Solution concepts for games with general coalitional structure

We introduce a theory of marginal values and their core stability for cooperative games with transferable utility and arbitrary set systems representing the set of feasible coalitions. The theory is based on the notion of strictly nested sets in a set system. For each maximal strictly nested set, we define a unique marginal contribution vector. Using these marginal contribution vectors several solutions concepts are introduced.

The gravity center or GC-solution of a game is defined as the average of the marginal vectors over all maximal strictly nested sets. For union stable set systems, buildings sets, the GC-solution differs from Myerson-type solutions. The half-space or HS-solution is defined as the average of the marginal vectors over the class of so-called half-space nested sets and is appropriate for example when feasible coalitions represent social networks. The normal tree or NT-solution is defined as the average of the marginal vectors over all so-called NT-nested sets and is appropriate when feasibility of a coalition is based on bilateral communication between players. For graphical building sets, the NT-solution is equal to the average tree solution. We also study core stability of the solutions and show that the conditions under which the HS- and NT-solutions belong to the core are weaker than conditions under which the GC-solution is stable.

For a more general set system, there exists a unique minimal building set containing the set system, its building covering. As solutions for games on an arbitrary set system of feasible coalitions we propose to take the solutions for its building covering with respect to the M-extension of the characteristic function of the game.