Bayesian Optimal No-Deficit Mechanism Design

  • Shuchi Chawla
  • Jason D. Hartline
  • Uday Rajan
  • R. Ravi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4286)


One of the most fundamental problems in mechanism design is that of designing the auction that gives the optimal profit to the auctioneer. For the case that the probability distributions on the valuations of the bidders are known and independent, Myerson [15] reduces the problem to that of maximizing the common welfare by considering the virtual valuations in place of the bidders’ actual valuations. The Myerson auction maximizes the seller’s profit over the class of all mechanisms that are truthful and individually rational for all the bidders; however, the mechanism does not satisfy ex post individual rationality for the seller. In other words, there are examples in which for certain sets of bidder valuations, the mechanism incurs a loss.

We consider the problem of merging the worst case no-deficit (or ex post seller individual rationality) condition with this average case Bayesian expected profit maximization problem. When restricting our attention to ex post incentive compatible mechanisms for this problem, we find that the Myerson mechanism is the optimal no-deficit mechanism for supermodular costs, that Myerson merged with a simple thresholding mechanism is optimal for all-or-nothing costs, and that neither mechanism is optimal for general submodular costs. Addressing the computational side of the problem, we note that for supermodular costs the Myerson mechanism is NP-hard to compute. Furthermore, we show that for all-or-nothing costs the optimal thresholding mechanism is NP-hard to compute. Finally, we consider relaxing the ex post incentive compatibility constraint and show that there is a Bayesian incentive compatible mechanism that achieves the same expected profit as Myerson, but never incurs a loss.


Combinatorial Auction Threshold Mechanism Optimal Mechanism Post Incentive Optimal Auction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Archer, A., Papadimitriou, C., Tawar, K., Tardos, E.: An Approximate Truthful Mechanism for Combinatorial Auctions with Single Parameter Agents. In: Proc. 14th Symp. on Discrete Alg. ACM/SIAM (2003)Google Scholar
  2. 2.
    Archer, A., Tardos, E.: Truthful mechanisms for one-parameter agents. In: Proc. of the 42nd IEEE Symposium on Foundations of Computer Science (2001)Google Scholar
  3. 3.
    Bulow, J., Roberts, J.: The Simple Economics of Optimal Auctions. The Journal of Political Economy 97, 1060–1090 (1989)CrossRefGoogle Scholar
  4. 4.
    Cornelli, F.: Optimal Selling Procedures with Fixed Costs. Journal of Economic Theory 71, 1–30 (1996)zbMATHCrossRefGoogle Scholar
  5. 5.
    Feigenbaum, J., Papadimitriou, C., Shenker, S.: Sharing the Cost of Multicast Transmissions. In: Proc. of 32nd Symposium Theory of Computing, pp. 218–226. ACM Press, New York (2000)Google Scholar
  6. 6.
    Fiat, A., Goldberg, A., Hartline, J., Karlin, A.: Competitive Generalized Auctions. In: Proc. 34th ACM Symposium on the Theory of Computing. ACM Press, New York (2002)Google Scholar
  7. 7.
    Goldberg, A.V., Hartline, J.D.: Competitiveness via Concensus. In: Proc. 14th Symp. on Discrete Alg. ACM/SIAM (2003)Google Scholar
  8. 8.
    Goldberg, A.V., Hartline, J.D., Wright, A.: Competitive Auctions and Digital Goods. In: Proc. 12th Symp. on Discrete Alg., pp. 735–744. ACM/SIAM (2001)Google Scholar
  9. 9.
    Hartline, J.: Stanford cs364b: Topics in algorithmic game theory. Course Notes (2006)Google Scholar
  10. 10.
    Iwata, S., Fujishige, S., Fleischer, L.: A Combinatorial, Strongly Polynomial-Time Algorithm for Minimizing Submodular Functions. Journal of the ACM 48(4), 761–777 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Krishna, V.: Auction theory. Academic Press, San Diego (2002)Google Scholar
  12. 12.
    Lehmann, D., O’Callaghan, L.I., Shoham, Y.: Truth Revelation in Approximately Efficient Combinatorial Auctions. In: Proc. of 1st ACM Conf. on E-Commerce, pp. 96–102. ACM Press, New York (1999)CrossRefGoogle Scholar
  13. 13.
    Mehta, A., Shenker, S., Vazirani, V.: Profit-Maximizing Multicast Pricing Via Approximate Fixed Points. In: Proc. of 4th ACM Conference on Electronic Commerce. ACM Press, New York (2003)Google Scholar
  14. 14.
    Moulin, H., Shenker, S.: Strategyproof Sharing of Submodular Costs: Budget Balance Versus Efficiency. Economic Theory 18, 511–533 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Myerson, R.: Optimal Auction Design. Mathematics of Operations Research 6, 58–73 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ronen, A.: On Approximating Optimal Auctions. In: Proc. of Third ACM Conference on Electronic Commerce. ACM Press, New York (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Shuchi Chawla
    • 1
  • Jason D. Hartline
    • 1
  • Uday Rajan
    • 2
  • R. Ravi
    • 3
  1. 1.Microsoft ResearchMountain View
  2. 2.Ross School of BusinessUniversity of MichiganAnn Arbor
  3. 3.Tepper School of BusinessCarnegie Mellon UniversityPittsburgh

Personalised recommendations