The Price of Anarchy for Network Formation in an Adversary Model

TL;DR

This paper analyzes the efficiency of network formation under adversarial link removal, establishing bounds on the price of anarchy and stability, and highlighting differences between unilateral and bilateral link formation strategies.

Contribution

It provides the first bounds on the price of anarchy in an adversarial network formation model, including specific bounds for different adversaries and link formation rules.

Findings

Price of stability is 1+o(1) under certain conditions.

Bounded price of anarchy for unilateral link formation, up to constant factors.

Divergent bounds for bilateral link formation depending on adversary and parameters.

Abstract

We study network formation with n players and link cost \alpha > 0. After the network is built, an adversary randomly deletes one link according to a certain probability distribution. Cost for player v incorporates the expected number of players to which v will become disconnected. We show existence of equilibria and a price of stability of 1+o(1) under moderate assumptions on the adversary and n \geq 9. As the main result, we prove bounds on the price of anarchy for two special adversaries: one removes a link chosen uniformly at random, while the other removes a link that causes a maximum number of player pairs to be separated. For unilateral link formation we show a bound of O(1) on the price of anarchy for both adversaries, the constant being bounded by 10+o(1) and 8+o(1), respectively. For bilateral link formation we show O(1+\sqrt{n/\alpha}) for one adversary (if \alpha > 1/2), and \Theta(n) for the other (if \alpha > 2 considered constant and n \geq 9). The latter is the worst that can happen for any adversary in this model (if \alpha = \Omega(1)). This points out substantial differences between unilateral and bilateral link formation.