Enumeration of minimal balanced collections

Enumerate all minimal balanced collections on a finite player set, thereby determining their number and structure; this enumeration is needed to fully support combinatorial and polyhedral analyses of balanced games.

Background

Minimal balanced collections are families of coalitions equipped with positive balancing weights whose incidence vectors balance every player, and they are central to the Bondareva–Shapley characterization of games with nonempty cores. The paper uses these collections to define the facet inequalities of the polyhedra of balanced games. Although the collections are finite for a fixed number of players, their enumeration is not resolved in the paper and is identified as an open combinatorial problem.