Bounded-Distance Network Creation Games

A \emph{network creation game} simulates a decentralized and non-cooperative building of a communication network. Informally, there are $n$ players sitting on the network nodes, which attempt to establish a reciprocal communication by activating, incurring a certain cost, any of their incident links. The goal of each player is to have all the other nodes as close as possible in the resulting network, while buying as few links as possible. According to this intuition, any model of the game must then appropriately address a balance between these two conflicting objectives. Motivated by the fact that a player might have a strong requirement about its centrality in the network, in this paper we introduce a new setting in which if a player maintains its (either \emph{maximum} or \emph{average}) distance to the other nodes within a given \emph{bound}, then its cost is simply equal to the \emph{number} of activated edges, otherwise its cost is unbounded. We study the problem of understanding the structure of pure Nash equilibria of the resulting games, that we call \textsc{MaxBD} and \textsc{SumBD}, respectively. For both games, we show that when distance bounds associated with players are \emph{non-uniform}, then equilibria can be arbitrarily bad. On the other hand, for \textsc{MaxBD}, we show that when nodes have a \emph{uniform} bound $R$ on the maximum distance, then the \emph{Price of Anarchy} (PoA) is lower and upper bounded by $2$ and $O\left(n^{\frac{1}{\lfloor\log_3 R\rfloor+1}}\right)$ for $R \ge 3$ (i.e., the PoA is constant as soon as $R$ is $\Omega(n^{\epsilon})$, for some $\epsilon>0$), while for the interesting case $R=2$, we are able to prove that the PoA is $\Omega(\sqrt{n})$ and $O(\sqrt{n \log n} )$. For the uniform \textsc{SumBD} we obtain similar (asymptotically) results, and moreover we show that the PoA becomes constant as soon as the bound on the average distance is $2^{\omega\big({\sqrt{\log n}}\big)}$.