On nash equilibria for a network creation game

We study a network creation game recently proposed by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker. In this game, each player (vertex) can create links (edges) to other players at a cost of fi per edge. The goal of every player is to minimize the sum consisting of (a) the cost of the links he has created and (b) the sum of the distances to all other players. Fabrikant et al. conjectured that there exists a constant A such that, for any fi > A, all non-transient Nash equilibria graphs are trees. They showed that if a Nash equilibrium is a tree, the price of anarchy is constant. In this paper we disprove the tree conjecture. More precisely, we show that for any positive integer n0, there exists a graph built by n ‚ n0 players which contains cycles and forms a non- transient Nash equilibrium, for any fi with 1 < fip n=2. Our construction makes use of some interesting results on finite affine planes. On the other hand we show that, for fi ‚ 12ndlogne, every Nash equilibrium forms a tree. Without relying on the tree conjecture, Fabrikant et al.