Polynomial Bottleneck Congestion Games with Optimal Price of Anarchy.

Authors: Costas Busch, Rajgopal Kannan, Athanasios Vasilakos
Subjects: Computer Science and Game Theory

link: http://arxiv.org/abs/1010.4812
Abstract

We study {\em bottleneck congestion games} where the social cost is
determined by the worst congestion of any resource. These games directly relate
to network routing problems and also job-shop scheduling problems. In typical
bottleneck congestion games, the utility costs of the players are determined by
the worst congested resources that they use. However, the resulting Nash
equilibria are inefficient, since the price of anarchy is proportional on the
number of resources which can be high. Here we show that we can get smaller
price of anarchy with the bottleneck social cost metric. We introduce the {\em
polynomial bottleneck games} where the utility costs of the players are
polynomial functions of the congestion of the resources that they use. In
particular, the delay function for any resource $r$ is $C_{r}^\M$, where $C_r$
is the congestion measured as the number of players that use $r$, and $\M \geq
1$ is an integer constant that defines the degree of the polynomial. The
utility cost of a player is the sum of the individual delays of the resources
that it uses. The social cost of the game remains the same, namely, it is the
worst bottleneck resource congestion: $\max_{r} C_r$. We show that polynomial
bottleneck games are very efficient and give price of anarchy
$O(|R|^{1/(\M+1)})$, where $R$ is the set of resources. This price of anarchy
is tight, since we demonstrate a game with price of anarchy
$\Omega(|R|^{1/(\M+1)})$, for any $\M \geq 1$. We obtain our tight bounds by
using two proof techniques: {\em transformation}, which we use to convert
arbitrary games to simpler games, and {\em expansion}, which we use to bound
the price of anarchy in a simpler game.

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