Methods for comparison of coalition influence on games in characteristic function form and their interrelationships

This paper extends two existent methods, called the blockability relation and the viability relation, for simple games to compare influence of coalitions, to those for games in characteristic function form, and shows that the newly defined relations satisfy transitivity and completeness. It is shown in this paper that for every game in characteristic function form the blockability relation and the viability relation have a complementary interrelationship.     

The desirability relation of simple games

We prove some properties of simple games with a complete desirability relation, and investigate the relationships between the desirability of a simple game G and that of some simple games that are derived from G. We also provide an example of a proper simple game that has a complete and acyclic desirability relation but is not a weighted majority 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.     

Fundamentals of simple games from a viewpoint of blockability relations

This paper shows some elementary facts on simple games with respect to blockability relations. It is verified in this paper that fundamental concepts on simple games as null players, dictators, veto players, and so on can be expressed in terms of blockability relations. More, some new concepts as “conflict-free” and so on, are introduced from the viewpoint of blockability relations into the framework of simple games.     

The collective value: a new solution for games with coalition structures

In this study, we provide a new solution for cooperative games with coalition structures. The collective value of a player is defined as the sum of the equal division of the pure surplus obtained by his coalition from the coalitional bargaining and of his Shapley value for the internal coalition. The weighted Shapley value applied to a game played by coalitions with coalition-size weights is assigned to each coalition, reflecting the size asymmetries among coalitions. We show that the collective value matches exogenous interpretations of coalition structures and provide an axiomatic foundation of this value. A noncooperative mechanism that implements the collective value is also presented.     

The lattice of embedded subsets

In cooperative game theory, games in partition function form are real-valued function on the set of the so-called embedded coalitions, that is, pairs (S,π) where S is a subset (coalition) of the set N of players, and π is a partition of N containing S. Despite the fact that many studies have been devoted to such games, surprisingly nobody clearly defined a structure (i.e., an order) on embedded coalitions, resulting in scattered and divergent works, lacking unification and proper analysis. The aim of the paper is to fill this gap, thus to study the structure of embedded coalitions (called here embedded subsets), and the properties of games in partition function form.     

On the use of binary decision diagrams for solving problems on simple games

Simple games are yes/no cooperative games which arise in many practical applications. Recently, we have used reduced ordered binary decision diagrams and quasi-reduced ordered binary decision diagrams (abbreviated as Robdds and Qobdds, respectively) for the representation of simple games and for the computation of some power indices. In the present paper, we continue this work. We show how further important computational problems on simple games can be solved using Qobdds, viz. the identification of some key players, the computation of the desirability relation on individuals, the test whether a simple game is proper and strong, respectively, and the computation of Qobdd-representations for the sets of all minimal winning coalitions, all shift-minimal winning coalitions and all blocking coalitions, respectively. Applications of these solutions include the computation of recent power indices based on shift-minimal winning coalitions and the test for linear separability of a directed simple game.     

Hierarchical organization structures and constraints on coalition formation

This paper studies the constraints in coalition formation that result from a hierarchical organization structure on the class of players in a cooperative game with transferable utilities. If one assumes that the superiors of a certain individual have to give permission to the actions undertaken by the individual, then one arrives at a limited collection of formable orautonomous coalitions. This resulting collection is a lattice of subsets on the player set. We show that if the collection of formable coalitions is limited to a lattice, the core allows for (infinite) exploitation of subordinates. For discerning lattices we are able to generalize the results of Weber (1988), namely the core is a subset of the convex hull of the collection of all attainable marginal contribution vectors plus a fixed cone. This relation is an equality if and only if the game is convex. This extends the results of Shapley (1971) and Ichiishi (1981).     

The Myerson value for directed graph games

A directed graph game consists of a cooperative game with transferable utility and a digraph which describes limited cooperation and the dominance relation among the players. Under the assumption that only coalitions of strongly connected players are able to fully cooperate, we introduce the digraph-restricted game in which a non-strongly connected coalition can only realize the sum of the worths of its strong components. The Myerson value for directed graph games is defined as the Shapley value of the digraph-restricted game. We establish axiomatic characterizations of the Myerson value for directed graph games by strong component efficiency and either fairness or bi-fairness.     

Simple games and weighted games: A theoretical and computational viewpoint

It is a well-known result in the theory of simple games that a game is weighted if and only if it is trade robust. In this paper we propose a variant of trade robustness, that we call invariant-trade robustness, which is enough to determine whether a simple game is weighted. To test whether a simple game is invariant-trade robust we do not need to consider all winning coalitions; a reduced subset of minimal winning coalitions is enough.We make a comparison between the two methods (trade robustness and invariant-trade robustness) to check whether a simple game is weighted. We also provide by means of algorithms a full classification using both methods, for simple games with less than 8 voters according to the maximum level of (invariant-)trade robustness they achieve.     

On connected coalitions in dominated simple games

In a book by Axelrod it is claimed that, in the presence of well defined policy order, only connected coalitions form. Here we investigate the compatability of Axelrod's hypothesis with several hypotheses (about coalition formation in dominated simple game) that were formulated by Peleg.     

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.     

Forms of representation for simple games: Sizes,conversions and equivalences

Simple games are cooperative games in which the benefit that a coalition may have is always binary, i.e., a coalition may either win or loose. This paper surveys different forms of representation of simple games, and those for some of their subfamilies like regular games and weighted games. We analyze the forms of representations that have been proposed in the literature based on different data structures for sets of sets. We provide bounds on the computational resources needed to transform a game from one form of representation to another one. This includes the study of the problem of enumerating the fundamental families of coalitions of a simple game. In particular we prove that several changes of representation that require exponential time can be solved with polynomial-delay and highlight some open problems.     

Cooperation in dividing the cake

This paper defines models of cooperation among players partitioning a completely divisible good (such as a cake or a piece of land). The novelty of our approach lies in the players’ ability to form coalitions before the actual division of the good with the aim to maximize the average utility of the coalition. A social welfare function which takes into account coalitions drives the division. In addition, we derive a cooperative game which measures the performance of each coalition. This game is compared with the game in which players start cooperating only after the good has been portioned and has been allocated among the players. We show that a modified version of the game played before the division outperforms the game played after the division.     

The path relation for directed planar graphs in rectangles, and its relation to the free diad

In this paper, we define the path relation of a directed graph to be the relation which relates two vertices if there is a path from the first to the second. We study the restriction of this relation to paths from sources to sinks, and consider the question of when two finite graphs embedded in a rectangle give the same relation. We find a set of local changes to these graphs which can be used to get between any two graphs for which this relation is the same. Furthermore, we classify the relations which can arise as this relation for a finite directed graph embedded in a rectangle as the triconvex relations between finite ordinals (defined in this paper).This work originated from some of the author’s work on category theory. It turns out that the category of finite ordinals and relations that can be the path relation of a directed graph embedded in a rectangle, is relevant to the study of diads—introduced by the author as a common generalisation of monads and comonads (note that the terms diad and dyad have been used to mean different things by other authors). More specifically, the referee of one of the author’s papers suggested that it would be useful to identify the category which plays the role for diads that the category of finite ordinals and order-preserving functions plays for monads. It turns out that the category of finite ordinals and relations that can be path relations of graphs embedded in a rectangle, is exactly the category that plays this role.     

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.     

Efficient firm groups: Allocative efficiency in cooperative games

The concept of efficiency in groups postulates that a coalition of firms has to record a smaller distance toward the aggregate technology frontier compared with the sum of individual distances. Efficiency analysis (either allocative or technical) is defined with respect to cooperative firm game in order to provide operational distance functions, the so-called pseudo-distance functions. These pseudo-distances belong to the core interior of the allocative firm game, in other terms, any given firm coalition may always improve its allocative efficiency. We prove that such a result is impossible for technical efficiency, i.e., the technical efficiency cannot increase for all possible coalitions.     

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.     

Cooperative games with coalition structures

Many game-theoretic solution notions have been defined or can be defined not only with reference to the all-player coalition, but also with reference to an arbitrary coalition structure. In this paper, theorems are established that connect a given solution notion, defined for a coalition structure ? with the same solution notion applied to appropriately defined games on each of the coalitions in ?. This is done for the kernel, nucleolus, bargaining set, value, core, and thevon Neumann-Morgenstern solution. It turns out that there is a single function that plays the central role in five out of the six solution notions in question, though each of these five notions is entirely different. This is an unusual instance of a game theoretic phenomenon that does not depend on a particular solution notion but holds across a wide class of such notions.     

Transshipment games with identical newsvendors and cooperation costs

In a transshipment game, supply chain agents cooperate to transship surplus products. Although the game has been well studied in the OR literature, the fundamental question whether the agents can afford cooperation costs to set up and maintain the game in the first place has not been addressed thus far. This paper addresses this question for the cooperative transshipment games with identical agents having normally distributed independent demands. We provide characterization of equal allocations which are in the core of symmetric games, and prove that not all transshipment games are convex. In particular, we prove that though individual allocations grow with the coalition size, the growth diminishes according to two rules of diminishing individual allocations. These results are the basis for studying the games with cooperation costs. We model the cooperation costs by the cooperation network topology and the cooperation cost per network link. We consider two network topologies, the clique and the hub, and prove bounds for the cost per link that render coalitions stable. These bounds always limit coalition size for cliques. However, the opposite is shown for hubs, namely newsvendors can afford cooperation costs only if their coalition is sufficiently large.