Hierarchical Simple Games: Representations and Weightedness

TL;DR

This paper explores hierarchical simple games, establishing dualities, canonical representations, and characterizations of weightedness, thereby advancing the understanding of structured coalition formation in access control and secret sharing.

Contribution

It introduces a duality between disjunctive and conjunctive hierarchical games and provides canonical representation theorems and characterizations of weightedness for these games.

Findings

Proves duality between disjunctive and conjunctive hierarchical games

Provides canonical representation theorems for hierarchical games

Characterizes disjunctive hierarchical games as complete with a unique shift-maximal losing coalition

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.