Network creation games with disconnected equilibria

Abstract

In this paper we extend a popular non-cooperative network creation game (NCG) [11] to allow for disconnected equilibrium networks. There are n players, each is a vertex in a graph, and a strategy is a subset of players to build edges to. For each edge a player must pay a cost α, and the individual cost for a player represents a trade-off between edge costs and shortest path lengths to all other players. We extend the model to a penalized game (PCG), for which we reduce the penalty for a pair of disconnected players to a finite value β. We prove that the PCG is not a potential game, but pure Nash equilibria always exist, and pure strong equilibria exist in many cases. We provide tight conditions under which disconnected (strong) Nash equilibria can evolve. Components of these equilibria must be (strong) Nash equilibria of a smaller NCG. But in contrast to the NCG, for the vast majority of parameter values no tree is a stable component. Finally, we show that the price of anarchy is Θ(n), several orders of magnitude larger than in the NCG. Perhaps surprisingly, the price of anarchy for strong equilibria increases only to at most 4. © 2008 Springer Berlin Heidelberg.