Skip to main content
Book cover

Gems of Combinatorial Optimization and Graph Algorithms pp 95–102Cite as

Selfish Routing and Proportional Resource Allocation

A Joint Bound on the Loss of Optimality

  • 1586 Accesses

Abstract

We consider two optimization problems, multicommodity flow and resource allocation, from a game-theoretic point of view. We show that two known bounds on the respective losses of optimality can be derived from the same geometric quantity, which yields a simple proof of either result.

Keywords

  • Nash Equilibrium
  • Equilibrium Allocation
  • Equilibrium Flow
  • Feasible Allocation
  • Large Rectangle

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-24971-1_9
  • Chapter length: 8 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   59.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-24971-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   79.99
Price excludes VAT (USA)
Hardcover Book
USD   109.99
Price excludes VAT (USA)
Learn about institutional subscriptions
Fig. 1
Fig. 2

Notes

  1. 1.

    Note that in (1) and (2), the cost of each path and arc, respectively, is fixed in advance according to its load experienced in some equilibrium. Such equilibrium costs are, in fact, unique, even if instances admit more than one equilibrium flow.

  2. 2.

    Note that this allocation cannot arise from an equilibrium; only the bid of Player 1 would be positive, who could bid less and still receive the entire resource.

  3. 3.

    Similarly to the case of uncoordinated routing in Sect. 2, the equilibrium allocation is known to be unique.

  4. 4.

    For (7), we also use that \(x_i \le 1\) and that \(U_i\) is non-negative for all i, which implies that \(\sum _{i=1}^n U_i(x_i) x_i \le \sum _{i=1}^n U_i(x_i)\).

References

  1. Beckmann, M.J., McGuire, C.B. Winsten, C.B.: Studies in the Economics of Transportation. Yale University Press (1956)

    Google Scholar 

  2. Braess, D.: Über ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung 12, 258–268 (1969). English translation: Braess, D., Nagurney, A., Wakolbinger, T.: On a paradox of traffic planning. Trans. Sci. 39, 446–450 (2005)

    Google Scholar 

  3. Correa, J.R., Schulz, A.S., Stier-Moses, N.E.: Selfish routing in capacitated networks. Math. Oper. Res. 29, 961–976 (2004)

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Correa, J.R., Schulz, A.S., Stier-Moses, N.E.: A geometric approach to the price of anarchy in nonatomic congestion games. Games Econ. Behav. 64, 457–469 (2008)

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. Correa, J.R., Schulz, A.S., Stier-Moses, N.E.: The price of anarchy of the proportional allocation mechanism revisited. In: Chen, Y., Immorlica, N. (eds.) Web and Internet Economics. Lecture Notes in Computer Science, Vol. 8289, pp. 109–120. Springer (2013)

    Google Scholar 

  6. Correa, J.R., Stier-Moses, N.E.: Wardrop equilibria. In: Wiley Encyclopedia of Operations Research and Management Science. Wiley (2011)

    Google Scholar 

  7. Dafermos, S.C.: Traffic equilibrium and variational inequalities. Trans. Sci. 14, 42–54 (1980)

    MathSciNet  CrossRef  Google Scholar 

  8. Dafermos, S.C., Sparrow, F.T.: The traffic assignment problem for a general network. J. Res. U.S. Nat. Bur. Stand. 73B, 91–118 (1969)

    Google Scholar 

  9. Florian, M.: Untangling traffic congestion: Application of network equilibrium models in transportation planning. OR/MS Today 26, 52–57 (1999)

    Google Scholar 

  10. Gardner, M.: Some surprising theorems about rectangles in triangles. Math. Horiz. 5, 18–22 (1997)

    Google Scholar 

  11. Haurie, A., Marcotte, P.: On the relationship between Nash-Cournot and Wardrop equilibria. Networks 15, 295–308 (1985)

    MathSciNet  CrossRef  MATH  Google Scholar 

  12. Johari, R., Tsitsiklis, J.N.: Efficiency loss in a network resource allocation game. Math. Oper. Res. 29, 407–435 (2004)

    MathSciNet  CrossRef  MATH  Google Scholar 

  13. Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. In: Meinel, C., Tison, S. (eds.) Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, Vol. 1563, pp. 404–413, Springer (1999)

    Google Scholar 

  14. Lange, L.H.: What is the biggest rectangle you can put inside a given triangle? Coll. Math. J. 24, 237–240 (1993)

    CrossRef  Google Scholar 

  15. Nash, J.: Equilibrium points in \(n\)-person games. Proc. Natl. Acad. Sci. 36, 48–49 (1950)

    MathSciNet  CrossRef  MATH  Google Scholar 

  16. Papadimitriou, C.H.: Algorithms, games, and the Internet. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 749–753. ACM Press (2001)

    Google Scholar 

  17. Pigou, A.C.: The Economics of Welfare. Macmillan (1920)

    Google Scholar 

  18. Roughgarden, T.: Potential functions and the inefficiency of equilibria. Proc. Intern. Congr. Math. III, 1071–1094 (2006)

    Google Scholar 

  19. Roughgarden, T., Tardos, É.: How bad is selfish routing? J. ACM 49, 236–259 (2002)

    MathSciNet  CrossRef  MATH  Google Scholar 

  20. Roughgarden, T., Tardos, É.: Introduction to the inefficiency of equilibria. Chapter 17 In: Nisan, N. et al. (eds.) Algorithmic Game Theory. Cambridge University Press (2007)

    Google Scholar 

  21. Smith, M.J.: The existence, uniqueness and stability of traffic equilibria. Trans. Res. B 13, 295–304 (1979)

    MathSciNet  CrossRef  Google Scholar 

  22. Wardrop, J.G.: Some theoretical aspects of road traffic research. Proc. Inst. Civil Eng. Part II 1, 325–378 (1952)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik und Fakultät für Wirtschaftswissenschaften, Technische Universität München, Arcisstr. 21, 80333, München, Germany

    Andreas S. Schulz

Authors
  1. Andreas S. Schulz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreas S. Schulz .

Editor information

Editors and Affiliations

  1. Technische Universität München, Munich, Germany

    Andreas S. Schulz

  2. Institute of Mathematics, Technische Universität Berlin, Berlin, Berlin, Germany

    Martin Skutella

  3. Technische Universität Braunschweig, Braunschweig, Germany

    Sebastian Stiller

  4. Institute of Theoretical Informatics, Karlsruher Institut für Technologie (KIT), Karlsruhe, Germany

    Dorothea Wagner

Rights and permissions

Reprints and Permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Schulz, A.S. (2015). Selfish Routing and Proportional Resource Allocation. In: Schulz, A., Skutella, M., Stiller, S., Wagner, D. (eds) Gems of Combinatorial Optimization and Graph Algorithms . Springer, Cham. https://doi.org/10.1007/978-3-319-24971-1_9

Download citation