Solution Concepts for Games with General Coalition Structure

Abstract

We introduce a theory on marginal values and their core stability for cooperative games with arbitrary coalition structure. The theory is based on the notion of nested sets and the complex of nested sets associated to an arbitrary set system and the M-extension of a game for this set. For a set system being a building set or partition system, the corresponding complex is a polyhedral complex, and the vertices of this complex correspond to maximal strictly nested sets. To each maximal strictly nested set is associated a rooted tree. Given characteristic function, to every maximal strictly nested set a marginal value is associated to a corresponding rooted tree as in [9]. We show that the same marginal value is obtained by using the M-extension for every permutation that is associated to the rooted tree. The GC-solution is dened as the average of the marginal values over all maximal strictly nested sets. The solution can be viewed as the gravity center of the image of the vertices of the polyhedral complex. The GC-solution diers from the Myerson-kind value dened in [2] for union stable structures. The HS-solution is defined as the average of marginal values over the subclass of so-called half-space nested sets. The NT-solution is another solution and is defined as the average of marginal values over the subclass of NT-nested sets. For graphical buildings the collection of NT-nested sets corresponds to the set of spanning normal trees on the underlying graph and the NT-solution coincides with the average tree solution. We also study core stability of the solutions and show that both the HS-solution and NT-solution belong to the core under half-space supermodularity, which is a weaker condition than convexity of the game.

For an arbitrary set system we show that there exists a unique minimal building set containing the set system. As solutions we take the solutions for this building covering by extending in a natural way the characteristic function to it by using its Mobius inversion.

Keywords: Core, polytope, building set, nested set complex, Mobius inversion, permutations

JEL Classification: C71