Tree Nash Equilibria in the Network Creation Game

In the network creation game with n vertices, every vertex (player) creates an (adjacent) edge and decides to which other vertices the created edge should go. Each created edge costs a fixed amount α > 0. Each player aims to have a good connection with the rest of the vertices and, at the same time, to pay as little as possible. Formally, the cost of each player in the resulting (created) graph is defined as α times the number of edges created by the player plus the sum of the distances to all other vertices. It has been conjectured that for α ≥ n, every Nash equilibrium of this game is a tree and has been confirmed for every α ≥ 273 · n. We improve on this bound and show that this is true for every α ≥ 65 · n. We also show that our approach cannot be used to show the desired bound, but we conjecture that a slightly worse bound α ≥ 1.3 · n can be achieved. Toward this conjecture, we show that if a Nash equilibrium has a cycle of length at most 10, then indeed α < 1.3 · n. We investigate our approach for a coalitional variant of a Nash equilibrium, which coalitions of two players cannot collectively improve, and show that if α ≥ 41 · n, then every such Nash equilibrium is a tree.