Solution Concepts for Games with General Coalitional Structure

We introduce several new solution concepts for cooperative games with arbitrary coalition structure. Of our main interest are coalitions structures being so-called building sets. A collection of sets is a building set if every singleton is a member and if the union of any two non-disjoint sets in the collection is also in the collection. For example, the vertex sets of all connected subgraphs of a connected graph form a building set. To a building set is associated a collection of strictly nested sets. These sets give rise to a polyhedral complex.Maximal strictly nested sets correspond to the vertices of this complex. For a given characteristic function on a building set a marginal payoff vector is associated to any strictly nested set. The GC-solution is defined as the average of the marginal payoff vectors over all maximal strictly nested sets. The solution can be viewed as the gravity center of the image of the vertices of the complex and differs for graphical building sets from the Myerson value. The HS-solution is defined as the average of marginal payoff vectors over the class of so-called half-space nested sets. The NT-solution is defined as the average of marginal payoff vectors over another specific class of maximal strictly nested sets, the class of so-called NT-nested sets. On graphical buildings the collection of NT-nested sets corresponds to the set of normal trees on the underlying graph and the NT-solution coincides with the average tree solution. We also study core stability of the solutions.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