Weightedness and structural characterization of hierarchical simple games

In this paper we give structural characterizations of disjunctive and conjunctive hierarchical simple games by characterizing them as complete games with a unique shift-maximal losing coalition, and a unique shift-minimal winning coalition, respectively. We prove canonical representation theorems for both types of hierarchical games and establish duality between them. We characterize the disjunctive and conjunctive hierarchical games that are weighted majority games. This paper was inspired by Beimel et al. (2008) and Farràs and Padró (2010) characterizations of ideal weighted threshold access structures of secret sharing schemes.