Advertisement

On the Price of Anarchy of Cost-Sharing in Real-Time Scheduling Systems

  • Eirini Georgoulaki
  • Kostas KolliasEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11920)

Abstract

We study cost-sharing games in real-time scheduling systems where the operation cost of the server at any given time is a function of its load. We focus on monomial cost functions and consider both the case when the degree is less than one (inducing positive externalities for the jobs) and when it is greater than one (inducing negative externalities for the jobs). For the former case, we provide tight price of anarchy bounds which show that the price of anarchy grows to infinity as a polynomial of the number of jobs in the game. For the latter, we observe that existing results provide constant and tight (asymptotically in the degree of the monomial) bounds on the price of anarchy. We then switch our attention to improving the price of anarchy by means of a simple coordination mechanism that has no knowledge of the instance. We show that our mechanism reduces the price of anarchy of games with n jobs and unit server costs from \(\varTheta (\sqrt{n})\) to 2. We also show that for a restricted class of instances a similar improvement is achieved for monomial server costs. This is not the case, however, for unrestricted instances of monomial costs for which we prove that the price of anarchy remains super-constant for our mechanism.

References

  1. 1.
    Aland, S., Dumrauf, D., Gairing, M., Monien, B., Schoppmann, F.: Exact price of anarchy for polynomial congestion games. SIAM J. Comput. 40(5), 1211–1233 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Albers, S.: Energy-efficient algorithms. Commun. ACM 53(5), 86–96 (2010)CrossRefGoogle Scholar
  3. 3.
    Anshelevich, E., Dasgupta, A., Kleinberg, J.M., 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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Awerbuch, B., Azar, Y., Epstein, A. The price of routing unsplittable flow. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, 22–24 May 2005, pp. 57–66 (2005)Google Scholar
  5. 5.
    Azar, Y., Fleischer, L., Jain, K., Mirrokni, V.S., Svitkina, Z.: Optimal coordination mechanisms for unrelated machine scheduling. Oper. Res. 63(3), 489–500 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bar-Noy, A., Guha, S., Naor, J., Schieber, B.: Approximating the throughput of multiple machines in real-time scheduling. SIAM J. Comput. 31(2), 331–352 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bhawalkar, K., Gairing, M., Roughgarden, T.: Weighted congestion games: the price of anarchy, universal worst-case examples, and tightness. ACM Trans. Econ. Comput. 2(4), 14:1–14:23 (2014)CrossRefGoogle Scholar
  8. 8.
    Caragiannis, I.: Efficient coordination mechanisms for unrelated machine scheduling. Algorithmica 66(3), 512–540 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chang, J., Gabow, H.N., Khuller, S.: A model for minimizing active processor time. Algorithmica 70(3), 368–405 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, H., Roughgarden, T., Valiant, G.: Designing network protocols for good equilibria. SIAM J. Comput. 39(5), 1799–1832 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Christodoulou, G., Koutsoupias, E.: The price of anarchy of finite congestion games. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, 22–24 May 2005, pp. 67–73 (2005)Google Scholar
  12. 12.
    Christodoulou, G., Koutsoupias, E., Nanavati, A.: Coordination mechanisms. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 345–357. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-27836-8_31CrossRefGoogle Scholar
  13. 13.
    Cole, R., Correa, J.R., Gkatzelis, V., Mirrokni, V.S., Olver, N.: Decentralized utilitarian mechanisms for scheduling games. Games Econ. Behav. 92, 306–326 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gairing, M., Kollias, K., Kotsialou, G.: Tight bounds for cost-sharing in weighted congestion games. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 626–637. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-47666-6_50CrossRefGoogle Scholar
  15. 15.
    Gkatzelis, V., Kollias, K., Roughgarden, T.: Optimal cost-sharing in general resource selection games. Oper. Res. 64(6), 1230–1238 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Goemans, M.X., Mirrokni, V.S., Vetta, A.: Sink equilibria and convergence. In: 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), 23–25 October 2005, Pittsburgh, PA, USA, Proceedings, pp. 142–154 (2005)Google Scholar
  17. 17.
    Gopalakrishnan, R., Marden, J.R., Wierman, A.: Potential games are necessary to ensure pure nash equilibria in cost sharing games. Math. Oper. Res. 39(4), 1252–1296 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Immorlica, N., Li, L.E., Mirrokni, V.S., Schulz, A.S.: Coordination mechanisms for selfish scheduling. Theor. Comput. Sci. 410(17), 1589–1598 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Irani, S., Pruhs, K.: Algorithmic problems in power management. SIGACT News 36(2), 63–76 (2005)CrossRefGoogle Scholar
  20. 20.
    Khandekar, R., Schieber, B., Shachnai, H., Tamir, T.: Real-time scheduling to minimize machine busy times. J. Sched. 18(6), 561–573 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Klimm, M., Schmand, D.: Sharing non-anonymous costs of multiple resources optimally. In: Paschos, V.T., Widmayer, P. (eds.) CIAC 2015. LNCS, vol. 9079, pp. 274–287. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-18173-8_20CrossRefGoogle Scholar
  22. 22.
    Kollias, K., Roughgarden, T.: Restoring pure equilibria to weighted congestion games. ACM Trans. Econ. Comput. 3(4), 21:1–21:24 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. Comput. Sci. Rev. 3(2), 65–69 (2009)CrossRefGoogle Scholar
  24. 24.
    Marden, J.R., Philips, M.: Optimizing the price of anarchy in concave cost sharing games. In: 2017 American Control Conference, ACC 2017, Seattle, WA, USA, 24–26 May 2017, pp. 5237–5242 (2017)Google Scholar
  25. 25.
    Marden, J.R., Wierman, A.: Distributed welfare games. Oper. Res. 61(1), 155–168 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rosenthal, R.W.: A class of games possessing pure-strategy nash equilibria. Int. J. Game Theory 2(1), 65–67 (1973)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Roughgarden, T.: Intrinsic robustness of the price of anarchy. J. ACM 62(5), 32:1–32:42 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Roughgarden, T., Schrijvers, O.: Network cost-sharing without anonymity. AACM Trans. Econ. Comput. 4(2), 8:1–8:24 (2016)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Tamir, T.: Cost-sharing games in real-time scheduling systems. In: Christodoulou, G., Harks, T. (eds.) WINE 2018. LNCS, vol. 11316, pp. 423–437. Springer, Cham (2018).  https://doi.org/10.1007/978-3-030-04612-5_28CrossRefGoogle Scholar
  30. 30.
    von Falkenhausen, P., Harks, T.: Optimal cost sharing for resource selection games. Math. Oper. Res. 38(1), 184–208 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of AthensAthensGreece
  2. 2.Google ResearchMountain ViewUSA

Personalised recommendations