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#### Stackelberg Routing in Arbitrary Networks

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##### Citation

Bonifaci, V., Harks, T., & Schäfer, G. (2010). Stackelberg Routing in Arbitrary
Networks.* Mathematics of Operations Research,* *35*(2),
330-346. doi:10.1287/moor.1100.0442v1.

Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-16EB-B

##### Abstract

We investigate the impact of \emph{Stackelberg routing} to reduce the price of
anarchy in network routing games. In this setting, an $\alpha$ fraction of the
entire demand is first routed centrally according to a predefined
\emph{Stackelberg strategy} and the remaining demand is then routed selfishly
by (nonatomic) players. Although several advances have been made recently in
proving that Stackelberg routing can in fact significantly reduce the price of
anarchy for certain network topologies, the central question of whether this
holds true in general is still open. We answer this question negatively by
constructing a family of single-commodity instances such that every Stackelberg
strategy induces a price of anarchy that grows linearly with the size of the
network.
Moreover, we prove upper bounds on the price of anarchy of the
Largest-Latency-First (LLF) strategy that only depend on the size of the
network. Besides other implications, this rules out the possibility to
construct constant-size networks to prove an unbounded price of anarchy.
In light of this negative result, we consider bicriteria bounds. We develop an
efficiently computable Stackelberg strategy that induces a flow whose cost is
at most the cost of an optimal flow with respect to demands scaled by a factor
of $1 + \sqrt{1-\alpha}$.
Finally, we analyze the effectiveness of an easy-to-implement Stackelberg
strategy, called SCALE. We prove bounds for a general class of latency
functions that includes polynomial latency functions as a special case. Our
analysis is based on an approach which is simple, yet powerful enough to obtain
(almost) tight bounds for SCALE in general networks.