Network Creation Games with Disconnected Equilibria

In this paper we extend a popular non-cooperative networkcreation game (NCG) [11] to allow for disconnected equilibriumnetworks. There are n players, each is a vertex in a graph, and astrategy is a subset of players to build edges to. For each edge aplayer must pay a cost ±, and the individual cost for aplayer represents a trade-off between edge costs and shortest pathlengths to all other players. We extend the model to a penalizedgame (PCG), for which we reduce the penalty for a pair ofdisconnected players to a finite value ². We prove that thePCG is not a potential game, but pure Nash equilibria always exist,and pure strong equilibria exist in many cases. We provide tightconditions under which disconnected (strong) Nash equilibria canevolve. Components of these equilibria must be (strong) Nashequilibria of a smaller NCG. But in contrast to the NCG, for thevast majority of parameter values no tree is a stable component.Finally, we show that the price of anarchy is ˜(n), severalorders of magnitude larger than in the NCG. Perhaps surprisingly,the price of anarchy for strong equilibria increases only to atmost 4.

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