Roughly weighted hierarchical simple games

Hierarchical simple games–both disjunctive and conjunctive–are natural generalizations of $k$ -out-of- $n$ games. They are ideal in the sense that they allow most efficient and secure secret sharing schemes to be defined on these games as access structures. Another important generalization of $k$ -out-of- $n$ games with origin in economics and politics are weighted and roughly weighted majority games. Weighted hierarchical games have been classified by Beimel et al. (2008) and Gvozdeva et al. (2012); it appeared that they cannot have more than two nontrivial levels in their hierarchy. In this paper we characterize roughly weighted hierarchical games and show that they cannot have more than three nontrivial levels. This shows that hierarchical games are rather far from weighted and even roughly weighted games, and hence provide an interesting set of examples for the theory of simple games. Our methods are purely game-theoretic.