 
Summary: On Nash Equilibria for a Network Creation Game
Susanne Albers
Stefan Eilts
Eyal EvenDar
Yishay Mansour §
Liam Roditty ¶
Abstract
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
per edge. The player's goal 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. [10] conjectured that there exists a constant A such that, for any > A, all non
transient Nash equilibria graphs are trees. 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 nontransient Nash equilibrium, for any with 1 < n/2. Our construction
makes use of some interesting results on finite affine planes. On the other hand we show that for
12n log n every Nash equilibrium forms a tree.
The main result of Fabrikant et al. [10] is an upper bound on the price of anarchy of O(
) where
