Combinatorics on Social Configurations

TL;DR

This paper explores the intersection of combinatorics and cooperative game theory, focusing on balanced collections and their relation to the core of the game, using combinatorial species to uncover new properties.

Contribution

It introduces a novel perspective linking balanced collections in cooperative game theory with uniform hypergraphs and combinatorial species, opening new research avenues.

Findings

Identifies connections between balanced collections and hypergraph theory.

Proposes using combinatorial species to analyze social configurations.

Highlights unexplored interactions between combinatorics and game theory.

Abstract

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.