Capacitated Network Design Games

We study a capacitated symmetric network design game, where each of n agents wishes to construct a path from a network’s source to its sink, and the cost of each edge is shared equally among the agents using it. The uncapacitated version of this problem has been introduced by Anshelevich et al. (2003) and has been extensively studied. We find that the consideration of edge capacities entails a significant effect on the quality of the obtained Nash equilibria (NE), under both the utilitarian and the egalitarian objective functions, as well as on the convergence rate to an equilibrium. The following results are established. First, we provide bounds for the price of anarchy (PoA) and the price of stability (PoS) measures with respect to the utilitarian (i.e., sum of costs) and egalitarian (i.e., maximum cost) objective functions. Our main result here is that unlike the uncapacitated version, the network topology is a crucial factor in the quality of NE. Specifically, a network topology has a bounded PoA if and only if it is series-parallel (SP), i.e., a network that is built inductively by series compositions and parallel compositions of SP networks. Second, we show that the convergence rate of best-response dynamics (BRD) may take Ω(n^1.5) steps. This is in contrast to the uncapacitated version, where convergence is guaranteed within at most n iterations.