  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 paper
considers one of the most-well studied models, originally proposed by Fabrikant
et al. In it, each of N agents corresponds to a vertex, which can create edges
to other vertices at a cost of alpha each, for some parameter alpha. 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 (alpha, 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 paper, we significantly sharpen some of
those results, proving that for all constant non-integral alpha > 2, the price
of anarchy is in fact 1+o(1), i.e., not only is it bounded by a constant, but
it tends to 1 as N tends to infinity. For constant integral alpha >= 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.
