E. Algaba

Cooperative game theory approach to allocating benefits of horizontal cooperation

Logistics costs in general, and transportation costs in particular, represent a large fraction of the operating costs of many companies. One way to try to reduce these costs is through horizontal cooperation among shippers. Thus, when the transportation needs of two or more companies are merged, their collective transportation requirements can be met at lower cost. The attainable cost savings are due to economies of scale, which translate into cheaper rates due to increased negotiation power, use of larger vehicles and bundling of shipments. In this paper, a linear model is presented and used to study the cost savings that different companies may achieve when they merge their transportation requirements. On the one hand, solving this optimization model for different collaboration scenarios allows testing and quantifying the synergies among different potential partners, thus identifying the most profitable collaboration opportunities. On the other, the problem of allocating the joint cost savings of the cooperation is tackled using cooperative game theory. The proposed approach is illustrated with an example in which different cooperative game solution concepts are compared. Extensive numerical experiments have also been carried out to gain insight into the properties of the corresponding cost savings game and the behavior of the different solution concepts.

Cooperative games on antimatroids

The aim of this paper is to introduce cooperative games with a feasible coalition system which is called antimatroid. These combinatorial structures generalize the permission structures, which have nice economical applications. With this goal, we first characterize the approaches from a permission structure with special classes of antimatroids. Next, we use the concept of interior operator in an antimatroid and we define the restricted game taking into account the limited possibilities of cooperation determined by the antimatroid. These games extend the restricted games obtained by permission structures. Finally, we provide a computational method to obtain the Shapley and Banzhaf values of the players in the restricted game, by using the worths of the original game.

Computing power indices in weighted multiple majority games

Abstract The Shapley–Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. The Banzhaf power index depends on the number of ways in which each voter can effect a swing. If the input size of the problem is n, then the function which measures the worst case running time for computing these indices is in O n2 n . We present a method based on generating functions to compute these power indices efficiently for weighted multiple majority games and we study the temporal complexity of the algorithms. Finally, we apply the algorithms obtained with this method to compute the Banzhaf and the Shapley–Shubik indices under the two decision rules adopted in the Nice European Union summit.

The Myerson value for union stable structures

We study cooperation structures with the following property: Given any two feasible coalitions with non-empty intersection, its union is a feasible coalition again. These combinatorial structures have a direct relationship with graph communication situations and conference structures à la Myerson. Characterizations of the Myerson value in this context are provided using the concept of basis for union stable systems. Moreover, TU-games restricted by union stable systems generalizes graph-restricted games and games with permission structures.

The distribution of power in the European Constitution

Abstract The aim of this paper is to analyze the distribution of voting power in the Constitution for the enlarged European Union. By using generating functions, we calculate the Banzhaf power indices for the European countries in the Council of Ministers under the decision rules prescribed by the Treaty of Nice and the new rules proposed by the European Constitution Treaty. Moreover, we analyze the power of the European citizens under the egalitarian model proposed by Felsenthal and Machover [D.S. Felsenthal, M. Machover, The measurement of voting power: Theory and practice, problems and paradoxes, Edward Elgar, Cheltenham, 1998].

The position value for union stable systems

Abstract. In this paper, we study the position value for games in which partial cooperation exist, that is based on a union stable coalition system. The concept of basis is introduced for these systems, allowing for a definition of the position value. Moreover, an axiomatic characterization of the position value is provided for a specific class of union stable systems. Conditions under which convexity is inherited from the underlying game to the conference game, and the position value is a core vector of the restricted game are provided.

Axiomatizations of the Shapley value for cooperative games on antimatroids

Abstract. Cooperative games on antimatroids are cooperative games restricted by a combinatorial structure which generalize the permission structure. So, cooperative games on antimatroids group several well-known families of games which have important applications in economics and politics. Therefore, the study of the rectricted games by antimatroids allows to unify criteria of various lines of research. The current paper establishes axioms that determine the restricted Shapley value on antimatroids by conditions on the cooperative game v and the structure determined by the antimatroid. This axiomatization generalizes the axiomatizations of both the conjunctive and disjunctive permission value for games with a permission structure. We also provide an axiomatization of the Shapley value restricted to the smaller class of poset antimatroids. Finally, we apply our model to auction situations.

Generating Functions for Computing the Myerson Value

The complexity of a computational problem is the order of computational resources which are necessary and sufficient to solve the problem. The algorithm complexity is the cost of a particular algorithm. We say that a problem has polynomial complexity if its computational complexity is a polynomial in the measure of input size. We introduce polynomial time algorithms based in generating functions for computing the Myerson value in weighted voting games restricted by a tree. Moreover, we apply the new generating algorithm for computing the Myerson value in the Council of Ministers of the European Union restricted by a communication structure.

An axiomatization of the Banzhaf value for cooperative games on antimatroids

Cooperative games on antimatroids are cooperative games in which coalition formation is restricted by a combinatorial structure which generalizes permission structures. These games group several well-known families of games which have important applications in economics and politics. The current paper establishes axioms that determine the restricted Banzhaf value for cooperative games on antimatroids. The set of given axioms generalizes the axiomatizations given for the Banzhaf permission values. We also give an axomatization of the restricted Banzhaf value for the smaller class of poset antimatroids. Finally, we apply the above results to auction situations.

An Iterative Design Method for Coalitional Control Networks with Constraints on the Shapley Value

Abstract In this work, we introduce a new iterative design method for a coalitional control scheme for linear systems recently proposed. In this scheme, the links in the network infrastructure are enabled or disabled depending on their contribution to the overall system performance. As a consequence, the local controllers are divided dynamically into sets or coalitions that cooperate in order to attain their control tasks. The new design method allows the control system designer to include new constraints regarding the game theoretical tools of the control architecture, while optimizing the matrices that define the controller.