Abstract

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.

MSC codes

  1. 68Q25
  2. 68W25
  3. 90B18
  4. 90B20
  5. 91A10

Keywords

  1. Braess's paradox
  2. selfish routing
  3. price of anarchy
  4. traffic networks
  5. approximation algorithms

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References

1.
Y. Azar and A. Epstein, The hardness of network design for unsplittable flow with selfish users, in Proceedings of the Third Workshop on Approximation and Online Algorithms (WAOA), 2005, pp. 41–54.
2.
R. Banner and A. Orda, Bottleneck routing games in communication networks, IEEE J. Sel. Area Commun., 25 (2007), pp. 1173–1179.
3.
M. J. Beckmann, C. B. McGuire, and C. B. Winsten, Studies in the Economics of Transportation, Yale University Press, New Haven, CT, 1956.
4.
D. Braess, Über ein Paradoxon aus der Verkehrsplanung, Unternehmensforschung, 12 (1968), pp. 258–268.
5.
C. Busch and M. Magdon-Ismail, Atomic routing games on maximum congestion, Theoret. Comput. Sci., 410 (2009), pp. 3337–3347.
6.
D. Chakrabarty, A. Mehta, V. Nagarajan, and V. Vazirani, Fairness and optimality in congestion games, in Proceedings of the 6th ACM Conference on Electronic Commerce (EC), 2005, pp. 67–73.
7.
C. K. Chau and K. M. Sim, The price of anarchy for non-atomic congestion games with symmetric cost maps and elastic demands, Oper. Res. Lett., 31 (2003), pp. 327–334.
8.
G. Christodoulou and E. Koutsoupias, The price of anarchy of finite congestion games, in Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC), 2005, pp. 67–73.
9.
V. Chvátal, Linear Programming, Freeman, New York, 1983.
10.
J. R. Correa, A. S. Schulz, and N. E. Stier Moses, Selfish routing in capacitated networks, Math. Oper. Res., 29 (2004), pp. 961–976.
11.
J. R. Correa, A. S. Schulz, and N. E. Stier Moses, Fast, fair, and efficient flows in networks, Oper. Res., 55 (2007), pp. 215–225.
12.
J. R. Correa, A. S. Schulz, and N. E. Stier Moses, On the inefficiency of equilibria in congestion games, Games Econom. Behav., 64 (2008), pp. 457–469.
13.
D. Dumrauf and M. Gairing, Price of anarchy for polynomial Wardrop games, in Proceedings of the 2nd Annual Workshop on Internet and Network Economics (WINE), Lecture Notes in Comput. Sci. 4286, Springer, Berlin, 2006, pp. 319–330.
14.
A. Epstein, M. Feldman, and Y. Mansour, Efficient graph topologies in network routing games, Games Econom. Behav., 66 (2009), pp. 115–125.
15.
M. A. Hall, Properties of the equilibrium state in transportation networks, Trans. Sci., 12 (1978), pp. 208–216.
16.
O. Jahn, R. H. Möhring, A. S. Schulz, and N. E. Stier Moses, System-optimal routing of traffic flows with user constraints in networks with congestion, Oper. Res., 53 (2005), pp. 600–616.
17.
H. Kameda, private communication, 2002.
18.
H. Kameda, How harmful the paradox can be in the Braess/Cohen-Kelly-Jeffries networks, in Proceedings of the 21st INFOCOM Conference, Vol. 1, 2002, pp. 437–445.
19.
E. Köhler and M. Skutella, Flows over time with load-dependent transit times, SIAM J. Optim., 15 (2005), pp. 1185–1202.
20.
E. Koutsoupias and C. H. Papadimitriou, Worst-case equilibria, in Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science (STACS), Lecture Notes in Comput. Sci. 1563, Springer, Berlin, 1999, pp. 404–413.
21.
H. Lin, T. Roughgarden, and É. Tardos, A stronger bound on Braess's paradox, in Proceedings of the 15th Annual Symposium on Discrete Algorithms (SODA), 2004, pp. 333–334.
22.
H. Lin, T. Roughgarden, É. Tardos, and A. Walkover, Braess's paradox, Fibonacci numbers, and exponential inapproximability, in Proceedings of the 32nd Annual International Colloquium on Automata, Languages, and Programming (ICALP), Lecture Notes in Comput. Sci. 3580, Springer, Berlin, 2005, pp. 497–512.
23.
N. Nisan, T. Roughgarden, É. Tardos, and V. Vazirani, eds., Algorithmic Game Theory, Cambridge University Press, Cambridge, UK, 2007.
24.
C. H. Papadimitriou, Algorithms, games, and the internet, in Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC), 2001, pp. 749–753.
25.
G. Perakis, The “price of anarchy” under nonlinear and asymmetric costs, Math. Oper. Res., 32 (2007), pp. 614–628.
26.
T. Roughgarden, How unfair is optimal routing?, in Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2002, pp. 203–204.
27.
T. Roughgarden, The price of anarchy is independent of the network topology, J. Comput. System Sci., 67 (2003), pp. 341–364.
28.
T. Roughgarden, The maximum latency of selfish routing, in Proceedings of the 15th Annual Symposium on Discrete Algorithms (SODA), 2004, pp. 973–974.
29.
T. Roughgarden, On the severity of Braess's Paradox: Designing networks for selfish users is hard, J. Comput. System Sci., 72 (2006), pp. 922–953.
30.
T. Roughgarden and É. Tardos, How bad is selfish routing?, J. ACM, 49 (2002), pp. 236–259.
31.
M. J. Smith, The existence, uniqueness and stability of traffic equilibria, Transportation Res. Part B, 13 (1979), pp. 295–304.
32.
J. G. Wardrop, Some theoretical aspects of road traffic research, in Proceedings of the Institute of Civil Engineers, Pt. II, Vol. 1, 1952, pp. 325–378.
33.
D. Weitz, The Price of Anarchy, manuscript, 2001.

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Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 1667 - 1686
ISSN (online): 1095-7146

History

Submitted: 1 September 2009
Accepted: 30 August 2011
Published online: 6 December 2011

MSC codes

  1. 68Q25
  2. 68W25
  3. 90B18
  4. 90B20
  5. 91A10

Keywords

  1. Braess's paradox
  2. selfish routing
  3. price of anarchy
  4. traffic networks
  5. approximation algorithms

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