In this article we show that the price of stability of shapley network design games on undirected graphs with k players is at most k3(k+1)/2-k21+k3(k+1)/2-k2hk=(1-?(1/k4))hk, where hk denotes the k-th harmonic number. This improves on the known upper bound of hk, which is also valid for directed graphs but for these, in contrast, is tight. Hence, we give the first non-trivial upper bound on the price of stability for undirected shapley network design games that is valid for an arbitrary number of players. Our bound is proved by analyzing the price of stability restricted to nash equilibria that minimize the potential function of the game. We also present a game with k=3 players in which such a restricted price of stability is 1.634. This shows that the analysis of bilò and bove (2011)  is tight. In addition, we give an example for three players that improves the lower bound on the (unrestricted) price of stability to 1.571.
Disser, Y., Feldmann, A. E., Klimm, M., & Mihalák, M. (2015). Improving the Hk-bound on the price of stability in undirected Shapley network design games. Theoretical Computer Science, 562, 557-564. https://doi.org/10.1016/j.tcs.2014.10.037