An $H_{n/2}$ Upper Bound on the Price of Stability of Undirected Network   Design Games
  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 $n \geq n(\epsilon)$. 
Akaki Mamageishvili, Mat\'u\v{s} Mihal\'ak, Simone Montemezzani
cs.GT