Skip to main content
SpringerLink
Log in

Price of Anarchy for Machines Models

2002; Czumaj, Vöcking

  • Reference work entry
Encyclopedia of Algorithms

  • 193 Accesses

Keywords and Synonyms

Worst-case coordination ratio ; Selfish routing                

Problem Definition

Notations

This entry considers a selfish routing model formally introduced by Koutsoupias and Papadimitriou [11], in which the goal is to route the traffic on parallel links with linear latency functions. One can describe this model as a scheduling problem with m independent machines with speeds \( { s_1, \dots, s_m } \) and n independent tasks with weights \( { w_1, \dots, w_n } \). The goal is to allocate the tasks to the machines to minimize the maximum load of the links in the system.

It is assumed that all tasks are assigned by non-cooperative agents. The set of pure strategies for task i is the set \( { \{1, \dots, m\} } \) and a mixed strategy is a distribution on this set.

Given a combination \( { (j_1, \dots, j_n) \in \{1, \dots, m\}^n } \) of pure strategies, one for each task, the cost for task i is \( { \sum_{j_k = j_i} \frac{w_k}{s_{j_i}} } \), which is the time needed for machine j...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 399.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    To simplify the notation, for any real \( { x \ge 0 } \), let \( { \log x } \) denote \( { \log x = \max \{\log_2 x, 1\} } \). Also, following standard convention, \( { \Gamma(N) } \) is used to denote the Gamma (factorial) function, which for any natural N is defined by \( { \Gamma(N+1) = N! } \) and for an arbitrary real \( { x > 0 } \) is \( { \Gamma(x)= \int_0^{\infty} t^{x-1} \mskip2mu\mathrm{e}^{-t} \mskip2mu\mathrm{d} t } \). For the inverse of the Gamma function, \( { \Gamma^{(-1)}(N) } \), it is known that \( { \Gamma^{(-1)} (N) = x } \) such that \( { \lfloor x \rfloor! \le N-1 \le \lceil x \rceil! } \). It is well known that \( { \Gamma^{(-1)} (N) = (\log N )/(\log\log N) (1+o(1)) } \).

Recommended Reading

  1. Awerbuch, B., Azar, Y., Richter, Y., Tsur, D.: Tradeoffs in worst-case equlibria. Theor. Comput. Sci. 361, 200–209 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Christodoulou, G., Koutsoupias, E., Nanavati, A.: Coordination mechanisms. In: Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP), pp. 345–357 (2004)

    Google Scholar 

  3. Czumaj, A.: Selfish routing on the Internet. In: Leung, J. (ed.) Handbook of Scheduling: Algorithms, Models, and Performance Analysis. CRC Press, Boca Raton, FL, USA (2004)

    Google Scholar 

  4. Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. ACM Trans. Algorithms 3(1) (2007)

    Google Scholar 

  5. Czumaj, A., Krysta, P., Vöcking, B.: Selfish traffic allocation for server farms. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), pp. 287–296 (2002)

    Google Scholar 

  6. Fischer, S., Vöcking, B.: On the structure and complexity of worst-case equilibria. Theor. Comput. Sci. 378(2), 165–174 (2007)

    Article  MATH  Google Scholar 

  7. Fotakis, D., Kontogiannis, S., Koutsoupias, E., Mavronicolas, M., Spirakis, P.: The structure and complexity of Nash equilibria for a selfish routing game. In Proceedings of the 29th International Colloquium on Automata, Languages and Programming (ICALP), pp. 123–134, (2002)

    Google Scholar 

  8. Gairing, M., Lücking, T., Mavronicolas, M., Monien, B.: The price of anarchy for polynomial social cost. Theor. Comput. Sci. 369(1-3), 116–135 (2006)

    Article  MATH  Google Scholar 

  9. Gonnet, G.: Expected length of the longest probe sequence in hash code searching. J. Assoc. Comput. Mach. 28(2), 289–304 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  10. Koutsoupias, E., Mavronicolas, M., Spirakis, P.: Approximate equilibria and ball fusion. Theor. Comput. Syst. 36(6), 683–693 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. In: Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science (STACS), pp. 404–413 (1999)

    Google Scholar 

  12. Mavronicolas, M., Spirakis, P.: The price of selfish routing. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC), pp. 510–519 (2001)

    Google Scholar 

  13. Nash Jr., J.F.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)

    Article  MathSciNet  Google Scholar 

  14. Vöcking, B.: Selfish load balancing. In: Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V. (eds.) Algorithmic Game Theory. Cambridge University Press, New York, NY, USA (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DIMAP and Computer Science, University of Warwick, Coventry, UK

    Artur Czumaj

  2. Department of Computer Science, RWTH Aachen University, Aachen, Germany

    Berthold Vöcking

Authors
  1. Artur Czumaj
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Berthold Vöcking
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag

About this entry

Cite this entry

Czumaj, A., Vöcking, B. (2008). Price of Anarchy for Machines Models. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_300

Download citation

Publish with us

Policies and ethics

Search

Navigation