广义核心及其公理化研究

General Core and Its Axiomatic Characterization

在经典合作博弈中,核心是满足所有联盟超量均非正的分配集合,是维持合作稳定性的重要解,在管理决策和公共决策的产品分发方面具有重要应用。基于文献中的多种核心变体,本文将核心推广到任意的二元组(Π,F)上,其中Π是一个拓扑空间,F是Π上的实值连续函数所组成的有限集。广义核心是定义在Π上,满足F的所有分量均非正的集合。广义核心拓展了核心的适用范围,使之可应用到具有冲突情形的博弈模型中。本文证明,当广义核心非空时,广义核子包含在广义核心中。此外,本文通过公理化方法,描述非差异性、非正冗余性、集合缩减性以及最大最小无关性,并证明满足这四个性质的解与广义核心具有一致性,以刻画广义核心的公平合理性。最后,本文给出广义核心与经典合作博弈解的联系与区别,在揭示合作博弈解内在联系方面具有重要的意义。

An important issue in cooperative games is to divide the worth of the grand coalition among all players.For this purpose,various solutions have been proposed.As a famous solution in the cooperative game theory,the core has captured a lot of attention.It is an imputation set in which all coalition excesses are non-positive.Since the coalition excessis a concept that is used to evaluate the complaint of coalitions towards a given imputation in a cooperative game,no coalition has the incentive to deviate from the cooperation once the players receive a core allocation.From this point of view,the core is an important solution to stabilize the cooperation.This solution has been successfully applied to solve the problems of interest or product distribution in fields including economics,political science and operational research.However,the limitations of the core will occur in the following circumstances:(1)Except for the coalition excess,one would want to access the complaint of a coalition via other different ways,such as the per-capita excess,the envy excess,the optimistic excess,etc.(2)Instead of evaluating the coalitional complaint,one would be more willing to intuitively evaluate the complaint of one player,which has actually been done by many scholars who proposed the player excess,the per-capita player excess,etc.(3)The complexity of reality creates the necessity to study the stabilization of other types of games,such as cooperative games with coalition structure,cooperative with nontransferable utilities,non-cooperative games,etc.In view of the above circumstances,this paper generalizes the core to an arbitrary pair(Π,F)∈Ω,whereΩis a class of potential“decision spaces”,Πis a topological space and for every j∈M,F j is a component of F,which is a real continuous function on the domainΠ.F j could be treated as a general complaint function.The general core is a set that collects the topological space such that every coordinate of F is nonpositive.As a consequence,we make it possible to apply the general core in several other conflict situations.Moreover,analogous to the conclusion that the nucleolus belongs to the core whenever the core is nonempty,we prove that the general core contains the general nucleolus whenever the general core is not empty.The main part of this paper is devoted to presenting an axiomatic characterization of the general core to illustrate its reasonability and fairness.The main result in this paper is Theorem 1,in which the general core is characterized by four axioms:Non-discrimination,Non-positive Redundancy,Reduction of the Scope,and Invariance with respect to Max/Min.Non-discrimination says that if M is an empty set,then the solution is made up of the whole topological space.Non-positive Redundancy argues that the solution keeps unchanged if we delete a non-positive coordinate of F.Alternatively,Non-positive Redundancy means that non-positive coordinates of F have no effect on the solution itself.Reduction of the Scope states that a set is never contained in the solution if there exists a coordinate of F on such a set that is never smaller than a given positive number.Invariance with respect to Max/Min implies that the solution is the same if we replace two coordinates of F with their maximum and minimum and keep the other coordinates unchanged.In order to make the above axioms be valid,we assume that the topological space is closed when we delete some coordinates of F or when we operate the maximal and minimal operators on F.Besides,we also prove the logical independence of the four mentioned axioms.Last,various applications of the general core are listed in Section 4.When F’s coordinatesare the coalition excesses or the per-capita excesses,Propositions 2,3,and 4 present respectively that the general core could(1)reduce to some classical solutions,such as the union between the k-core and k-anticore,etc,(2)reduce to some sets,such as the pre-imputation set or the imputation set,etc,(3)reduce to the core on various games,such as the cooperative games with coalition structures,the cooperative games with graph structures or the cooperative games with nontransferable utilities,etc.Proposition 5 gives a refinement of the core when F is restricted to the vector whose coordinates are the sum excesses.These propositions show the relationships between the general core and cooperative game solutions,which is of great significance in revealing the inner connection of these cooperative game solutions.As have shown,a characterization of the core is given for abstract function F,the abstract spaceΠand the abstract dimension restriction M in this paper.One future work could focus on the characterizations for some specific functions,specific spaces,and specific dimension restrictions.We could also study some other general solutions,such as the general bargaining set,the general stable set,and so on.