The Price of Anarchy in Network Creation Games Is (Mostly) Constant

We study the price of anarchy and the structure of equilibria in network creation games. A network creation game is played by n players {1,2,aEuro broken vertical bar,n}, each identified with a vertex of a graph (network), where the strategy of player i, i=1,aEuro broken vertical bar,n, is to build some edges adjacent to i. The cost of building an edge is alpha > 0, a fixed parameter of the game. The goal of every player is to minimize its creation cost plus its usage cost. The creation cost of player i is alpha times the number of built edges. In the SumGame variant, the usage cost of player i is the sum of distances from i to every node of the resulting graph. In the MaxGame variant, the usage cost is the eccentricity of i in the resulting graph of the game. In this paper we improve previously known bounds on the price of anarchy of the game (of both variants) for various ranges of alpha, and give new insights into the structure of equilibria for various values of alpha. The two main results of the paper show that for alpha > 273a <...n all equilibria in SumGame are trees and thus the price of anarchy is constant, and that for alpha > 129 all equilibria in MaxGame are trees and the price of anarchy is constant. For SumGame this answers (almost completely) one of the fundamental open problems in the field-is price of anarchy of the network creation game constant for all values of alpha?-in an affirmative way, up to a tiny range of alpha.