Interaction indices for games on combinatorial structures with forbidden coalitions

The notion of interaction among a set of players has been defined on the Boolean lattice and Cartesian products of lattices. The aim of this paper is to extend this concept to combinatorial structures with forbidden coalitions. The set of feasible coalitions is supposed to fulfil some general conditions. This general representation encompasses convex geometries, antimatroids, augmenting systems and distributive lattices. Two axiomatic characterizations are obtained. They both assume that the Shapley value is already defined on the combinatorial structures. The first one is restricted to pairs of players and is based on a generalization of a recursivity axiom that uniquely specifies the interaction index from the Shapley value when all coalitions are permitted. This unique correspondence cannot be maintained when some coalitions are forbidden. From this, a weak recursivity axiom is defined. We show that this axiom together with linearity and dummy player are sufficient to specify the interaction index. The second axiomatic characterization is obtained from the linearity, dummy player and partnership axioms. An interpretation of the interaction index in the context of surplus sharing is also proposed. Finally, our interaction index is instantiated to the case of games under precedence constraints.

► We consider interaction indices for coalitions structures that are formed of maximal chains.
► The concepts of reduce game and restricted game are defined for such coalitions structures.
► The interaction indices for pairs of players are characterized by the Linearity, Dummy player and Recursivity axioms.
► The interaction indices for any coalition are characterized by the Linearity, Dummy player and Partnership axioms.
► A simple combinatorial formula is obtained for games under precedence constraints.