Annual Report 2010 - Fachgruppe Informatik an der RWTH Aachen ...

Aspects of Wardrop Equilibria
Lars Olbrich
Global communication networks like the Internet often lack a central authority that monitors
and regulates network traffic. Network users may behave selfishly according to their private
interest without regard to the overall system performance. Such highly complex environments
prompted a paradigm shift in computer science. Whereas traditional concepts are designed for
stand-alone machines and manageable networks, a profound understanding of large-scale
communication systems with strategic users requires to combine methods from theoretical
computer science with game-theoretic techniques. In this thesis, we study equilibrium
situations in Wardrop's traffic model. In Wardrop's model a rate of traffic between each pair
of vertices of a network is modeled as network flow, i.e., traffic is allowed to split into
arbitrary pieces. The resources are the network edges with latency functions quantifying the
time needed to traverse an edge. The latency of an edge depends on the congestion. It
increases the more flow traverses that edge. A common interpretation of the Wardrop model
is that flow is controlled by an infinite number of agents each of which is responsible to route
an infinitesimal amount of traffic between its origin and destination vertex. Each agent plays a
pure strategy in choosing one path from its origin to its destination, where the agent's
disutility is the sum of edge latencies on this path. A Wardrop equilibrium denotes a strategy
profile in which all used paths between a given origin-destination pair have equal and
minimal latency. Wardrop equilibria are also Nash equilibria as no agent can decrease its
experienced latency by unilaterally deviating to another path. Like Nash equilibria in general,
Wardrop equilibria do not optimize any global objective per se. In particular, the total latency
of all agents is not minimized at Wardrop equilibrium. Addressing this issue, Roughgarden
and Tardos gave tight bounds on the price of anarchy measuring the worst-possible
inefficiency of equilibria with respect to the incurred latency. Further, the famous Braess's
paradox states that adding edges to a network may in fact worsen the unique equilibrium. The
primary goal of this thesis is to provide a deeper understanding of Wardrop equilibria. We
identify several problems whose solution captures the essence of Wardrop equilibria. First,
we study natural and innovative means to reduce the price of anarchy. Secondly, we analyze
the stability of equilibria regarding modifications of the network environment. Finally, we
propose a distributed algorithm for computing approximate equilibria. The inefficiency of
equilibria motivates our first line of research. In Wardrop's model, imposing marginal cost
taxes on every edge completely eliminates the inefficiency of selfish routing. We concentrate
on optimal taxes for the crucial and more realistic case in which only a given subset of the
edges can be taxed. We establish NP-hardness of this optimization problem in general
networks. On the positive side, we provide a polynomial time algorithm for computing
optimal taxes in parallel link networks with linear latency functions. We also propose a novel
approach to improve the performance of selfish flow in networks by additionally routing
flow, called auxiliary flow. We focus on the computational complexity for the optimal
utilization of auxiliary flow and present strong inapproximability results. In particular, the
minimal amount of auxiliary flow needed to induce an optimal flow as the outcome of selfish
behavior cannot be approximated by any subexponential factor. Further, we study the
sensitivity of Wardrop equilibria. From both the practical and the theoretical perspective it is
a natural and intriguing question, how equilibria respond to slight modifications of either the
network topology or the traffic flow. We show positive and negative results on the stability of
flow pattern and flow characteristics at equilibrium. As it is fundamental for the above studies
that selfish behavior in network routing games yields an equilibrium, it is not clear how the
set of agents can attain an equilibrium in the first place. In previous work it was shown that an
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