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On the Formation of Steady Coalitions

Dylan Laplace Mermoud

TL;DR

This work addresses the problem of grand coalition formation in transferable utility games under internal coalition dynamics. It develops a domination-based framework and introduces cooperahedra, a new family of polyhedra that generalize permutohedra and removahedra, to characterize when a dominating core element can steer the game toward global cooperation. The authors provide a nonemptiness criterion for cooperahedra via balancedness and effective coalitions, and a complete characterization of blind spots where core dominance is impossible, with applications to flow and market games. The results yield a geometric, polyhedral lens on coalition formation and suggest algorithmic checks through minimal balanced collections.

Abstract

This paper studies the formation of the grand coalition of a cooperative game by investigating its possible internal dynamics. Each coalition is capable of forcing all players to reconsider the current state of the game when it does not provide sufficient payoff. Different coalitions may ask for contradictory evolutions, leading to the impossibility of the grand coalition forming. In this paper, we give a characterization of the impossibility, for a given state, of finding a new state dominating the previous one such that each aggrieved coalition has a satisfactory payoff. To do so, we develop new polyhedral tools related to a new family of polyhedra, appearing in numerous situations in cooperative game theory.

On the Formation of Steady Coalitions

TL;DR

This work addresses the problem of grand coalition formation in transferable utility games under internal coalition dynamics. It develops a domination-based framework and introduces cooperahedra, a new family of polyhedra that generalize permutohedra and removahedra, to characterize when a dominating core element can steer the game toward global cooperation. The authors provide a nonemptiness criterion for cooperahedra via balancedness and effective coalitions, and a complete characterization of blind spots where core dominance is impossible, with applications to flow and market games. The results yield a geometric, polyhedral lens on coalition formation and suggest algorithmic checks through minimal balanced collections.

Abstract

This paper studies the formation of the grand coalition of a cooperative game by investigating its possible internal dynamics. Each coalition is capable of forcing all players to reconsider the current state of the game when it does not provide sufficient payoff. Different coalitions may ask for contradictory evolutions, leading to the impossibility of the grand coalition forming. In this paper, we give a characterization of the impossibility, for a given state, of finding a new state dominating the previous one such that each aggrieved coalition has a satisfactory payoff. To do so, we develop new polyhedral tools related to a new family of polyhedra, appearing in numerous situations in cooperative game theory.
Paper Structure (7 sections, 11 theorems, 31 equations, 2 figures)

This paper contains 7 sections, 11 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. How to characterize the steadiness of a coalition?
  4. Cooperahedra
  5. Blind spots
  6. Applications to market games
  7. Concluding remarks

Key Result

Theorem 1

The core of a game $(N,v)$ is nonempty if and only if $(N, v)$ is balanced.

Figures (2)

  • Figure 1: Directed graph describing a flow game on $N = \{a, b, c, d\}$.
  • Figure 2: Directed graph describing a flow game on $N = \{a, b, c\}$.

Theorems & Definitions (34)

  • Definition 1: von1944theory
  • Definition 2: bondareva1963some, shapley1967balanced
  • Example 1
  • Theorem 1: bondareva1963some, shapley1967balanced
  • Definition 3: von1944theory
  • Theorem 2
  • proof
  • Definition 4: von1944theory
  • Definition 5
  • Proposition 1
  • ...and 24 more