A fundamental study for partially defined cooperative games

The payoff of each coalition has been assumed to be known precisely in the conventional cooperative games. However, we may come across situations where some coalitional values remain unknown. This paper treats cooperative games whose coalitional values are not known completely. In the cooperative games it is assumed that some of coalitional values are known precisely but others remain unknown. Some complete games associated with such incomplete games are proposed. Solution concepts are studied in a special case where only values of the grand coalition and singleton coalitions are known. Through the investigations of solutions of complete games associated with the given incomplete game, we show a focal point solution suggested commonly from different viewpoints.     

Game-theoretic analysis of cooperation among supply chain agents: Review and extensions

This paper surveys some applications of cooperative game theory to supply chain management. Special emphasis is placed on two important aspects of cooperative games: profit allocation and stability. The paper first describes the construction of the set of feasible outcomes in commonly seen supply chain models, and then uses cooperative bargaining models to find allocations of the profit pie between supply chain partners. In doing so, several models are analyzed and surveyed, and include suppliers selling to competing retailers, and assemblers negotiating with component manufacturers selling complementary components. The second part of the paper discusses the issue of coalition formation among supply chain partners. An exhaustive survey of commonly used stability concepts is presented. Further, new ideas such as farsightedness among supply chain players are also discussed and analyzed. The paper also opens some avenues of future research in applying cooperative game theory to supply chain management.     

Cooperative games of choosing partners and forming coalitions in the marketplace

Two games of interacting between a coalition of players in a marketplace and the residual players acting there are discussed, along with two approaches to fair imputation of gains of coalitions in cooperative games that are based on the concepts of the Shapley vector and core of a cooperative game. In the first game, which is an antagonistic one, the residual players try to minimize the coalition's gain, whereas in the second game, which is a noncooperative one, they try to maximize their own gain as a coalition. A meaningful interpretation of possible relations between gains and Nash equilibrium strategies in both games considered as those played between a coalition of firms and its surrounding in a particular marketplace in the framework of two classes of n-person games is presented. A particular class of games of choosing partners and forming coalitions in which models of firms operating in the marketplace are those with linear constraints and utility functions being sums of linear and bilinear functions of two corresponding vector arguments is analyzed, and a set of maximin problems on polyhedral sets of connected strategies which the problem of choosing a coalition for a particular firm is reducible to are formulated based on the firm models of the considered kind.     

供应商成本信息不对称下装配系统协同问题研究

本文研究的装配系统由两个供应商和一个制造商组成,其中一个供应商的成本信息为公共信息,另一个供应商的成本信息为其私有信息,制造商通过设计契约来揭示该私有信息。作为装配系统的主导者,制造商一方面以自身利润最大化为目标,另一方面希望通过系统协同来获得更多的利润。研究结果表明,制造商对两个供应商都采用批发价契约时,无法揭示供应商的私有信息;制造商对信息公开的供应商采用批发价契约,对拥有私有信息的供应商采用价格补贴契约时,可以揭示供应商的私有信息,但不能实现装配系统的整体协同;制造商对两个供应商都采用价格补贴契约时,在成功揭示供应商的私有信息的同时,也实现了装配系统的整体协同。数值分析结果表明,同制造商对两个供应商分别提供批发价契约和价格补贴契约时相比,制造商对两个供应商都提供价格补贴契约时,私有信息的揭示区间会变大,系统总利润和拥有私有信息的供应商的利润会增加,信息公开的供应商的利润会降低,制造商的利润可能增加也可能降低。     

基于预售的生鲜供应链销售模式选择与定价研究

研究预售下由供应商与零售商构成的生鲜供应链的销售模式选择与定价策略,其中包括分散与集中两种销售模式,撇脂与渗透两种定价策略。结果表明,在网络消费者对实体渠道偏好较低时,分散与集中模式均会选择撇脂定价,反之选择渗透定价。撇脂定价下,集中模式中零售商可以通过调节两个渠道的价格占有更大的市场,获取高于分散模式下的利润。而分散模式中,由于两个零售商会存在竞争,因此为了获得竞争优势该模式下的定价会存在低价销售的情形。比较销售模式可知,在网络消费者占比少或占比多但对实体渠道偏好越小时,供应商在分散模式下能获得更高的利润,反之,集中模式对供应商更有利。但是,对零售商而言,分散模式更有利于实体渠道,而集中模式更有利于网络渠道。     

基于多维线性扩展的模糊联盟合作对策tau值性质与计算方法

研究模糊联盟合作对策tau值的计算方法及其性质. 利用多维线性扩展方法定义了模糊联盟合作对策的tau值, 证明了其存在性、唯一性等性质, 并推导出基于多维线性扩展凸模糊联盟合作对策tau值的计算公式. 研究结果发现, 基于多维线性扩展的模糊联盟合作对策tau值是对清晰联盟合作对策tau值的扩展, 而清晰联盟合作对策tau值仅是其特例. 特别地, 对于凸模糊联盟合作对策, 利用其tau值计算公式, 可进一步简化求解过程.     

Multisided matching games with complementarities

The paper considers multisided matching games with transfereable utility using the approach of cooperative game theory. Stable matchings are shown to exist when characteristic functions are supermodular, i.e., agents' abilities to contribute to the value of a coalition are complementary across types. We analyze the structure of the core of supermodular matching games and suggest an algorithm for constructing its extreme payoff vectors. Received February 1997/Final version September 1998     

Approximating the least core value and least core of cooperative games with supermodular costs

We study the approximation of the least core value and the least core of supermodular cost cooperative games. We provide a framework for approximation based on oracles that approximately determine maximally violated constraints. This framework yields a 3-approximation algorithm for computing the least core value of supermodular cost cooperative games, and a polynomial-time algorithm for computing a cost allocation in the 2-approximate least core of these games. This approximation framework extends naturally to submodular profit cooperative games. For scheduling games, a special class of supermodular cost cooperative games, we give a fully polynomial-time approximation scheme for computing the least core value. For matroid profit games, a special class of submodular profit cooperative games, we give exact polynomial-time algorithms for computing the least core value as well as a least core cost allocation.     

Values for Interior Operator Games

The aim of this paper is to study a new class of cooperative games called interior operator games. These games are additive games restricted by antimatroids. We consider several types of cooperative games as peer group games, big boss games, clan games and information market games and show that all of them are interior operator games. Next, we analyze the properties of these games and compute the Shapley, Banzhaf and Tijs values.     

Totally balanced combinatorial optimization games

Combinatorial optimization games deal with cooperative games for which the value of every subset of players is obtained by solving a combinatorial optimization problem on the resources collectively owned by this subset. A solution of the game is in the core if no subset of players is able to gain advantage by breaking away from this collective decision of all players. The game is totally balanced if and only if the core is non-empty for every induced subgame of it.?We study the total balancedness of several combinatorial optimization games in this paper. For a class of the partition game [5], we have a complete characterization for the total balancedness. For the packing and covering games [3], we completely clarify the relationship between the related primal/dual linear programs for the corresponding games to be totally balanced. Our work opens up the question of fully characterizing the combinatorial structures of totally balanced packing and covering games, for which we present some interesting examples: the totally balanced matching, vertex cover, and minimum coloring games. Received: November 5, 1998 / Accepted: September 8, 1999?Published online February 23, 2000     

On ordinal equivalence of the Shapley and Banzhaf values for cooperative games

In this paper I consider the ordinal equivalence of the Shapley and Banzhaf values for TU cooperative games, i.e., cooperative games for which the preorderings on the set of players induced by these two values coincide. To this end I consider several solution concepts within semivalues and introduce three subclasses of games which are called, respectively, weakly complete, semicoherent and coherent cooperative games. A characterization theorem in terms of the ordinal equivalence of some semivalues is given for each of these three classes of cooperative games. In particular, the Shapley and Banzhaf values as well as the segment of semivalues they limit are ordinally equivalent for weakly complete, semicoherent and coherent cooperative games.     

Subgame Consistent Cooperative Solution of Dynamic Games with Random Horizon

In cooperative dynamic games, a stringent condition—that of subgame consistency—is required for a dynamically stable cooperative solution. In particular, under a subgame-consistent cooperative solution an extension of the solution policy to a subgame starting at a later time with a state brought about by prior optimal behavior will remain optimal. This paper extends subgame-consistent solutions to dynamic (discrete-time) cooperative games with random horizon. In the analysis, new forms of the Bellman equation and the Isaacs–Bellman equation in discrete-time are derived. Subgame-consistent cooperative solutions are obtained for this class of dynamic games. Analytically tractable payoff distribution mechanisms, which lead to the realization of these solutions, are developed. This is the first time that subgame-consistent solutions for cooperative dynamic games with random horizon are presented.     

The Compromise Value for Cooperative Games with Random Payoffs

This paper introduces and studies the compromise value for cooperative games with random payoffs, that is, for cooperative games where the payoff to a coalition of players is a random variable. This value is a compromise between utopia payoffs and minimal rights and its definition is based on the compromise value for NTU games and the τ-value for TU games. It is shown that the nonempty core of a cooperative game with random payoffs is bounded by the utopia payoffs and the minimal rights. Consequently, for such games the compromise value exists. Further, we show that the compromise value of a cooperative game with random payoffs coincides with the τ-value of a related TU game if the players have a certain type of preferences. Finally, the compromise value and the marginal value, which is defined as the average of the marginal vectors, coincide on the class of two-person games. This results in a characterization of the compromise value for two-person games.I thank Peter Borm, Ruud Hendrickx and two anonymous referees for their valuable comments.     

Cores and related solution concepts for multi-choice games

A multi-choice game is a generalization of a cooperative game in which each player has several activity levels. Cooperative games form a subclass of the class of multi-choice games.This paper extends some solution concepts for cooperative games to multi-choice games. In particular, the notions of core, dominance core and Weber set are extended. Relations between cores and dominance cores and between cores and Weber sets are extensively studied. A class of flow games is introduced and relations with non-negative games with non-empty cores are investigated.     

基于区间与模糊Shapley值的合作收益分配策略

将经典Shapley值三条公理进行拓广,提出具有模糊支付合作对策的Shapley值公理体系。研究一种特殊的模糊支付合作对策,即具有区间支付的合作对策,并且给出了该区间Shapley值形式。根据模糊数和区间数的对应关系,提出模糊支付合作对策的Shapley值,指出该模糊Shapley值是区间支付模糊合作对策的自然模糊延拓。结果表明:对于任意给定置信水平α,若α=1,则模糊Shapley值对应经典合作对策的Shapley值,否则对应具有区间支付合作对策的区间Shapley值。通过模糊数的排序,给出了最优的分配策略。由于对具有模糊支付的合作对策进行比较系统的研究,从而为如何求解局中人参与联盟程度模糊化、支付函数模糊化的合作对策,奠定了一定的基础。     

Survey of cooperative inventory games and extensions

We examine cooperative games in supply-chain management termed Inventory Games. Supply-chain management has non-cooperative and cooperative interactions between the participating players. We provide a concise survey of cooperative inventory games in the form of extensions on two basic problems. For deterministic games, Economic Order Quantity-like policies with joint replenishment are of primary interest. For stochastic games we examine Newsvendor-like centralization games and their extensions. We conclude with a short summary of a dynamic Newsvendor realization game and directions for further research.     

Cooperation in assembly systems: The role of knowledge sharing networks

Process improvement plays a significant role in reducing production costs over the life cycle of a product. We consider the role of process improvement in a decentralized assembly system in which a buyer purchases components from several first-tier suppliers. These components are assembled into a finished product, which is sold to the downstream market. The assembler faces a deterministic demand/production rate and the suppliers incur variable inventory costs and fixed setup production costs. In the first stage of the game, which is modeled as a non-cooperative game among suppliers, suppliers make investments in process improvement activities to reduce the fixed production costs. Upon establishing a relationship with the suppliers, the assembler establishes a knowledge sharing network – this network is implemented as a series of meetings among suppliers and also mutual visits to their factories. These meetings facilitate the exchange of best practices among suppliers with the expectation that suppliers will achieve reductions in their production costs from the experiences learned through knowledge sharing. We model this knowledge exchange as a cooperative game among suppliers in which, as a result of cooperation, all suppliers achieve reductions in their fixed costs. In the non-cooperative game, the suppliers anticipate the cost allocation that results from the cooperative game in the second stage by incorporating the effect of knowledge sharing in their cost functions. Based on this model, we investigate the benefits and challenges associated with establishing a knowledge sharing network. We identify and compare various cost allocation mechanisms that are feasible in the cooperative game and show that the system optimal investment levels can be achieved only when the most efficient supplier receives the incremental benefits of the cost reduction achieved by other suppliers due to the knowledge transfer.     

有限合作博弈的Shapley分配

以Myerson关于有限合作的图博弈模型为基础,结合经典合作博弈的相关结论,建立了有限合作博弈的Shapley分配,讨论了分配的相关性质.同时在支付函数满足链递增性的假设下,进一步研究了有限合作关系变化对收益分配的影响,给出了相关的研究结论.     

On some families of cooperative fuzzy games

In a fuzzy cooperative game the players may choose to partially participate in a coalition. A fuzzy coalition consists of a group of participating players along with their participation level. The characteristic function of a fuzzy game specifies the worth of each such coalition. This paper introduces well-known properties of classical cooperative games to the theory of fuzzy games, and studies their interrelations. It deals with convex games, exact games, games with a large core, extendable games and games with a stable core.     

图限制下合作对策的τ值

首先研究了图限制下合作对策的r值,这个单值解是由Tijs提出的经典合作对策τ值的推广.并且当合作图为完全图时,准均衡图对策的τ值与经典合作对策下的准均衡对策的τ值一致.其次利用分支有效性,S-均衡下的相对不变性和限制成比例性讨论了τ值的公理化方法.最后介绍了两类特殊图对策的τ值.