Shapley mappings and the cumulative value for n-person games with fuzzy coalitions

In this paper we prove existence and uniqueness of the so-called Shapley mapping, which is a solution concept for a class of n-person games with fuzzy coalitions whose elements are defined by the specific structure of their characteristic functions. The Shapley mapping, when it exists, associates to each fuzzy coalition in the game an allocation of the coalitional worth satisfying the efficiency, the symmetry, and the null-player conditions. It determines a “cumulative value” that is the “sum” of all coalitional allocations for whose computation we provide an explicit formula.     

分销供应链中竞争零售商联盟的稳定性

针对由单一供应商和三个相互竞争零售商组成的两层分销供应链系统,在三种不同的博弈框架下,采用合作博弈论中短视的Nash稳定性概念与远视的最大一致集(LCS)概念研究了供应商与不同零售商联盟间的定价博弈,分别讨论了不同类型零售商联盟的稳定性。发现不论是在供应商处于领导地位,还是在零售商处于领导地位的市场中,当竞争强度较弱时,大联盟不是短视零售商联盟的稳定结构,却有可能是远视零售商联盟的稳定结构;当竞争强度较强时,则无论是短视零售商还是远视零售商都以大联盟为稳定结构,但是,在供应商处于领导地位的市场中,远视零售商形成大联盟的阈值较高;在供应商和零售商地位相同的市场中,大联盟则是远视零售商和短视零售商共同的稳定结构。     

The consistent Shapley value for hyperplane games

A new value is defined for n-person hyperplane games, i.e., non-sidepayment cooperative games, such that for each coalition, the Pareto optimal set is linear. This is a generalization of the Shapley value for side-payment games. It is shown that this value is consistent in the sense that the payoff in a given game is related to payoffs in reduced games (obtained by excluding some players) in such a way that corrections demanded by coalitions of a fixed size are cancelled out. Moreover, this is the only consistent value which satisfies Pareto optimality (for the grand coalition), symmetry and covariancy with respect to utility changes of scales. It can be reached by players who start from an arbitrary Pareto optimal payoff vector and make successive adjustments.     

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.     

Weighted multiple majority games with unions: Generating functions and applications to the European Union

An a priori system of unions or coalition structure is a partition of a finite set of players into disjoint coalitions which have made a prior commitment to cooperate in playing a game. This paper provides “ready-to-apply” procedures based on generating functions that are easily implementable to compute coalitional power indices in weighted multiple majority games. As an application of the proposed procedures, we calculate and compare coalitional power indices under the decision rules prescribed by the Treaty of Nice and the new rules proposed by the Council of the European Union.     

微分对策中的联盟解

提出时间区间[t_0,∞)上的n人微分对策两阶段联盟解. 在第一阶段不能形成大联盟的假设是自然的,即源于这一思想. 在第一阶段以联盟作为局中人的对策中计算得到其纳什均衡,之后对每个联盟的收益按Shapley值进行分配. 一个n人微分减排模型的例子阐明了上述结果.     

An outranking-based general approach to solving group multi-objective optimization problems

This paper presents a general approach to solving multi-objective programming problems with multiple decision makers. The proposal is based on optimizing a bi-objective measure of “collective satisfaction”. Group satisfaction is understood as a reasonable balance between the strengths of an agreeing and an opposing coalition, considering also the number of decision makers not belonging to any of these coalitions. Accepting the vagueness of “collective satisfaction”, even the vagueness of “person satisfaction”, fuzzy outranking relations and other fuzzy logic models are used.     

Axiomatizations of symmetrically weighted solutions

If the excesses of the coalitions in a transferable utility game are weighted, then we show that the arising weighted modifications of the well-known (pre)nucleolus and (pre)kernel satisfy the equal treatment property if and only if the weight system is symmetric in the sense that the weight of a subcoalition of a grand coalition may only depend on the grand coalition and the size of the subcoalition. Hence, the symmetrically weighted versions of the (pre)nucleolus and the (pre)kernel are symmetric, i.e., invariant under symmetries of a game. They may, however, violate anonymity, i.e., they may depend on the names of the players. E.g., a symmetrically weighted nucleolus may assign the classical nucleolus to one game and the per capita nucleolus to another game.     

An acyclic relation for comparison of bargaining powers of coalitions and its interrelationship with bargaining set

This paper proposes a method to compare bargaining power of coalitions within the framework of games in coalition form with transferable utility. The method is expressed by a relation on the set of all coalitions in a game, the relation which is defined based on the players’ bargaining power. It is shown in this paper that the newly defined relation satisfies acyclicity. Also, it is verified in this paper that the set of all individually rational payoff configurations under which all coalitions have the equal bargaining power coincides with the bargaining set. Some examples demonstrate how the newly proposed method works.     

Cooperative games with large cores

A payoff vector in ann-person cooperative game is said to be acceptable if no coalition can improve upon it. The core of a game consists of all acceptable vectors which are feasible for the grand coalition. The core is said to be large if for every acceptable vectory there is a vectorx in the core withx?y. This paper examines the class of games with large cores.     

Balancedness conditions for exact games

We provide two new characterizations of exact games. First, a game is exact if and only if it is exactly balanced; and second, a game is exact if and only if it is totally balanced and overbalanced. The condition of exact balancedness is identical to the one of balancedness, except that one of the balancing weights may be negative, while for overbalancedness one of the balancing weights is required to be non-positive and no weight is put on the grand coalition. Exact balancedness and overbalancedness are both easy to formulate conditions with a natural game-theoretic interpretation and are shown to be useful in applications. Using exact balancedness we show that exact games are convex for the grand coalition and we provide an alternative proof that the classes of convex and totally exact games coincide. We provide an example of a game that is totally balanced and convex for the grand coalition, but not exact. Finally we relate classes of balanced, totally balanced, convex for the grand coalition, exact, totally exact, and convex games to one another.     

Strong equilibria in claim games corresponding to convex games

This paper deals with a specific aspect of the problem of coalition formation in a situation described by a TU-game. First, we define a very simple normal form game which models the process of coalition formation. To define the payoff functions of the players we use an allocation rule for TU-games. The main objective of this paper is ascertain what conditions of the allocation rule lead to the grand coalition being a strong equilibrium of the normal form game, when the original TU-game is convex. Received January 1996/Revised version December 1996/Final version May 1997     

The aspiration approach to predicting coalition formation and payoff distribution in sidepayment games

This paper presents the aspiration approach to coalition formation and payoff distribution in games with sidepayments. The approach is based on the idea that players set prices for their participation within coalitions. The solution space which is appropriate for price-setting players is different from that of the usual solution concepts and is called the space of aspirations. Solution concepts defined on the space of aspirations correspond to notions of how players bargain over their prices. Once the players choose a vector of prices, the coalitions which can afford to pay these prices are the coalitions which are predicted to form in the game.     

Coalitions and payoffs in three‐person sequential games: Initial tests of two formal models

In this paper we develop two formal models predicting coalitions and payoffs among rank striving players in a sequential three‐person game. We test the models’ predictions with data from a laboratory study of eleven male triads. Each triad plays a sequence of games; in each game a two‐person coalition forms and divides the coalition's point value between the two coalition partners. Participants know that the sequence of games will end without warning at a randomly chosen time; at the sequence's end each player's monetary payoff is a linear function of the rank of his accumulated point score, relative to those of the other members of his triad. The complexity of this situation prevents players and analysts from representing it as a single game; thus they are unable to use n‐person game theory to identify optimal strategies. Consequently, we assume that players, unable to develop strategies that are demonstrably optimal in the long run, adopt certain bargaining heuristics and surrogate short run objectives.

The two models follow the same basic outline; they differ, however, in the planning horizon they assume players to use. Proceeding from a priori assumptions concerning each player's decision calculus and the bargaining process, the two models state the probability that each coalition forms and predict the point divisions in the winning coalition. The laboratory data provide consistently strong support for the predictions of both models.     

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.     

基于平均树值的无圈图博弈有效解

本文对无圈图博弈进行了研究,考虑了大联盟收益不小于各分支收益之和的情况。通过引入剩余公平分配性质,也就是任意两个分支联盟的平均支付变化相等,给出了一个基于平均树值的无圈图博弈有效解。同时,结合有效性和分支公平性对该有效解进行了刻画。特别地,若无圈图博弈满足超可加性时,证明了该有效解一定是核中的元素,说明此时的解是稳定的。最后,通过一案例分析了该有效解的特点,即越大的分支分得的剩余越多,并且关键参与者,也就是具有较大度的参与者可获得相对多的支付。     

The Harsanyi paradox and the “right to talk” in bargaining among coalitions

We describe a coalitional value from a non-cooperative point of view, assuming coalitions are formed for the purpose of bargaining. The idea is that all the players have the same chances to make proposals. This means that players maintain their own “right to talk” when joining a coalition. The resulting value coincides with the weighted Shapley value in the game between coalitions, with weights given by the size of the coalitions. Moreover, the Harsanyi paradox (forming a coalition may be disadvantageous) disappears for convex games.     

Potentials and reduced games for share functions

A value function for cooperative games with transferable utility assigns to every game a distribution of the payoffs. A value function is efficient if for every such a game it exactly distributes the worth that can be obtained by all players cooperating together. An approach to efficiently allocate the worth of the ‘grand coalition’ is using share functions which assign to every game a vector whose components sum up to one. Every component of this vector is the corresponding players’ share in the total payoff that is to be distributed. In this paper we give characterizations of a class of share functions containing the Shapley share function and the Banzhaf share function using generalizations of potentials and of Hart and Mas-Colell's reduced game property.     

A new basis and the Shapley value

The purpose of this paper is to introduce a new basis of the set of all TU games. Shapley (1953) introduced the unanimity game in which cooperation of all players in a given coalition yields payoff. We introduce the commander game in which only one player in a given coalition yields payoff. The set of the commander games forms a basis and has two properties. First, when we express a game by a linear combination of the basis, the coefficients related to singletons coincide with the Shapley value. Second, the basis induces the null space of the Shapley value.     

Coalition formation in the triad when two are weak and one is strong

Eighteen groups of subjects each participated in five different computer-controlled superadditive 3-person characteristic function games with sidepayments, that modeled negotiable conflicts in which two of the players are weak and one is considerably stronger. Both the degree to which the strong player was powerful and the type of communication were experimentally manipulated. The 90 game outcomes rejected any solution concept that predicts a single payoff vector for a given coalition structure, but supported the recently developed single-parameter α-power model that allows range predictions. Both the degree of power and type of communication were found to affect game outcomes and to determine the predictive power of models that make point predictions in 3-person games.