Network-creation games have been studied recently in many different settings. These games are motivated by social networks in which selfish agents want to construct a connection graph among themselves. Each node wants to minimize its average or maximum distance to the others, without paying much to construct the network. Many generalizations have been considered, including nonuniform interests between nodes, general graphs of allowable edges, and bounded-budget agents. In all of these settings, there is no known constant bound on the price of anarchy. In fact, in many cases, the price of anarchy can be very large, namely, a constant power of the number of agents. This means that we have no control over the behavior of a network when agents act selfishly. On the other hand, the price of stability in all these models is constant, which means that there is chance that agents act selfishly and we end up with a reasonable social cost.
In this paper, we show how to use an advertising campaign (as introduced in [Balcan et al., “Improved Equilibria via Public Service Advertising.”) to find efficient equilibria in an $(n, k)$-uniform bounded-budget connection game; our result holds for $k = \omega(\log(n))$. More formally, we present advertising strategies such that if an $\alpha$ fraction of the agents agree to cooperate in the campaign, the social cost will be at most $O(1/\alpha)$ times the optimum cost. This is the first constant bound on the price of anarchy that interestingly can be adapted to different settings. We also generalize our method to work in cases in which $\alpha$ is not known in advance. Also, we do not need to assume that the cooperating agents spend all their budget on the campaign; even a small fraction ($\beta$ fraction) of their budget is sufficient to obtain a constant price of anarchy.
"Constant Price of Anarchy in Network-Creation Games via Public-Service Advertising." Internet Math. 8 (1-2) 29 - 45, 2012.