Individually Stable Dynamics in Coalition Formation over Graphs

TL;DR

This paper studies how coalition formation dynamics over graphs converge to stable outcomes, focusing on preferences and graph structures that guarantee convergence, especially in acyclic graphs like trees, paths, and stars.

Contribution

It provides a comprehensive analysis of convergence conditions for individually stable coalition formation dynamics across various preferences and acyclic graph topologies.

Findings

Convergence is guaranteed under certain preference classes and acyclic graph structures.

A hierarchy of preferences influences the convergence behavior.

Acyclic graphs like trees, paths, and stars ensure convergence in the studied dynamics.

Abstract

Coalition formation over graphs is a well studied class of games whose players are vertices and feasible coalitions must be connected subgraphs. In this setting, the existence and computation of equilibria, under various notions of stability, has attracted a lot of attention. However, the natural process by which players, starting from any feasible state, strive to reach an equilibrium after a series of unilateral improving deviations, has been less studied. We investigate the convergence of dynamics towards individually stable outcomes under the following perspective: what are the most general classes of preferences and graph topologies guaranteeing convergence? To this aim, on the one hand, we cover a hierarchy of preferences, ranging from the most general to a subcase of additively separable preferences, including individually rational and monotone cases. On the other hand, given that convergence may fail in graphs admitting a cycle even in our most restrictive preference class, we analyze acyclic graph topologies such as trees, paths, and stars.