On the Price of Anarchy for High-Price Links

We study Nash equilibria and the price of anarchy in the classic model of Network Creation Games introduced by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker in 2003. This is a selfish network creation model where players correspond to nodes in a network and each of them can create links to the other $n-1$ players at a prefixed price $\alpha > 0$. The player's goal is to minimise the sum of her cost buying edges and her cost for using the resulting network. One of the main conjectures for this model states that the price of anarchy, i.e. the relative cost of the lack of coordination, is constant for all $\alpha$. This conjecture has been confirmed for $\alpha = O(n^{1-\delta})$ with $\delta \geq 1/\log n$ and for $\alpha > 4n-13$. The best known upper bound on the price of anarchy for the remaining range is $2^{O(\sqrt{\log n})}$. We give new insights into the structure of the Nash equilibria for $\alpha > n$ and we enlarge the range of the parameter $\alpha$ for which the price of anarchy is constant. Specifically, we prove that for any small $\epsilon>0$, the price of anarchy is constant for $\alpha > n(1+\epsilon)$ by showing that any biconnected component of any non-trivial Nash equilibrium, if it exists, has at most a constant number of nodes.