On the vertices of the k-additive core

Michel Grabisch (1) , Pedro Miranda (2)

1 CES - Centre d'économie de la Sorbonne
2 UCM - Universidad Complutense de Madrid = Complutense University of Madrid [Madrid]

The core of a game v on N, which is the set of additive games φ dominating v such that φ(N)=v(N), is a central notion in cooperative game theory, decision making and in combinatorics, where it is related to submodular functions, matroids and the greedy algorithm. In many cases however, the core is empty, and alternative solutions have to be found. We define the k-additive core by replacing additive games by k-additive games in the definition of the core, where k-additive games are those games whose Möbius transform vanishes for subsets of more than k elements. For a sufficiently high value of k, the k-additive core is nonempty, and is a convex closed polyhedron. Our aim is to establish results similar to the classical results of Shapley and Ichiishi on the core of convex games (corresponds to Edmonds' theorem for the greedy algorithm), which characterize the vertices of the core.

Mots clés: Cooperative games; Core; k-additive games; Vertices

Domaines: Mathématique discrète [cs.DM]; Informatique et théorie des jeux [cs.GT]; Economies et finances

Michel Grabisch, Pedro Miranda. On the vertices of the k-additive core. Discrete Mathematics, 2008, 308 (22), pp.5204-5217. ⟨10.1016/j.disc.2007.09.042⟩. ⟨hal-00321625⟩