The ring design game with fair cost allocation

In this paper we study the network design game when the underlying network is a ring. In a network design game we have a set of players, each of them aims at connecting nodes in a network by installing links and equally sharing the cost of the installation with other users. The ring design game is the special case in which the potential links of the network form a ring. It is well known that in a ring design game the price of anarchy may be as large as the number of players. Our aim is to show that, despite the worst case, the ring design game always possesses good equilibria. In particular, we prove that the price of stability of the ring design game is at most 3/2, and such bound is tight. Moreover, we observe that the worst Nash equilibrium cannot cost more than 2 times the optimum if the price of stability is strictly larger than 1. We believe that our results might be useful for the analysis of more involved topologies of graphs, e.g., planar graphs.