Individually Stable Dynamics in Coalition Formation over Graphs

TL;DR

The paper tackles convergence of individually stable dynamics in graph-restricted hedonic games, focusing on how graph topology and preference structure affect reaching IS partitions from arbitrary starts. It provides a taxonomy linking general, IR, monotone, and LAS preferences to specific graphs (paths, stars, trees) and proves convergence (or non-convergence) results under these combinations. Key contributions include a complete convergence picture for paths under monotone and IR/LAS settings, polynomial-time convergence on stars under IR/starting-IR conditions, and LAS-based convergence on trees with a nuanced complexity landscape (including exponential lower bounds). The findings advance understanding of decentralized coalition formation on networks and have implications for designing stable, scalable coalition protocols in networked settings.

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.

Figures (10)

• Figure 1: Claim (i) of the proof by induction.
• Figure 2: Claim (ii) of the proof by induction.
• Figure 3: Claim (i) of the induction base. Coalitions of $P$ are displayed from left to right, separated with dotted vertical lines. The evolution of the coalitions over time is shown from top to bottom, with time steps given on the extreme left.
• Figure 4: Claim (ii) of the induction base.
• Figure 5: Claim (i) of the induction step.

Theorems & Definitions (49)

• Definition 2.1
• Example 2.2
• Lemma 3.1
• Theorem 3.2
• proof : Proof sketch
• Example 3.3
• Theorem 3.4
• Lemma 4.1
• Theorem 4.2
• Theorem 4.3