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 ≤ n − 1, the price of anarchy is \(\Theta({n/\sqrt{\alpha}})\) and in this range equilibria always exist. In range α ∈ (n − 1,n(n − 1)) the price of anarchy is \(\Theta({\sqrt{\alpha}})\), and for α ≥ n(n − 1) it is constant. For the MAX-game, we show that the price of anarchy is either \(\Theta({1 + n/\sqrt{\alpha}})\), for α ≥ 1, or else 1. Given a graph with girth of at least 4α, equilibria always exist. Concerning the dynamics, both games are not potential games. For the SUM-game, we even show that it is not weakly acyclic.