Hierarchical Simple Games: Representations and Weightedness

Abstract: In many situations, both in human and artificial societies, cooperating agents have different status with respect to the activity and it is not uncommon that certain actions are only allowed to coalitions that satisfy certain criteria, e.g., to sufficiently large coalitions or coalitions which involve players of sufficient seniority. Simmons (1988) formalized this idea in the context of secret sharing schemes by defining the concept of a (disjunctive) hierarchical access structure. Tassa (2007) introduced their conjunctive counterpart. From the game theory perspective access structures in secret sharing schemes are simple games. In this paper we prove the duality between disjunctive and conjunctive hierarchical games. We introduce a canonical representation theorem for both types of hierarchical games and characterize disjunctive ones as complete games with a unique shift-maximal losing coalition. We give a short combinatorial proof of the Beimel-Tassa-Weinreb (2008) characterization of weighted disjunctive hierarchical games. By duality we get similar theorems for conjunctive hierarchical games.