  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.
