Jesús Mario Bilbao

Cooperative Games on Combinatorial Structures

Preface. 1. Structures. 2. Linear optimization methods. 3. Discrete convex analysis. 4. Computational complexity. 5. Restricted games by partition systems. 6. Restricted games by union stable systems. 7. Values for games on convex geometries. 8. Values for games on matroids. 9. The core, the selectope and the Weber set. 10. Simple games on closure spaces. 11. Voting power. 12. Computing values with Mathematica. Bibliography. Index.

Voting power in the European Union enlargement

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 there are n players in a voting situation, then the function which measures the worst case running time for computing these indices is in O( n 2 n ). We present a combinatorial method based in generating functions to compute these power indices efficiently in weighted double or triple majority games and we study the time complexity of the algorithms. Moreover, we calculate these power indices for the countries in the Council of Ministers of the European Union under the new decision rules prescribed by the Treaty of Nice.

Generating functions for computing power indices efficiently

TheShapley-Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. TheBanzhaf power index depends on the number of ways in which each voter can effect a swing. We introduce a combinatorial method based ingenerating functions for computing these power indices efficiently and we study thetime 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.

The Shapley value on convex geometries

Abstract A game on a convex geometry is a real-valued function defined on the family L of the closed sets of a closure operator which satisfies the finite Minkowski–Krein–Milman property. If L is the boolean algebra 2 N then we obtain an n -person cooperative game. Faigle and Kern investigated games where L is the distributive lattice of the order ideals of the poset of players. We obtain two classes of axioms that give rise to a unique Shapley value for games on convex geometries.

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].

Axioms for the Shapley value on convex geometries

The purpose of this article is an extension of Shapleys value for games with restricted cooperation. The classical model of cooperative game where every subset of players is a feasible coalition may be unrealistic. The feasible coalitions in our model will be the convex sets, i.e., those subsets of players belonging to a convex geometry L. In the last section, we apply this model to several examples about the power in the Council of Ministers of the European Union.

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.

The core of games on convex geometries

Abstract A game on a convex geometry is a real-valued function defined on the family L of the closed sets of a closure operator which satisfies the finite Minkowski–Krein–Milman property. If L is the Boolean algebra 2N then we obtain a n-person cooperative game. We will introduce convex and quasi-convex games on convex geometries and we will study some properties of the core and the Weber set of these games.