Combinatorics on Social Configurations

In cooperative game theory, the social configurations of players are modeled
by balanced collections. The Bondareva-Shapley theorem, perhaps the most
fundamental theorem in cooperative game theory, characterizes the existence of
solutions to the game that benefit everyone using balanced collections. Roughly
speaking, if the trivial set system of all players is one of the most efficient
balanced collections for the game, then the set of solutions from which each
coalition benefits, the so-called core, is non-empty.
In this paper, we discuss some interactions between combinatorics and
cooperative game theory that are still relatively unexplored. Indeed, the
similarity between balanced collections and uniform hypergraphs seems to be a
relevant point of view to obtain new properties on those collections through
the theory of combinatorial species.