## AbstractWe 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 $\alpha$ 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 $\alpha > A$, 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 $n_0$, there exists a graph built by $n\geq n_0$ players which contains cycles and forms a non-transient Nash equilibrium, for any $\alpha$ with $1< \alpha \leq \sqrt{n/2}$. Our construction makes use of some interesting results on finite affine planes. On the other hand we show that, for $\alpha \geq 12n\lceil\log n\rceil$, every Nash equilibrium forms a tree. Without relying on the tree conjecture, Fabrikant et al.\ proved an upper bound on the price of anarchy of $O(\sqrt{\alpha})$, where $\alpha \in [2,n^2]$. We improve this bound. Specifically, we derive a constant upper bound for $\alpha \in O(\sqrt{n})$ and for $\alpha \geq 12n\lceil\log n\rceil$. For the intermediate values we derive an improved bound of $O(1+ (\min\{{\alpha^2\over n}, {n^2\over \alpha}\})^{1/3})$. Additionally, we develop characterizations of Nash equilibria and extend our results to a weighted network creation game as well as to scenarios with cost sharing.
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