Algorithms, games, and the internet
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Journal of the ACM (JACM)
How unfair is optimal routing?
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
The price of anarchy is independent of the network topology
Journal of Computer and System Sciences - STOC 2002
A stronger bound on Braess's Paradox
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
The maximum latency of selfish routing
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Selfish Routing in Capacitated Networks
Mathematics of Operations Research
The price of anarchy of finite congestion games
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Fairness and optimality in congestion games
Proceedings of the 6th ACM conference on Electronic commerce
Flows over Time with Load-Dependent Transit Times
SIAM Journal on Optimization
On the severity of Braess's paradox: designing networks for selfish users is hard
Journal of Computer and System Sciences - Special issue on FOCS 2001
Algorithmic Game Theory
The “Price of Anarchy” Under Nonlinear and Asymmetric Costs
Mathematics of Operations Research
Fast, Fair, and Efficient Flows in Networks
Operations Research
Atomic routing games on maximum congestion
Theoretical Computer Science
STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
Price of anarchy for polynomial wardrop games
WINE'06 Proceedings of the Second international conference on Internet and Network Economics
Braess's paradox, fibonacci numbers, and exponential inapproximability
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
The hardness of network design for unsplittable flow with selfish users
WAOA'05 Proceedings of the Third international conference on Approximation and Online Algorithms
The price of anarchy for non-atomic congestion games with symmetric cost maps and elastic demands
Operations Research Letters
Bottleneck Routing Games in Communication Networks
IEEE Journal on Selected Areas in Communications
The price of anarchy for selfish ring routing is two
WINE'12 Proceedings of the 8th international conference on Internet and Network Economics
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We give several new upper and lower bounds on the worst-case severity of Braess's paradox and the price of anarchy of selfish routing with respect to the maximum latency objective. In single-commodity networks with arbitrary continuous and nondecreasing latency functions, we prove that this worst-case price of anarchy is exactly $n-1$, where $n$ is the number of network vertices. For Braess's paradox in such networks, we prove that removing at most $c$ edges from a network decreases the common latency incurred by traffic at equilibrium by at most a factor of $c+1$. In particular, the worst-case severity of Braess's paradox with a single edge removal is maximized in Braess's original four-vertex network. In multicommodity networks, we exhibit an infinite family of two-commodity networks, related to the Fibonacci numbers, in which both the worst-case severity of Braess's paradox and the price of anarchy for the maximum latency objective grow exponentially with the network size. This construction demonstrates that numerous known selfish routing results for single-commodity networks have no analogues in networks with two or more commodities. We also prove an upper bound on both of these quantities that is exponential in the network size and independent of the network latency functions, showing that our construction is close to optimal. Finally, we use our family of two-commodity networks to exhibit a natural network design problem with intrinsically exponential (in)approximability.