On the Existence of Pure Strategy Nash Equilibria in Integer–Splittable Weighted Congestion Games

  • Long Tran-Thanh
  • Maria Polukarov
  • Archie Chapman
  • Alex Rogers
  • Nicholas R. Jennings
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6982)


We study the existence of pure strategy Nash equilibria (PSNE) in integer–splittable weighted congestion games (ISWCGs), where agents can strategically assign different amounts of demand to different resources, but must distribute this demand in fixed-size parts. Such scenarios arise in a wide range of application domains, including job scheduling and network routing, where agents have to allocate multiple tasks and can assign a number of tasks to a particular selected resource. Specifically, in an ISWCG, an agent has a certain total demand (aka weight) that it needs to satisfy, and can do so by requesting one or more integer units of each resource from an element of a given collection of feasible subsets. Each resource is associated with a unit–cost function of its level of congestion; as such, the cost to an agent for using a particular resource is the product of the resource unit–cost and the number of units the agent requests.

While general ISWCGs do not admit PSNE [(Rosenthal, 1973b)], the restricted subclass of these games with linear unit–cost functions has been shown to possess a potential function [(Meyers, 2006)], and hence, PSNE. However, the linearity of costs may not be necessary for the existence of equilibria in pure strategies. Thus, in this paper we prove that PSNE always exist for a larger class of convex and monotonically increasing unit–costs. On the other hand, our result is accompanied by a limiting asumption on the structure of agents’ strategy sets: specifically, each agent is associated with its set of accessible resources, and can distribute its demand across any subset of these resources.

Importantly, we show that neither monotonicity nor convexity on its own guarantees this result. Moreover, we give a counterexample with monotone and semi–convex cost functions, thus distinguishing ISWCGs from the class of infinitely–splittable congestion games for which the conditions of monotonicity and semi–convexity have been shown to be sufficient for PSNE existence [(Rosen, 1965)]. Furthermore, we demonstrate that the finite improvement path property (FIP) does not hold for convex increasing ISWCGs. Thus, in contrast to the case with linear costs, a potential function argument cannot be used to prove our result. Instead, we provide a procedure that converges to an equilibrium from an arbitrary initial strategy profile, and in doing so show that ISWCGs with convex increasing unit–cost functions are weakly acyclic.


Cost Function Nash Equilibrium Smart Grid Pure Strategy Forward Move 
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  1. Beier, R., Czumaj, A., Krysta, P., Vöcking, B.: Computing equilibria for congestion games with (im)perfect information. In: Proc. of SODA 2004, pp. 746–755 (2004)Google Scholar
  2. Byde, A., Polukarov, M., Jennings, N.R.: Games with congestion-averse utilities. In: Mavronicolas, M., Papadopoulou, V.G. (eds.) SAGT 2009. LNCS, vol. 5814, pp. 220–232. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. Cominetti, R., Correa, J.R., Stier-Moses, N.E.: The impact of oligopolistic competition in networks. Operations Research 57(6), 1421–1437 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Dunkel, J., Schulz, A.: On the complexity of pure-strategy nash equilibria in congestion and local-effect games. Mathemathics of Operations Research 33(4), 851–868 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Koutsoupias, E., Papadimitriou, C.H.: Worst–case equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, p. 404. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. Krysta, P., Sanders, P., Vöcking, B.: Scheduling and traffic allocation for tasks with bounded splittability. In: Proc. of the 28th International Symposium on Mathematical Foundations of Computer Science, pp. 500–510 (2003)Google Scholar
  7. Leslie, D.S., Collins, E.J.: Generalised weakened fictitious play. Games and Economic Behavior 56, 285–298 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Leyton-Brown, K., Tennenholtz, M.: Local-effect games. In: Proc. of IJCAI 2003, pp. 772–780 (2003)Google Scholar
  9. Marden, J.R., Arslan, G., Shamma, J.S.: Regret based dynamics: Convergence in weakly acyclic games. In: Proc. of AAMAS 2007, pp. 194–201 (2007)Google Scholar
  10. Meyers, C.: Network flow problems and congestion games: complexity and approximation results. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, USA. AAI0809430 (2006)Google Scholar
  11. Milchtaich, I.: Congestion games with player–specific payoff functions. Games and Economic Behavior 13(1), 111–124 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Monderer, D., Shapley, L.S.: Potential games. Games and Economic Behavior 14, 124–143 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Orda, A., Rom, R., Shimkin, N.: Competitive routing in multiuser communication networks. IEEE/ACM Trans. Networking 1(5), 510–521 (1993)CrossRefGoogle Scholar
  14. Penn, M., Polukarov, M., Tennenholtz, M.: Congestion games with load-dependent failures: Identical resources. Games and Economic Behavior 67(1), 156–173 (2009a)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Penn, M., Polukarov, M., Tennenholtz, M.: Random order congestion games. Mathematics of Op. Res. 34(3), 706–725 (2009b)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Rosen, J.: Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 33, 520–534 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Int. J. of Game Theory 2, 65–67 (1973a)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Rosenthal, R.W.: The network equilibrium problem in integers. Networks 3, 53–59 (1973b)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Roughgarden, T., Tardos, E.: How bad is selfish routing? J. ACM 49(2), 236–259 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Shachnai, H., Tamir, T.: Multiprocessor scheduling with machine allotment and parallelism constraints. Algorithmica 32, 651–678 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Voice, T.D., Polukarov, M., Jennings, N.R.: On the impact of strategy and utility structures on congestion-averse games. In: Leonardi, S. (ed.) WINE 2009. LNCS, vol. 5929, pp. 600–607. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Long Tran-Thanh
    • 1
  • Maria Polukarov
    • 1
  • Archie Chapman
    • 2
  • Alex Rogers
    • 1
  • Nicholas R. Jennings
    • 1
  1. 1.School of Electronics and Computer ScienceUniversity of SouthamptonUK
  2. 2.The University of Sydney Business SchoolSydneyAustralia

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