Topological Price of Anarchy Bounds for Clustering Games on Networks

TL;DR

This paper investigates how network structure influences the inefficiency of clustering games, providing topological bounds on the Price of Anarchy that are more precise than traditional worst-case bounds, especially on random graphs.

Contribution

It introduces topological bounds on the Price of Anarchy for clustering games, offering a more nuanced understanding of inefficiency based on network properties, including tight bounds for Erdős-Rényi graphs.

Findings

Topological bounds improve understanding of game inefficiency.

Price of Anarchy varies significantly with network density.

Tight bounds established for Erdős-Rényi random graphs.

Abstract

We consider clustering games in which the players are embedded in a network and want to coordinate (or anti-coordinate) their strategy with their neighbors. The goal of a player is to choose a strategy that maximizes her utility given the strategies of her neighbors. Recent studies show that even very basic variants of these games exhibit a large Price of Anarchy: A large inefficiency between the total utility generated in centralized outcomes and equilibrium outcomes in which players selfishly try to maximize their utility. Our main goal is to understand how structural properties of the network topology impact the inefficiency of these games. We derive topological bounds on the Price of Anarchy for different classes of clustering games. These topological bounds provide a more informative assessment of the inefficiency of these games than the corresponding (worst-case) Price of Anarchy bounds. As one of our main results, we derive (tight) bounds on the Price of Anarchy for clustering games on Erd\H{o}s-R\'enyi random graphs (where every possible edge in the network is present with a fixed probability), which, depending on the graph density, stand in stark contrast to the known Price of Anarchy bounds.