Duality on combinatorial structures. An application to cooperative games
AbstractThis paper deals with cooperative games in which only certain coalitions are feasible. Several combinatorial structures have been used to represent feasible coalition systems for different situations: poset of ideals, connected sets in graphs, convex geometries or antimatroids. Augmenting systems are combinatorial structures generalizing antimatroids defined in the framework of the game theory. In this paper, we introduced decreasing systems as dual structures of the augmenting systems. We connect the classical concept of dual game with the duality between these structures. The Shapley and the Banzhaf values are studied and compared for augmenting and decreasing systems.