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A characterization of absorbing sets in coalition formation games

Agustin G. Bonifacio, Elena Inarra, Pablo Neme

TL;DR

The paper addresses absorbing sets in coalition formation via a novel reduced form that aggregates the essential structure of absorbing sets into generalized rings, fixed components, and singletons. It proves a bijection between absorbing sets and reduced forms, enabling analysis of convergence to stability without enumerating all coalition structures. The approach yields necessary and sufficient conditions for convergence under various economic environments, including bargaining and rationing rules, and clarifies how properties like pairwise alignment and common ranking influence stability. The results unify and extend known outcomes in roommate problems and provide a framework for future work on dynamic rules and indifferences.

Abstract

Given a standard myopic dynamic process among coalition structures, an absorbing set is a minimal collection of such structures that is never left once entered through that process. Absorbing sets are an important solution concept in coalition formation games, but they have drawbacks: they can be large and hard to obtain. In this paper, we characterize an absorbing set in terms of a collection consisting of a small number of sets of coalitions that we refer to as a "reduced form" of a game. We apply our characterization to study convergence to stability in several economic environments.

A characterization of absorbing sets in coalition formation games

TL;DR

The paper addresses absorbing sets in coalition formation via a novel reduced form that aggregates the essential structure of absorbing sets into generalized rings, fixed components, and singletons. It proves a bijection between absorbing sets and reduced forms, enabling analysis of convergence to stability without enumerating all coalition structures. The approach yields necessary and sufficient conditions for convergence under various economic environments, including bargaining and rationing rules, and clarifies how properties like pairwise alignment and common ranking influence stability. The results unify and extend known outcomes in roommate problems and provide a framework for future work on dynamic rules and indifferences.

Abstract

Given a standard myopic dynamic process among coalition structures, an absorbing set is a minimal collection of such structures that is never left once entered through that process. Absorbing sets are an important solution concept in coalition formation games, but they have drawbacks: they can be large and hard to obtain. In this paper, we characterize an absorbing set in terms of a collection consisting of a small number of sets of coalitions that we refer to as a "reduced form" of a game. We apply our characterization to study convergence to stability in several economic environments.

Paper Structure

This paper contains 13 sections, 15 theorems, 15 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A characterization of absorbing sets
  4. Generalized rings
  5. The reduced form of a coalition formation game
  6. The characterization result
  7. Reduced forms in roommate problems
  8. Application: Convergence to stability
  9. Coalition formation games and bargaining solutions
  10. Coalition formation games and rationing rules
  11. Final remarks
  12. Relation between rings and cycles
  13. Proof of Theorem \ref{['bijection']}

Key Result

Theorem 1

For each coalition formation game, each absorbing set generates a reduced form and, conversely, each reduced form generates an absorbing set.

Figures (3)

  • Figure 1: Non-trivial absorbing set of game.
  • Figure 2: Non-trivial absorbing set of Example \ref{['ejemplo ring']}.
  • Figure 3: The absorbing set of Example \ref{['ejemplo tipo 2']}.

Theorems & Definitions (33)

  • Definition 1
  • Remark 1
  • Example 1
  • Definition 2
  • Example 2
  • Example 3
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • ...and 23 more