  In the network design game with $n$ players, every player chooses a path in
an edge-weighted graph to connect her pair of terminals, sharing costs of the
edges on her path with all other players fairly. We study the price of
stability of the game, i.e., the ratio of the social costs of a best Nash
equilibrium (with respect to the social cost) and of an optimal play. It has
been shown that the price of stability of any network design game is at most
$H_n$, the $n$-th harmonic number. This bound is tight for directed graphs. For
undirected graphs, the situation is dramatically different, and tight bounds
are not known. It has only recently been shown that the price of stability is
at most $H_n \left(1-\frac{1}{\Theta(n^4)} \right)$, while the worst-case known
example has price of stability around 2.25. In this paper we improve the upper
bound considerably by showing that the price of stability is at most $H_{n/2} +
\epsilon$ for any $\epsilon$ starting from some suitable $n \geq n(\epsilon)$.
