Network Cost-Sharing without Anonymity

We consider network cost-sharing games with nonanonymous cost functions, where the cost of each edge is a submodular function of its users, and this cost is shared using the Shapley value. Nonanonymous cost functions model asymmetries between the players, which can arise from different bandwidth requirements, durations of use, services needed, and so on. These games can possess multiple Nash equilibria of wildly varying quality. The goal of this article is to identify well-motivated equilibrium refinements that admit good worst-case approximation bounds. Our primary results are tight bounds on the cost of strong Nash equilibria and potential function minimizers in network cost-sharing games with nonanonymous cost functions, parameterized by the set C of allowable submodular cost functions. These two worst-case bounds coincide for every set C, and equal the summability parameter introduced in Roughgarden and Sundararajan [2009] to characterize efficiency loss in a family of cost-sharing mechanisms. Thus, a single parameter simultaneously governs the worst-case inefficiency of network cost-sharing games (in two incomparable senses) and cost-sharing mechanisms. This parameter is always at most the kth Harmonic number Hk ≈ ln k, where k is the number of players, and is constant for many function classes of interest.