SUMMARY In this paper, we present a control scheme in which a set of agents switches between different network topologies in order to regulate a set of unconstrained linear systems. The problems of how to decide the time‐varying communication strategy and the corresponding control strategies are addressed. A design method that guarantees closed‐loop stability for the proposed scheme is also provided. In addition, the network topology optimization problem is posed as a cooperative game, so that tools from game theory can be used to study the relevance of the different links and the agents. Finally, the proposed scheme is illustrated through a simulation example. Copyright © 2013 John Wiley & Sons, Ltd.

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 …rst 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 de…ne 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.

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 e¤ect a swing. We introduce a combinatorial method based in generating functions for computing these power indices e¢ciently and we study the time complexity of the algorithms. We also analyze the meet of two weighted voting games. Finally, we compute the voting power in the Council of Ministers of the European Union with the generating functions algorithms and we present its implementation in the system Mathematica. Mathematics Subject Classi…cation 2000: 91A12.

Abstract. In the classical model of cooperative games, it is considered that each coalition of players can form and cooperate to obtain its worth. However, we can think that in some situations this assumption is not real, that is, all the coalitions are not feasible. This suggests that it is necessary to rise the whole question of generalizing the concept of cooperative game, and therefore to introduce appropriate solution concepts. We propose a model for games on a matroid, based in several important properties of this combinatorial structure and we introduce the probabilistic Shapley value for games on matroids.

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. Mathematics Subject Classification 2000: 91A12