Anarchy is free in network creation

TL;DR

This paper analyzes the efficiency of network formation in a game-theoretic model, showing that for most parameters, Nash equilibria are nearly optimal as the network size grows, with some exceptions.

Contribution

It sharpens previous bounds on the price of anarchy in network creation games, proving near-optimality for non-integral alpha > 2 as network size increases.

Findings

Price of anarchy approaches 1 for non-integral alpha > 2 as N increases

For integral alpha >= 2, the price of anarchy remains bounded away from 1

Quantitative convergence rates are provided for the bounds

Abstract

The Internet has emerged as perhaps the most important network in modern computing, but rather miraculously, it was created through the individual actions of a multitude of agents rather than by a central planning authority. This motivates the game theoretic study of network formation, and our paper considers one of the most-well studied models, originally proposed by Fabrikant et al. In it, each of N agents corresponds to a vertex, which can create edges to other vertices at a cost of alpha each, for some parameter alpha. Every edge can be freely used by every vertex, regardless of who paid the creation cost. To reflect the desire to be close to other vertices, each agent's cost function is further augmented by the sum total of all (graph theoretic) distances to all other vertices. Previous research proved that for many regimes of the (alpha, N) parameter space, the total social cost (sum of all agents' costs) of every Nash equilibrium is bounded by at most a constant multiple of the optimal social cost. In algorithmic game theoretic nomenclature, this approximation ratio is called the price of anarchy. In our paper, we significantly sharpen some of those results, proving that for all constant non-integral alpha > 2, the price of anarchy is in fact 1+o(1), i.e., not only is it bounded by a constant, but it tends to 1 as N tends to infinity. For constant integral alpha >= 2, we show that the price of anarchy is bounded away from 1. We provide quantitative estimates on the rates of convergence for both results.