How do coalitions get built? Evidence from an extensive form coalition game with and without communication

In the laboratory, we investigate a non-cooperative three-person coalition game with externalities and the opportunity to extend existing coalitions. One bargainer, the builder, can propose and build a coalition over two stages. We examine the hypothesis that both absolute and relative payoffs affect the coalition formation process (and outcome). We observe many inefficient two-person final coalitions, and that the distribution of outcomes is sensitive to the constellation of both absolute and relative payoffs. Relative payoffs appear to be applied more myopically than has been observed in bilateral sequential bargaining games or suggested by social preference models. We hypothesize that the prevalence of two-person coalitions stems from builder’s uncertainty about individual acceptance thresholds. In fact, allowing nonbinding communication among the bargainers increases the prevalence of efficient coalitions. The main implication is that efficient coalition building involves strategies for mitigating the strategic uncertainty inherent in building coalitions.     

The Shapley value for arbitrary families of coalitions

We address the problem of finding a suitable definition of a value similar to that of Shapley’s, when the games are defined on a subfamily of coalitions with no structure. We present two frameworks: one based on the familiar efficiency, linearity and null player axioms, and the other on linearity and the behavior on unanimity games. We give several properties and examples in each case, and give necessary and sufficient conditions on the family of coalitions for the approaches to coincide.     

Coalitional unanimity versus strategy-proofness in coalition formation problems

This paper examines coalition formation problems from the viewpoint of mechanism design. We consider the case where (i) the list of feasible coalitions (those coalitions which are permitted to form) is given in advance; and (ii) each individual’s preference is a ranking over those feasible coalitions which include this individual. We are interested in requiring the mechanism to guarantee each coalition the “right” of forming that coalition at least when every member of the coalition ranks the coalition at the top. We name this property coalitional unanimity. We examine the compatibility between coalitional unanimity and incentive requirements, and prove that if the mechanism is strategy-proof and respects coalitional unanimity, then for each preference profile, there exists at most one strictly core stable partition, and the mechanism chooses such a partition whenever available. Further, the mechanism is coalition strategy-proof and respects coalitional unanimity if, and only if, the strictly core stable partition uniquely exists for every preference profile.     

Axiomatizations of the Shapley value for games on augmenting systems

This paper deals with cooperative games in which only certain coalitions are allowed to form. There have been previous models developed to confront the problem of unallowable coalitions. Games restricted by a communication graph were introduced by Myerson and Owen. In their model, the feasible coalitions are those that induce connected subgraphs. Another type of model is introduced in Gilles, Owen and van den Brink. In their model, the possibilities of coalition formation are determined by the positions of the players in a so-called permission structure. Faigle proposed another model for cooperative games defined on lattice structures. We introduce a combinatorial structure called augmenting system which is a generalization of the antimatroid structure and the system of connected subgraphs of a graph. In this framework, the Shapley value of games on augmenting systems is introduced and two axiomatizations of this value are showed.     

Caplow's model: A reformulation†

This paper examines Caplow's model of coalitions in the triad and identifies several shortcomings. Retaining the basic structure of Caplow's arguments, it recasts the problem as one of interdependent rational decision making. The reformulation allows increased generality in several directions, including a more general notion of strength, relaxation of the additivity assumption about strength, and the adoption of probabilistic choice to allow predictions about the relative frequencies of the various coalitions.     

超图结构上合作博弈的赋权Position值

在合作博弈的一般模型中总是假设所有联盟都能形成。不过,在实际中由于受到一些因素的制约,有些联盟是不能形成的。基于此,Myerson提出了具有图通讯结构的合作博弈。Myerson值和Position值是超图博弈上的两个重要分配规则。2005年,Slikker给出了在图博弈上Position值的公理化刻画。但超图博弈上Position值的公理化刻画一直悬而未决。本文通过引入“赋权平衡超边贡献公理”,并结合经典的“分支有效性”,提出了超图博弈上赋权Position值的公理化刻画。作为推论,解决了超图博弈上Position值的公理化刻画问题。     

Exploring the VCG mechanism in combinatorial auctions: The threshold revenue and the threshold-price rule

We explore interesting potential extensions of the Vickrey–Clarke–Groves (VCG) rule under the assumption of players with independent and private valuations and no budget constraints. First, we apply the VCG rule to a coalition of bidders in order to compute the second price of the coalition. Then, we introduce and formulate the problem of determining that partition of players into coalitions which maximize the auctioneer’s revenue in the case whereby such coalitions take part to a VCG auction each one as a single agent; in particular, we provide an integer linear formulation of this problem. We also generalize this issue by allowing players to simultaneously belong to distinct coalitions in the case that players’ valuation functions are separable. Finally, we propose some applications of these theoretical results. For instance, we exploit them to provide a class of new payment rules and to decide which bids should be defined as the highest losing ones in combinatorial auctions.     

Dynamic resource allocation in fuzzy coalitions: a game theoretic model

We introduce an efficient and dynamic resource allocation mechanism within the framework of a cooperative game with fuzzy coalitions (cooperative fuzzy game). A fuzzy coalition in a resource allocation problem can be so defined that membership grades of the players in it are proportional to the fractions of their total resources. We call any distribution of the resources possessed by the players, among a prescribed number of coalitions, a fuzzy coalition structure and every membership grade (equivalently fraction of the total resources), a resource investment. It is shown that this resource investment is influenced by the satisfaction of the players in regard to better performance under a cooperative setup. Our model is based on the real life situations, where possibly one or more players compromise on their resource investments in order to help forming coalitions.     

A new binary relation to compare viability of winning coalitions and its interrelationships to desirability relation and blockability relation

This paper aims to propose a new type of binary relations, called the viability relation, defined on the set of all coalitions in a simple game for a comparison of coalition influence, and to investigate its properties, especially its interrelationships to the desirability relation and the blockability relation. The viability relation is defined to compare coalitions based on their robustness over deviation of their members for complementing the inability of the desirability relation and the blockability relation to make a distinguishable comparison among winning coalitions. It is verified in this paper that the viability relation on a simple game is always transitive and is complete if and only if the simple game is S-unanimous for a coalition S. Examples show that there are no general inclusion relations among the desirability relation, the blockability relation and the viability relation. It is also verified that the viability relation and the blockability relation are complementary to each other. Specifically, the blockability relation between two coalitions is equivalent to the inversed viability relation between the complements of the two coalitions.     

A characterization of concordance relations

The notion of concordance is central to many multiple criteria techniques relying on ordinal information, e.g. outranking methods. It leads to compare alternatives by pairs on the basis of a comparison of coalitions of attributes in terms of “importance”. This paper proposes a characterization of the binary relations that can be obtained using such comparisons within a general framework for conjoint measurement that allows for intransitive preferences. We show that such relations are mainly characterized by the very rough differentiation of preference differences that they induce on each attribute.     

Two-sided coalitional matchings

In a two-sided coalitional matching problem agents on each side of the market simultaneously form coalitions which then are matched to coalitions from the other market side. We assume that each agent has preferences over groups on his own market side and over groups on the opposite market side. These preferences are combined lexicographically as to examine how the existence of core stable partitions on the distinct market sides, the restriction of agents’ preferences over groups to strict orderings, and the extent to which individual preferences respect common rankings shape the existence of core stable coalitional matchings.     

A note on balancedness and nonemptiness of the core in voting games

Consider a society with a finite number,n, of individuals who have to choose an alternative from a setA in them-dimensional Euclidean space IR ^m . Assuming that the preference relation overA of every individual is convex and continuous, Greenberg (1979) showed some that if the set of winning coalitions (i.e. those that have the veto power) consists of all coalitions with more thanmn/m + 1 individuals the core of the induced game is nonempty. Greenberg and Weber (1984) have strengthened this result by showing that the induced game is in fact balanced. On the other hand Le Breton (1987), Schofield (1984a) and Strnad (1985) have generalized Greenberg's theorem to arbitrary voting games. The purpose of this note is to show that however the induced game is not generally balanced.     

Balancedness,nonemptiness of the core and the existence of a majority equilibrium

Consider a society with a finite number,n, of individuals who have to choose an alternative from a setA in them-dimensional Euclidean Space, R ^m . Assuming that the preference relation overA of every individual is convex and continuous,Greenberg [1979] showed that if the set of winning coalitions (i.e. those that have the veto power) consists of all coalitions with more thanmn/(m+1) individuals, then there exists an alternativea ^* ?A which is a majority equilibrium, or equivalently, that the core of the induced game is nonempty. In the present paper we strengthen this result by showing that this game not only has a nonempty core but is also balanced.     

Strongly essential coalitions and the nucleolus of peer group games

Most of the known efficient algorithms designed to compute the nucleolus for special classes of balanced games are based on two facts: (i) in any balanced game, the coalitions which actually determine the nucleolus are essential; and (ii) all essential coalitions in any of the games in the class belong to a prespecified collection of size polynomial in the number of players. We consider a subclass of essential coalitions, called strongly essential coalitions, and show that in any game, the collection of strongly essential coalitions contains all the coalitions which actually determine the core, and in case the core is not empty, the nucleolus and the kernelcore. As an application, we consider peer group games, and show that they admit at most 2n−1 strongly essential coalitions, whereas the number of essential coalitions could be as much as 2ⁿ⁻¹. We propose an algorithm that computes the nucleolus of an n-player peer group game in time directly from the data of the underlying peer group situation.Research supported in part by OTKA grant T030945. The authors thank a referee and the editor for their suggestions on how to improve the presentation     

The restricted core of games on distributive lattices: how to share benefits in a hierarchy

Finding a solution concept is one of the central problems in cooperative game theory, and the notion of core is the most popular solution concept since it is based on some rationality condition. In many real situations, not all possible coalitions can form, so that classical TU-games cannot be used. An interesting case is when possible coalitions are defined through a partial ordering of the players (or hierarchy). Then feasible coalitions correspond to teams of players, that is, one or several players with all their subordinates. In these situations, the core in its usual formulation may be unbounded, making its use difficult in practice. We propose a new notion of core, called the restricted core, which imposes efficiency of the allocation at each level of the hierarchy, is always bounded, and answers the problem of sharing benefits in a hierarchy. We show that the core we defined has properties very close to the classical case, with respect to marginal vectors, the Weber set, and balancedness.     

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.     

大型群体决策问题中的简单多数规则（3个评选对象场合）

一个群体决策问题取决于两个因素,一个是群体决策的规则,另一个是投票。当选定群体决策规则时,一个群体决策问题由投票完全决定,此时,群体决策问题与投票之间一一对应。简单多数规则是个简单且被广泛采用的群体决策规则,但它有缺陷,我们可举出些群体决策问题使用简单多数规则没法从投票得到最后群体决策的结果。这里我们将给出一个简单多数规则的有趣性质,即在3个评选对象场合,使用简单多数规则没法从投票得到最后群体决策结果的n个评选人的群体决策问题的个数与所有n个评选人的群体决策问题的个数之比当评选人个数n趋向无穷时趋于零,这说明3个评选对象的大型群体决策场合,简单多数规则的缺陷不严重。     

Solving multiple criteria problems by interactive decomposition

An interactive decomposition method is developed for solving the multiple criteria (MC) problem. Based on nonlinear programming duality theory, the MC problem is decomposed into a series of subproblems and relaxed master problems. Each subproblem is a bicriterion problem, and each relaxed master problem is a standard linear program. The prime objective of the decomposition is to simplify and facilitate the process of making preference assessments and tradeoffs across many conflicting objectives. Therefore, the decision-maker's preference function is not assumed to be known explicitly; rather, the decision maker is required to make only limited local preference assessments in the context of feasible and nondominated alternatives. Also, the preference assessments are of the form of ordinal paired comparisons, and in most of them only two criteria are allowed to change their values simultaneously, while the remaining (l–2) criteria are held fixed at certain levels.     

Arc-based integer programming formulations for three variants of proportional symbol maps

Proportional symbol maps are a cartographic tool that employs scaled symbols to represent data associated with specific locations. The symbols we consider are opaque disks, which may be partially covered by other overlapping disks. We address the problem of creating a suitable drawing of the disks that maximizes one of two quality metrics: the total and the minimum visible length of disk boundaries. We study three variants of this problem, two of which are known to be NP-hard and another whose complexity is open. We propose novel integer programming formulations for each problem variant and test them on real-world instances with a branch-and-cut algorithm. When compared with state-of-the-art models from the literature, our models significantly reduce computation times for most instances.     

The core and the Weber set of games on augmenting systems

This paper deals with cooperative games in which only certain coalitions are allowed to form. There have been previous models developed to confront the problem of unallowable coalitions. Games restricted by a communication graph were introduced by Myerson and Owen. In their model, the feasible coalitions are those that induce connected subgraphs. Another type of model is introduced in Gilles, Owen and van den Brink. In their model, the possibilities of coalition formation are determined by the positions of the players in a so-called permission structure. Faigle proposed a general model for cooperative games defined on lattice structures. In this paper, the restrictions to the cooperation are given by a combinatorial structure called augmenting system which generalizes antimatroid structure and the system of connected subgraphs of a graph. In this framework, the core and the Weber set of games on augmenting systems are introduced and it is proved that monotone convex games have a non-empty core. Moreover, we obtain a characterization of the convexity of these games in terms of the core of the game and the Weber set of the extended game.