Two Dimensional Optimal Mechanism Design for a Sequencing Problem

  • Ruben Hoeksma
  • Marc Uetz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7801)

Abstract

We propose an optimal mechanism for a sequencing problem where the jobs’ processing times and waiting costs are private. Given public priors for jobs’ private data, we seek to find a scheduling rule and incentive compatible payments that minimize the total expected payments to the jobs. Here, incentive compatible refers to a Bayes-Nash equilibrium. While the problem can be efficiently solved when jobs have single dimensional private data, we here address the problem with two dimensional private data. We show that the problem can be solved in polynomial time by linear programming techniques, answering an open problem in [13]. Our implementation is randomized and truthful in expectation. The main steps are a compactification of an exponential size linear program, and a combinatorial algorithm to decompose feasible interim schedules. In addition, in computational experiments with random instances, we generate some more insights.

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References

  1. 1.
    Alaei, S., Fu, H., Haghpanah, N., Hartline, J., Malekian, A.: Bayesian Optimal Auctions via Multi- to Single-agent Reduction. In: Proc. 13th EC, p. 17 (2012)Google Scholar
  2. 2.
    Cai, Y., Daskalakis, C., Weinberg, S.M.: An Algorithmic Characterization of Multi-Dimensional Mechanisms. In: Proc. 44th STOC (2012)Google Scholar
  3. 3.
    Cai, Y., Daskalakis, C., Weinberg, S.M.: Optimal Multi-Dimensional Mechanism Design: Reducing Revenue to Welfare Maximization. In: Proc. 53rd FOCS (2012)Google Scholar
  4. 4.
    Conitzer, V., Sandholm, T.: Complexity of mechanism design. In: Proc. 18th Annual Conference on Uncertainty in Artificial Intelligence, UAI 2002, pp. 103–110 (2002)Google Scholar
  5. 5.
    Duives, J., Heydenreich, B., Mishra, D., Müller, R., Uetz, M.: Optimal Mechanisms for Single Machine Scheduling (2012) (manuscript)Google Scholar
  6. 6.
    Dyer, M.E., Wolsey, L.A.: Formulating the single machine sequencing problem with release dates as a mixed integer program. Disc. Appl. Math. 26, 255–270 (1990)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Edmonds, J.: Matroids and the greedy algorithm. Math. Prog. 1, 127–136 (1971)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Fishburn, P.C.: Induced binary probabilities and the linear ordering polytope: A status report. Mathematical Social Sciences 23, 67–80 (1992)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Fonlupt, J., Skoda, A.: Strongly polynomial algorithm for the intersection of a line with a polymatroid. In: Research Trends in Combinatorial Optimization, pp. 69–85. Springer (2009)Google Scholar
  10. 10.
    Gershkov, A., Goeree, J.K., Kushnir, A., Moldovanu, B., Shi, X.: On the Equivalence of Bayesian and Dominant Strategy Implementation. Econometrica 81, 197–220 (2013)CrossRefGoogle Scholar
  11. 11.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization. Algorithms and combinatorics. Springer (1988)Google Scholar
  12. 12.
    Hartline, J.D., Karlin, A.: Profit Maximization in Mechanism Design. In: Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V. (eds.) Algorithmic Game Theory, ch. 13. Cambridge University Press (2007)Google Scholar
  13. 13.
    Heydenreich, B., Mishra, D., Müller, R., Uetz, M.: Optimal Mechanisms for Single Machine Scheduling. In: Papadimitriou, C., Zhang, S. (eds.) WINE 2008. LNCS, vol. 5385, pp. 414–425. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Kenis, P.: Waiting lists in Dutch health care: An analysis from an organization theoretical perspective. J. Health Organization and Mgmt. 20, 294–308 (2006)CrossRefGoogle Scholar
  15. 15.
    Manelli, A.M., Vincent, D.R.: Bayesian and Dominant Strategy Implementation in the Independent Private Values Model. Econometrica 78, 1905–1938 (2010)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Myerson, R.B.: Optimal Auction Design. Math. OR 6, 58–73 (1981)MathSciNetMATHGoogle Scholar
  17. 17.
    Nisan, N.: Introduction to Mechanism Design (for Computer Scientists). In: Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V. (eds.) Algorithmic Game Theory, ch. 9. Cambridge University Press (2007)Google Scholar
  18. 18.
    Queyranne, M.: Structure of a simple scheduling polyhedron. Math. Prog. 58, 263–285 (1993)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Queyranne, M., Schulz, A.S.: Polyhedral Approaches to Machine Scheduling. TU Berlin Technical Report 408/1994Google Scholar
  20. 20.
    Sandholm, T.W.: Automated Mechanism Design: A New Application Area for Search Algorithms. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 19–36. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  21. 21.
    Smith, W.E.: Various optimizers for single-stage production. Naval Research Logistics Quarterly 3, 59–66 (1956)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Vohra, R.: Optimization and mechanism design. Math. Prog. 134, 283–303 (2012)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Yasutake, S., Hatano, K., Kijima, S., Takimoto, E., Takeda, M.: Online Linear Optimization over Permutations. In: Asano, T., Nakano, S.-i., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 534–543. Springer, Heidelberg (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ruben Hoeksma
    • 1
  • Marc Uetz
    • 1
  1. 1.Dept. Applied MathematicsUniversity of TwenteEnschedeThe Netherlands

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