Resumen de Minimum dimension and codimension for simple games

Josep Freixas Bosch , Dorota Marciniak

• The concept of dimension is based on the fact each simple game can be expressed as a nite intersection of weighted simple games. The question of eciency leads to the de nition of dimension. Given a particular subclass of simple games, we rstly study whether it is possible to express every simple game as an intersection of games in that subclass. We make an exhaustive study for signi cant subclasses of simple games like e.g.: homogeneous, linear, weakly linear, strong or proper games. By duality we extend this concept of dimension to that of codimension, which allows to eciently expressing a simple game as union of games. We secondly investigate the existence of minimal subclasses of weighted games for which every simple game can be expressed as the intersection of games belonging to the subclass. We prove that there is a minimum subclass with this property, which leads us to introduce a new concept of dimension.