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Multidimensional Mechanism Design: Finite-Dimensional Approximations and Efficient Computation

Author

Listed:
  • Alexandre Belloni

    (Fuqua School of Business, Duke University, Durham, North Carolina 27708)

  • Giuseppe Lopomo

    (Fuqua School of Business, Duke University, Durham, North Carolina 27708)

  • Shouqiang Wang

    (Fuqua School of Business, Duke University, Durham, North Carolina 27708)

Abstract

Multidimensional mechanism design problems have proven difficult to solve by extending techniques from the one-dimensional case. This paper considers mechanism design problems with multidimensional types when the seller's cost function is not separable across buyers. By adapting results obtained by Border [Border, K. 1991. Implementation of reduced form auctions: A geometric approach. Econometrica 59 1175--1187], we transform the seller's problem into a representation that only involves “interim” variables and eliminates the dimensionality dependence on the number of buyers. We show that the associated infinite-dimensional optimization problem posed by the theoretical model can be approximated arbitrarily well by a sequence of finite-dimensional linear programming problems.We provide an efficient---i.e., terminating in polynomial time in the problem size---method to compute the separation oracle associated with the Border constraints and incentive compatibility constraints. This implies that our finite-dimensional approximation is solvable in polynomial time.Finally, we illustrate how the numerical solutions of the finite-dimensional approximations can provide insights into the nature of optimal solutions to the infinite-dimensional problem in particular cases.

Suggested Citation

  • Alexandre Belloni & Giuseppe Lopomo & Shouqiang Wang, 2010. "Multidimensional Mechanism Design: Finite-Dimensional Approximations and Efficient Computation," Operations Research, INFORMS, vol. 58(4-part-2), pages 1079-1089, August.
    • Handle: RePEc:inm:oropre:v:58:y:2010:i:4-part-2:p:1079-1089
    DOI: 10.1287/opre.1100.0824
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    References listed on IDEAS

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    1. Armstrong, Mark, 1996. "Multiproduct Nonlinear Pricing," Econometrica, Econometric Society, vol. 64(1), pages 51-75, January.
    2. Sandro Brusco & Giuseppe Lopomo & Leslie M. Marx, 2011. "The Economics of Contingent Re-auctions," American Economic Journal: Microeconomics, American Economic Association, vol. 3(2), pages 165-193, May.
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    4. Eric Maskin & John Riley, 1984. "Monopoly with Incomplete Information," RAND Journal of Economics, The RAND Corporation, vol. 15(2), pages 171-196, Summer.
    5. Border, Kim C, 1991. "Implementation of Reduced Form Auctions: A Geometric Approach," Econometrica, Econometric Society, vol. 59(4), pages 1175-1187, July.
    6. Myerson, Roger B, 1979. "Incentive Compatibility and the Bargaining Problem," Econometrica, Econometric Society, vol. 47(1), pages 61-73, January.
    7. Manelli, Alejandro M. & Vincent, Daniel R., 2007. "Multidimensional mechanism design: Revenue maximization and the multiple-good monopoly," Journal of Economic Theory, Elsevier, vol. 137(1), pages 153-185, November.
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    Citations

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    Cited by:

    1. Braulio Calagua, 2023. "Reducing incentive constraints in bidimensional screening," The Journal of Mechanism and Institution Design, Society for the Promotion of Mechanism and Institution Design, University of York, vol. 8(1), pages 107-150, December.
    2. Sandro Brusco & Giuseppe Lopomo & Leslie M. Marx, 2011. "The Economics of Contingent Re-auctions," American Economic Journal: Microeconomics, American Economic Association, vol. 3(2), pages 165-193, May.
    3. Goeree, Jacob K. & Kushnir, Alexey, 2016. "Reduced form implementation for environments with value interdependencies," Games and Economic Behavior, Elsevier, vol. 99(C), pages 250-256.
    4. Alexey Kushnir & James Michelson, 2022. "Optimal Multi-Dimensional Auctions: Conjectures and Simulations," Papers 2207.01664, arXiv.org.
    5. Sameer Mehta & Milind Dawande & Ganesh Janakiraman & Vijay Mookerjee, 2022. "An Approximation Scheme for Data Monetization," Production and Operations Management, Production and Operations Management Society, vol. 31(6), pages 2412-2428, June.
    6. Wei Chen & Milind Dawande & Ganesh Janakiraman, 2018. "Optimal Procurement Auctions Under Multistage Supplier Qualification," Manufacturing & Service Operations Management, INFORMS, vol. 20(3), pages 566-582, July.
    7. Saeed Alaei & Hu Fu & Nima Haghpanah & Jason Hartline & Azarakhsh Malekian, 2019. "Efficient Computation of Optimal Auctions via Reduced Forms," Mathematics of Operations Research, INFORMS, vol. 44(3), pages 1058-1086, August.
    8. Kenneth Judd & Garrett van Ryzin, 2010. "Preface to the Special Issue on Computational Economics," Operations Research, INFORMS, vol. 58(4-part-2), pages 1035-1036, August.

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