或合作水平,并结合Haeringer提出用于推广Shapley值的权重系统,并由此定义了具有超图合作结构的赋权Position值。我们证明了具有超图合作结构的赋权Position值可以由"分支有效性"、"冗余超边性"、"超边可分解性"、"拟可加性"、"弱积极性"和"弱能转换"六个性质所唯一确定,并且发现参与者获得的支付随其度值的增加而增加,参与者分摊的成本随其度值的增加而降低。
A transferable utility cooperative game with hypergraph communication structure,also called hypergraph game,is a triple( N,v,H),where( N,v) is a cooperative game with transferable utility or simply a TU-game and( N,H) is a hypergraph( or hypernetwork). In addition to the Myerson value( Myerson),the position value( Meessen) is another important allocation rules for hypergraph games. In Meessen's model,the Shapley value of each hyperlink in a hypergraph structure is divided equally among the palyers in the hyperglink,but it ignores the communicative strength or the level of cooperation of each node in a hyperlink. The purpose of this paper is to present the weighted position value in the hypergraph cooperation structure with a weight scheme. In this new model,we introduce the degree of nodes in order to measure the communicative strength or the level of cooperation of each node in a hyperlink. By employing the weight system proposed by Haeringer that was used to generalize the Shapley value,we define the weighted position value for hypergraph communication situations. We show that the weighted position value for hypergraph communication situations can be uniquely characterized by six axioms: "component efficiency","quasi-additivity","weak positivity","weak power inversion",hyperlink decomposability"and "superfluous hyperlink property". It can be concluded that the payoffs of the players are increasing with respect to the degree value,while the costs of the players are decreasing.
A transferable utility cooperative game with hypergraph communication structure,also called hypergraph game,is a triple( N,v,H),where( N,v) is a cooperative game with transferable utility or simply a TU-game and( N,H) is a hypergraph( or hypernetwork). In addition to the Myerson value( Myerson),the position value( Meessen) is another important allocation rules for hypergraph games. In Meessen's model,the Shapley value of each hyperlink in a hypergraph structure is divided equally among the palyers in the hyperglink,but it ignores the communicative strength or the level of cooperation of each node in a hyperlink. The purpose of this paper is to present the weighted position value in the hypergraph cooperation structure with a weight scheme. In this new model,we introduce the degree of nodes in order to measure the communicative strength or the level of cooperation of each node in a hyperlink. By employing the weight system proposed by Haeringer that was used to generalize the Shapley value,we define the weighted position value for hypergraph communication situations. We show that the weighted position value for hypergraph communication situations can be uniquely characterized by six axioms: "component efficiency","quasi-additivity","weak positivity","weak power inversion",hyperlink decomposability"and "superfluous hyperlink property". It can be concluded that the payoffs of the players are increasing with respect to the degree value,while the costs of the players are decreasing.
引文
[1] Branzei R,Dimitrov D,Tijs S. Models in cooperative game theory[M]. Berlin:Springer,2008.
[2] Myerson R B. Graphs and cooperation in games[J].Mathematics of Operations Research, 1977, 2(3):225-229.
[3] Borm P,Owen G,Tijs S. On the position value for communication situations[J]. Society for Industrial and Applied Mathematics,1992,5(3):305-320.
[4] Myerson R B. Conference structures and fair allocation rules[J]. International Journal of Game Theory,1980,9(3):169-182.
[5] van den Nouweland A,Borm P,Tijs S. Allocation rules for hypergraph communication situations[J]. International Journal of Game Theory,1992,20(3):255-268.
[6] Meessen R. Communication games[J]. Master’s thesis,Department of Mathematics,University of Nijmegen,The Netherlands,1988.
[7] Shapley L S. A value for n-person games[J]. Contributions to the Theory of Games,1953,2(28):307-317.
[8] Shapley L S. Additive and non-additive set functions[M]. Princeton University,1953.
[9] Owen G. A note on the Shapley value[J]. Management Science,1968,14(11):731-731.
[10] Kalai E,Samet D. On weighted Shapley values[J].International Journal of Game Theory,1987,16(3):205-222.
[11] Chun Y. On the symmetric and weighted Shapley values[J]. International Journal of Game Theory,1991,20(2):183-190.
[12] Haeringer G. A new weight scheme for the shapley value[J]. Mathematical Social Sciences,2006,52(1):88-98.
[13] Haeringer G. Weighted myerson value[J]. International Game Theory Review,1999,1(02):187-192.
[14] Slikker M, van den Nouweland A. Communication situations with asymmetric players[J]. Mathematical Methods of Operations Research,2000,52(1):39-56.
[15] van den Brink R,van der Laan G,Pruzhansky V.Harsanyi power solutions for graph-restricted games[J].International Journal of Game Theory,2011,40(1):87-110.
[2] Myerson R B. Graphs and cooperation in games[J].Mathematics of Operations Research, 1977, 2(3):225-229.
[3] Borm P,Owen G,Tijs S. On the position value for communication situations[J]. Society for Industrial and Applied Mathematics,1992,5(3):305-320.
[4] Myerson R B. Conference structures and fair allocation rules[J]. International Journal of Game Theory,1980,9(3):169-182.
[5] van den Nouweland A,Borm P,Tijs S. Allocation rules for hypergraph communication situations[J]. International Journal of Game Theory,1992,20(3):255-268.
[6] Meessen R. Communication games[J]. Master’s thesis,Department of Mathematics,University of Nijmegen,The Netherlands,1988.
[7] Shapley L S. A value for n-person games[J]. Contributions to the Theory of Games,1953,2(28):307-317.
[8] Shapley L S. Additive and non-additive set functions[M]. Princeton University,1953.
[9] Owen G. A note on the Shapley value[J]. Management Science,1968,14(11):731-731.
[10] Kalai E,Samet D. On weighted Shapley values[J].International Journal of Game Theory,1987,16(3):205-222.
[11] Chun Y. On the symmetric and weighted Shapley values[J]. International Journal of Game Theory,1991,20(2):183-190.
[12] Haeringer G. A new weight scheme for the shapley value[J]. Mathematical Social Sciences,2006,52(1):88-98.
[13] Haeringer G. Weighted myerson value[J]. International Game Theory Review,1999,1(02):187-192.
[14] Slikker M, van den Nouweland A. Communication situations with asymmetric players[J]. Mathematical Methods of Operations Research,2000,52(1):39-56.
[15] van den Brink R,van der Laan G,Pruzhansky V.Harsanyi power solutions for graph-restricted games[J].International Journal of Game Theory,2011,40(1):87-110.