Abstract
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.
This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Center “On-The-Fly Computing” (SFB 901), by the EU within FET project MULTIPLEX under contract no. 317532, and the International Graduate School “Dynamic Intelligent Systems”.
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Abshoff, S., Cord-Landwehr, A., Jung, D., Skopalik, A. (2014). Multilevel Network Games. In: Liu, TY., Qi, Q., Ye, Y. (eds) Web and Internet Economics. WINE 2014. Lecture Notes in Computer Science, vol 8877. Springer, Cham. https://doi.org/10.1007/978-3-319-13129-0_36
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