On Strong Equilibria and Improvement Dynamics in Network Creation Games

TL;DR

This paper analyzes strong equilibria in network creation games, characterizing their structure for different edge costs, resolving conjectures about star equilibria, and examining the convergence of improvement dynamics.

Contribution

It provides a complete characterization of strong equilibria for all edge cost parameters and resolves a conjecture about star equilibria, also analyzing convergence properties of improvement dynamics.

Findings

Strong equilibria are characterized for all α in (0,2).

The conjecture that all star graphs are strong equilibria is confirmed.

Non-star trees can be strong equilibria for large α (≥ 2n).

Abstract

We study strong equilibria in network creation games. These form a classical and well-studied class of games where a set of players form a network by buying edges to their neighbors at a cost of a fixed parameter $\alpha$. The cost of a player is defined to be the cost of the bought edges plus the sum of distances to all the players in the resulting graph. We identify and characterize various structural properties of strong equilibria, which lead to a characterization of the set of strong equilibria for all $\alpha$ in the range $(0,2)$. For $\alpha > 2$, Andelman et al. (2009) prove that a star graph in which every leaf buys one edge to the center node is a strong equilibrium, and conjecture that in fact any star is a strong equilibrium. We resolve this conjecture in the affirmative. Additionally, we show that when $\alpha$ is large enough ($\geq 2n$) there exist non-star trees that are strong equilibria. For the strong price of anarchy, we provide precise expressions when $\alpha$ is in the range $(0,2)$, and we prove a lower bound of $3/2$ when $\alpha \geq 2$. Lastly, we aim to characterize under which conditions (coalitional) improvement dynamics may converge to a strong equilibrium. To this end, we study the (coalitional) finite improvement property and (coalitional) weak acyclicity property. We prove various conditions under which these properties do and do not hold. Some of these results also hold for the class of pure Nash equilibria.