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Multilevel Network Games

We consider a multilevel network game, where nodes can improve their communication costs by connecting to a high-speed network. The $n$ nodes are connected by a static network and each node can decide individually to become a gateway to the high-speed network. The goal of a node $v$ is to minimize its private costs, i.e., the sum (SUM-game) or maximum (MAX-game) of communication distances from $v$ to all other nodes plus a fixed price $α> 0$ if it decides to be a gateway. Between gateways the communication distance is $0$, and gateways also improve other nodes' distances by behaving as shortcuts. For the SUM-game, we show that for $α\leq n-1$, the price of anarchy is $Θ(n/\sqrtα)$ and in this range equilibria always exist. In range $α\in (n-1,n(n-1))$ the price of anarchy is $Θ(\sqrtα)$, and for $α\geq n(n-1)$ it is constant. For the MAX-game, we show that the price of anarchy is either $Θ(1 + n/\sqrtα)$, for $α\geq 1$, or else $1$. Given a graph with girth of at least $4α$, equilibria always exist. Concerning the dynamics, both the SUM-game and the MAX-game are not potential games. For the SUM-game, we even show that it is not weakly acyclic.