Resource Competition on Integral Polymatroids

  • Tobias Harks
  • Max Klimm
  • Britta Peis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8877)


We study competitive resource allocation problems in which players distribute their demands integrally over a set of resources subject to player-specific submodular capacity constraints. Each player has to pay for each unit of demand a cost that is a non-decreasing and convex function of the total allocation of that resource. This general model of resource allocation generalizes both singleton congestion games with integer-splittable demands and matroid congestion games with player-specific costs. As our main result, we show that in such general resource allocation problems a pure Nash equilibrium is guaranteed to exist by giving a pseudo-polynomial algorithm computing a pure Nash equilibrium.


Rank Function Resource Competition Strategy Space Congestion Game Pure Nash Equilibrium 
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  1. 1.
    Ackermann, H., Röglin, H., Vöcking, B.: Pure Nash equilibria in player-specific and weighted congestion games. Theoret. Comput. Sci. 410(17), 1552–1563 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, É., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. SIAM J. Comput. 38(4), 1602–1623 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Antonakopoulos, S., Chekuri, C., Shepherd, F.B., Zhang, L.: Buy-at-bulk network design with protection. Math. Oper. Res. 36(1), 71–87 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Beckmann, M., McGuire, C., Winsten, C.: Studies in the Economics and Transportation. Yale University Press, New Haven (1956)Google Scholar
  5. 5.
    Chen, H., Roughgarden, T.: Network design with weighted players. Theory Comput. Syst. 45(2), 302–324 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chen, H.L., Roughgarden, T., Valiant, G.: Designing network protocols for good equilibria. SIAM J. Comput. 39(5), 1799–1832 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Dunkel, J., Schulz, A.: On the complexity of pure-strategy Nash equilibria in congestion and local-effect games. Math. Oper. Res. 33(4), 851–868 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Faigle, U.: The greedy algorithm for partially ordered sets. Discrete Math. 28(2), 153–159 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    von Falkenhausen, P., Harks, T.: Optimal cost sharing for resource selection games. Math. Oper. Res. 38(1), 184–208 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Fotakis, D., Kontogiannis, S., Spirakis, P.: Selfish unsplittable flows. Theoret. Comput. Sci. 348(2-3), 226–239 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Gairing, M., Monien, B., Tiemann, K.: Routing (un-)splittable flow in games with player-specific linear latency functions. ACM Trans. Algorithms 7(3), 1–31 (2011)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Harks, T., Klimm, M.: Congestion games with variable demands. In: Apt, K. (ed.) Proc. 13th Conf. Theoret. Aspects of Rationality and Knowledge, pp. 111–120 (2011)Google Scholar
  13. 13.
    Harks, T., Klimm, M.: On the existence of pure Nash equilibria in weighted congestion games. Math. Oper. Res. 37(3), 419–436 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Haurie, A., Marcotte, P.: On the relationship between Nash-Cournot and Wardrop equilibria. Networks 15, 295–308 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Helgason, T.: Aspects of the theory of hypermatroids. In: Hypergraph Seminar, pp. 191–213. Springer (1974)Google Scholar
  16. 16.
    Ieong, S., McGrew, R., Nudelman, E., Shoham, Y., Sun, Q.: Fast and compact: A simple class of congestion games. In: Proc. 20th Natl. Conf. Artificial Intelligence and the 17th Innovative Appl. Artificial Intelligence Conf., pp. 489–494 (2005)Google Scholar
  17. 17.
    Johari, R., Tsitsiklis, J.N.: A scalable network resource allocation mechanism with bounded efficiency loss. IEEE J. Sel. Area Commun. 24(5), 992–999 (2006)CrossRefGoogle Scholar
  18. 18.
    Kelly, F., Maulloo, A., Tan, D.: Rate control in communication networks: Shadow prices, proportional fairness, and stability. J. Oper. Res. Soc. 49, 237–252 (1998)CrossRefzbMATHGoogle Scholar
  19. 19.
    Krysta, P., Sanders, P., Vöcking, B.: Scheduling and traffic allocation for tasks with bounded splittability. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 500–510. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  20. 20.
    Meyers, C.: Network Flow Problems and Congestion Games: Complexity and Approximation Results. Ph.D. thesis, MIT, Operations Research Center (2006)Google Scholar
  21. 21.
    Milchtaich, I.: Congestion games with player-specific payoff functions. Games Econom. Behav. 13(1), 111–124 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Milchtaich, I.: The equilibrium existence problem in finite network congestion games. In: Spirakis, P., Mavronicolas, M., Kontogiannis, S. (eds.) WINE 2006. LNCS, vol. 4286, pp. 87–98. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  23. 23.
    Rosenthal, R.: A class of games possessing pure-strategy Nash equilibria. Internat. J. Game Theory 2(1), 65–67 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Rosenthal, R.: The network equilibrium problem in integers. Networks 3, 53–59 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Roughgarden, T.: Selfish Routing and the Price of Anarchy. MIT Press, Cambridge (2005)Google Scholar
  26. 26.
    Schrijver, A.: Combinatorial optimization: Polyhedra and efficiency, vol. 24. Springer (2003)Google Scholar
  27. 27.
    Srikant, R.: The Mathematics of Internet Congestion Control. Birkhäuser, Basel (2003)Google Scholar
  28. 28.
    Tran-Thanh, L., Polukarov, M., Chapman, A., Rogers, A., Jennings, N.R.: On the existence of pure strategy Nash equilibria in integer–splittable weighted congestion games. In: Persiano, G. (ed.) SAGT 2011. LNCS, vol. 6982, pp. 236–253. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  29. 29.
    Wardrop, J.: Some theoretical aspects of road traffic research. Proc. Inst. Civil Engineers 1(Part II), 325–378 (1952)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Tobias Harks
    • 1
  • Max Klimm
    • 2
  • Britta Peis
    • 3
  1. 1.Department of Quantitative EconomicsMaastricht UniversityThe Netherlands
  2. 2.Department of MathematicsTechnische Universität BerlinGermany
  3. 3.School of Business and EconomicsRWTH Aachen UniversityGermany

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