Weightedness and structural characterization of hierarchical simple games

In this paper we give structural charaterizations for disjunctive and conjunctive hierarchical simple games by characterizing them as complete games with a unique shift-maximal losing and, respectively, shift-minimal winning coalitions. We prove canonical representation theorems for both types of hierarchical games and establish duality between them. We characterize those disjunctive and conjunctive hierarchical games which 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.