In the network creation game, each player purchases incident undirected edges at price $\alpha>0$ per edge and pays the sum of its distances to all players. Disconnected pairs have infinite distance. A pure Nash equilibrium admits no unilateral strategy change that reduces the changing player's cost.

This submission proves Theorem 3 of Fabrikant, Luthra, Maneva, Papadimitriou, and Shenker, On a Network Creation Game (PODC 2003, p. 350): every tree Nash equilibrium has social cost strictly less than five times the social optimum. The optimum ranges over all strategy profiles, and the bound holds for every positive edge price and every number of players at least two.

The concept package separates strategy profiles, individual cost, Nash equilibrium, social cost, the inefficiency ratio, and the theorem statement. The proof compares the distance savings obtained when both endpoints of a tree path buy a link to its midpoint. The resulting distance bound, together with unique edge ownership at equilibrium and a lower bound on every profile's social cost, gives the strict factor-five inequality.