# Multilevel Network Games

Abshoff, Sebastian; Cord-Landwehr, Andreas; Jung, Daniel; Skopalik, Alexander
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 \\$\backslash alpha > 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 \\$\backslash alpha \backslash leq n-1\\$, the price of anarchy is \\$\backslash Theta(n/\backslash sqrt\{\backslash alpha\})\\$ and in this range equilibria always exist. In range \\$\backslash alpha \backslash in (n-1,n(n-1))\\$ the price of anarchy is \\$\backslash Theta(\backslash sqrt\{\backslash alpha\})\\$, and for \\$\backslash alpha \backslash geq n(n-1)\\$ it is constant. For the MAX-game, we show that the price of anarchy is either \\$\backslash Theta(1 + n/\backslash sqrt\{\backslash alpha\})\\$, for \\$\backslash alpha\backslash geq 1\\$, or else \\$1\\$. Given a graph with girth of at least \\$4\backslash alpha\\$, 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.
Research areas:
Year:
2014
Type of Publication:
Article
Journal:
Lecture Notes in Computer Science
Volume:
8877
Pages:
435-440
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