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Combinatorics on Social Configurations

Dylan Laplace Mermoud, Pierre Popoli

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.

Combinatorics on Social Configurations

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.

Paper Structure

This paper contains 4 sections, 5 theorems, 6 equations, 3 figures, 1 table.

Table of Contents

  1. Cooperative Game Theory
  2. Hypergraphs
  3. Species of structures
  4. Decompositions in minimally uniform hypergraphs

Key Result

Theorem 1

The core of a game is nonempty if and only if $\{N\}$ belongs to the set of maximally efficient balanced collections.

Figures (3)

  • Figure 1: Construction of the species Gr of simple graphs.
  • Figure 2: All $2$-uniform hypergraphs of size $3$ with no more than $3$ nodes.
  • Figure 3: Example of a non-unique partition.

Theorems & Definitions (9)

  • Definition 1: von Neumann and Morgenstern von1944theory
  • Theorem 1: Bondareva bondareva1963some, Shapley shapley1967balanced
  • Proposition 1
  • Example 1
  • Theorem 2
  • Corollary 1
  • Example 2
  • Proposition 2
  • Example 3