Cooperation Games With Probabilistic Graph Structures And Its Applications

The classic transferable utility cooperative game believes that any players can cooperate effectively to form a feasible coalition.However,some coalitions can not be truly formed because cooperation in real life is often restricted by a variety of factors.In order to make the research of cooperative games more realistic,cooperative games with restricted coalition(also called cooperative games with cooperative structure)has attracted the attention of scholars,and there are two main cooperation structures in the form of cooperation among players,called the(priori)coalition structure and the(communication)graph structure.Shapley value and Banzhaf value are two well-known allocation rules of cooperative games.Moreover,the above two values are also extended to the cooperative game with graph structure,later called Myerson value and Banzhaf graph value and the axiomatic characterizations of both values are given.And Owen value and the collective value(later called Ka value)are two important allocation rules of cooperative games with coalition structure,which are later extended to cooperative games with coalition and graph structure,namely Owen graph value,the partition-graph value and the graph-partition value.However,considering that the two directly connected players(that is,adjacent players)may not be able to achieve 100% cooperation,but only a certain degree of cooperation,or the graph cooperation structure formed by the grand coalition is uncertain.Based on this idea,people introduced the generalized probabilistic graph structure to reflect the uncertainty of cooperation,and extended the Myerson value and the Position value to the cooperative game model,later called the probabilistic Myerson value and the probabilistic Position value.Furthermore,the axiomatic characterizations of the above two values are given.Of course,cooperation may be affected by two or more cooperation structures at the same time in practice,that is,cooperative games with coalition and generalized probabilistic graph structures may be formed.To this end,this paper considers cooperative games with generalized probabilistic graph structures and multi-cooperative structures,and extends some important distribution rules in the literature,such as Banzhaf value and Owen value,to cooperative games with generalized probabilistic graph structures or cooperative games with coalition and generalized probabilistic graph structures.Finally,axiomatic characterizations of these values are given.In the deterministic graph,the effective solution of the Owen graph value is also defined,that is,the effective equal division Owen value,and the characterization of the value is given.In addition,based on the measurement function on the graph structure(such as the degree function),three weight component-wise solutions under the cooperative game with graph structure are defined and characterized.Finally,the applications of these values are also discussed.This article mainly focuses on theoretical innovation and the main work is described as follows:(1)The probabilistic Banzhaf value is defined on the cooperative game with generalized probabilistic graph structures.In determine graph structure,the value deduced to the Banzhaf graph value.Two axiomatic characterizations of the value are provided by the axioms of probabilistic component total power,probabilistic fairness and probabilistic balanced contributions.Furthermore,we define the probabilistic player potential function and give an alternative characterization of the value by using the probabilistic player potential function.Finally,we employ an example of election to compare and analyze the probabilistic Banzhaf value with other values.(2)The partition-graph value and graph-partition value are extended to cooperative games with coalition and probabilistic graph structures(later called the partitionprobabilistic graph value and probabilistic graph-partition value).In determine graph structure,the values deduced to the partition-graph value and graph-partition value.When the coalition structure is the grand coalition or the trivial coalition structure,the values deduced to the probabilistic Myerson value.Moreover,the partition-probabilistic graph value is uniquely determined by the axioms of probabilistic partition component efficiency and probabilistic fairness,we give three characterizations of the probabilistic graphpartition value by employing the axioms of probabilistic graph efficiency,probabilistic balanced contributions,probabilistic collective balanced contributions,probabilistic balanced per capita contributions,probabilistic fairness for joining the grand coalition and probabilistic population solidarity within unions.Finally,we apply this value to China’s railway network and compare it with other values.(3)The Owen graph value is extended to the cooperative games with coalition and probabilistic graph structures(later called the probabilistic Owen value).In determine graph structure,the values deduced to the Owen graph value.When the coalition structure is the grand coalition or the trivial coalition structure,the values deduced to the probabilistic Myerson value.Moreover,three characterizations of the probabilistic Owen value are provided by using probabilistic versions of component efficiency,fairness,balanced contributions,probabilistic fairness in the quotient,probabilistic quotient game property,and balanced contributions for the unions.Finally,we compare the probabilistic Owen value with other values by using an example of corporate voting election.(4)In deterministic graph structure,an efficient extension of the Owen graph value is defined on cooperative games with coalition and graph structure.The value is an effective promotion of the effective value theory on cooperative games with coalition and graph structures.Moreover,three characterizations of the efficient Owen graph value is provided by using the axioms of efficiency,coherence with the Owen value for connected graphs,Quasi-link fairness in the quotient,balanced contributions for the unions,fairness of the surplus,fair distribution of surplus and fair distribution of surplus within component.Considering the influence of the graph structure on the allocation of players,the component-wise w-proportional value,the component-wise w-proportional surplus solution and the two-step component-wise w-proportional surplus solution are defined and proposed three axioms: equal w-proportion within component,equal w-proportional distribution of individual surplus within component and w-proportional distribution of the surplus.Moreover,axiomatic characterizations of the above three values are given.Finally,the component-wise w-proportional value,the component-wise w-proportional surplus solution and the two-step component-wise w-proportional surplus solution and the efficient Owen graph value are compared with other values by an example of research fund allocation.