The Internet has emerged as perhaps the most important network in modern computing, but rather miraculously, it was created through the individual actions of a multitude of agents rather than by a central planning authority. This motivates the game-theoretic study of network formation, and our article considers one of the most well-studied models, originally proposed by Fabrikant et al. In the model, each of n agents corresponds to a vertex, which can create edges to other vertices at a cost of α each, for some parameter α. Every edge can be freely used by every vertex, regardless of who paid the creation cost. To reflect the desire to be close to other vertices, each agent’s cost function is further augmented by the sum total of all (graph-theoretic) distances to all other vertices. Previous research proved that for many regimes of the (α, n ) parameter space, the total social cost (sum of all agents’ costs) of every Nash equilibrium is bounded by at most a constant multiple of the optimal social cost. In algorithmic game-theoretic nomenclature, this approximation ratio is called the price of anarchy. In our article, we significantly sharpen some of those results, proving that for all constant nonintegral α > 2, the price of anarchy is in fact 1 + o (1); that is, not only is it bounded by a constant, but also it tends to 1 as n → ∞. For constant integral α ⩾ 2, we show that the price of anarchy is bounded away from 1. We provide quantitative estimates on the rates of convergence for both results.