The VCG Mechanism for Bayesian Scheduling

  • Yiannis Giannakopoulos
  • Maria Kyropoulou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9470)


We study the problem of scheduling m tasks to n selfish, unrelated machines in order to minimize the makespan, where the execution times are independent random variables, identical across machines. We show that the VCG mechanism, which myopically allocates each task to its best machine, achieves an approximation ratio of \(O\left( \frac{\ln n}{\ln \ln n}\right) \). This improves significantly on the previously best known bound of \(O\left( \frac{m}{n}\right) \) for prior-independent mechanisms, given by Chawla et al. [STOC’13] under the additional assumption of Monotone Hazard Rate (MHR) distributions. Although we demonstrate that this is in general tight, if we do maintain the MHR assumption, then we get improved, (small) constant bounds for \(m\ge n\ln n\) i.i.d. tasks, while we also identify a sufficient condition on the distribution that yields a constant approximation ratio regardless of the number of tasks.


  1. 1.
    Archer, A., Tardos, É.: Truthful mechanisms for one-parameter agents. In: FOCS, pp. 482–491 (2001)Google Scholar
  2. 2.
    Ashlagi, I., Dobzinski, S., Lavi, R.: Optimal lower bounds for anonymous scheduling mechanisms. Math. Oper. Res. 37(2), 244–258 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aven, T.: Upper (lower) bounds on the mean of the maximum (minimum) of a number of random variables. J. Appl. Probab. 22(3), 723–728 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barlow, R.E., Marshall, A.W., Proschan, F.: Properties of probability distributions with monotone hazard rate. Ann. Math. Stat. 34(2), 375–389 (1963)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Berenbrink, P., Friedetzky, T., Hu, Z., Martin, R.: On weighted balls-into-bins games. Theor. Comput. Sci. 409(3), 511–520 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chawla, S., Immorlica, N., Lucier, B.: On the limits of black-box reductions in mechanism design. In: STOC, pp. 435–448 (2012)Google Scholar
  7. 7.
    Chawla, S., Hartline, J.D., Malec, D., Sivan, B.: Prior-independent mechanisms for scheduling. In: STOC, pp. 51–60 (2013)Google Scholar
  8. 8.
    Christodoulou, G., Kovács, A.: A deterministic truthful PTAS for scheduling related machines. SIAM J. Comput. 42(4), 1572–1595 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Christodoulou, G., Koutsoupias, E., Vidali, A.: A lower bound for scheduling mechanisms. Algorithmica 55(4), 729–740 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Clarke, E.H.: Multipart pricing of public goods. Public Choice 11(1), 17–33 (1971)CrossRefGoogle Scholar
  11. 11.
    Daskalakis, C., Weinberg, S.M.: Bayesian truthful mechanisms for job scheduling from bi-criterion approximation algorithms. In: SODA, pp. 1934–1952 (2015)Google Scholar
  12. 12.
    Devanur, N., Hartline, J., Karlin, A., Nguyen, T.: Prior-independent multi-parameter mechanism design. In: Chen, N., Elkind, E., Koutsoupias, E. (eds.) WINE 2011. LNCS, vol. 7090, pp. 122–133. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  13. 13.
    Dhangwatnotai, P., Dobzinski, S., Dughmi, S., Roughgarden, T.: Truthful approximation schemes for single-parameter agents. SIAM J. Comput. 40(3), 915–933 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dhangwatnotai, P., Roughgarden, T., Yan, Q.: Revenue maximization with a single sample. Games Econ. Behav. 91, 318–333 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Dughmi, S., Roughgarden, T., Sundararajan, M.: Revenue submodularity. Theory Comput. 8(1), 95–119 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Giannakopoulos, Y., Kyropoulou, M.: The VCG mechanism for bayesian scheduling. CoRR, abs/1509.07455 (2015).
  17. 17.
    Groves, T.: Incentives in teams. Econometrica 41(4), 617–631 (1973)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hall, L.A.: Approximation algorithms for scheduling. In: Hochbaum, D.S. (ed.) Approximation Algorithms for NP-Hard Problems, pp. 1–45. PWS, Boston (1997)Google Scholar
  19. 19.
    Hartline, J.D., Roughgarden, T.: Simple versus optimal mechanisms. In: EC, pp. 225–234 (2009)Google Scholar
  20. 20.
    Koutsoupias, E., Vidali, A.: A lower bound of \(1+\varphi \) for truthful scheduling mechanisms. Algorithmica 66(1), 211–223 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lavi, R., Swamy, C.: Truthful mechanism design for multidimensional scheduling via cycle monotonicity. Games Econ. Behav. 67(1), 99–124 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lenstra, J.K., Shmoys, D.B., Tardos, É.: Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46, 259–271 (1990)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lu, P.: On 2-player randomized mechanisms for scheduling. In: Leonardi, S. (ed.) WINE 2009. LNCS, vol. 5929, pp. 30–41. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  24. 24.
    Lu, P., Yu, C.: An improved randomized truthful mechanism for scheduling unrelated machines. In: STACS, pp. 527–538 (2008)Google Scholar
  25. 25.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995) CrossRefGoogle Scholar
  26. 26.
    Nisan, N., Ronen, A.: Algorithmic mechanism design. Games Econ. Behav. 35(1/2), 166–196 (2001)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Nisan, N., Ronen, A.: Computationally feasible VCG mechanisms. J. Artif. Int. Res. 29(1), 19–47 (2007)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Raab, M., Steger, A.: “Balls into Bins” — a simple and tight analysis. In: Luby, M., Rolim, J.D.P., Serna, M. (eds.) RANDOM 1998. LNCS, vol. 1518, pp. 159–170. Springer, Heidelberg (1998) CrossRefGoogle Scholar
  29. 29.
    Roughgarden, T., Talgam-Cohen, I., Yan, Q.: Supply-limiting mechanisms. In: EC, pp. 844–861 (2012)Google Scholar
  30. 30.
    Sivan, B.: Prior Robust Optimization. Ph.D. thesis, University of Wisconsin-Madison (2013)Google Scholar
  31. 31.
    Vickrey, W.: Counterspeculation, auctions, and competitive sealed tenders. J. Finance 16(1), 8–37 (1961)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Yu, C.: Truthful mechanisms for two-range-values variant of unrelated scheduling. Theor. Comput. Sci. 410(21–23), 2196–2206 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 2.5 International License (, which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Authors and Affiliations

  1. 1.University of LiverpoolLiverpoolUK
  2. 2.University of OxfordOxfordUK

Personalised recommendations