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...
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 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
Awerbuch, B., Azar, Y., Richter, Y., Tsur, D.: Tradeoffs in worst-case equlibria. Theor. Comput. Sci. 361, 200–209 (2006)
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)
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)
Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. ACM Trans. Algorithms 3(1) (2007)
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)
Fischer, S., Vöcking, B.: On the structure and complexity of worst-case equilibria. Theor. Comput. Sci. 378(2), 165–174 (2007)
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)
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)
Gonnet, G.: Expected length of the longest probe sequence in hash code searching. J. Assoc. Comput. Mach. 28(2), 289–304 (1981)
Koutsoupias, E., Mavronicolas, M., Spirakis, P.: Approximate equilibria and ball fusion. Theor. Comput. Syst. 36(6), 683–693 (2003)
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)
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)
Nash Jr., J.F.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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
DOI: https://doi.org/10.1007/978-0-387-30162-4_300
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-30770-1
Online ISBN: 978-0-387-30162-4
eBook Packages: Computer ScienceReference Module Computer Science and Engineering