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A geometric approach to mechanism design

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  • Jacob K. Goeree
  • Alexey Kushnir

Abstract

We develop a novel geometric approach to mechanism design using an important result in convex analysis: the duality between a closed convex set and its support function. By deriving the support function for the set of feasible interim values we extend the wellknown Maskin-Riley-Matthews-Border conditions for reduced-form auctions to social choice environments. We next refine the support function to include incentive constraints using a geometric characterization of incentive compatibility. Borrowing results from majorization theory that date back to the work of Hardy, Littlewood, and Polya (1929) we elucidate the "ironing" procedure introduced by Myerson (1981) and Mussa and Rosen (1978). The inclusion of Bayesian and dominant strategy incentive constraints result in the same support function, which establishes equivalence between these implementation concepts. Using Hotelling's lemma we next derive the optimal mechanism for any social choice problem and any linear objective, including revenue and surplus maximization. We extend the approach to include general concave objectives by providing a fixed-point condition characterizing the optimal mechanism. We generalize reduced-form implementation to environments with multi-dimensional, correlated types, non-linear utilities, and interdependent values. When value interdependencies are linear we are able to include incentive constraints into the support function and provide a condition when the second-best allocation is ex post incentive compatible.

Suggested Citation

  • Jacob K. Goeree & Alexey Kushnir, 2011. "A geometric approach to mechanism design," ECON - Working Papers 056, Department of Economics - University of Zurich, revised Jun 2013.
  • Handle: RePEc:zur:econwp:056
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    File URL: http://www.econ.uzh.ch/static/wp/econwp056.pdf
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    References listed on IDEAS

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    1. Jacob K. Goeree & Alexey Kushnir, 2011. "On the equivalence of Bayesian and dominant strategy implementation in a general class of social choice problems," ECON - Working Papers 021, Department of Economics - University of Zurich.
    2. Martin F. Hellwig, 2003. "Public-Good Provision with Many Participants," Review of Economic Studies, Oxford University Press, vol. 70(3), pages 589-614.
    3. Simo Puntanen, 2011. "Inequalities: Theory of Majorization and Its Applications, Second Edition by Albert W. Marshall, Ingram Olkin, Barry C. Arnold," International Statistical Review, International Statistical Institute, vol. 79(2), pages 293-293, August.
    4. Alex Gershkov & Benny Moldovanu & Xianwen Shi, 2011. "Bayesian and Dominant Strategy Implementation Revisited," Working Papers tecipa-422, University of Toronto, Department of Economics.
    5. Hernando-Veciana, Ángel & Michelucci, Fabio, 2011. "Second best efficiency and the English auction," Games and Economic Behavior, Elsevier, vol. 73(2), pages 496-506.
    6. Alejandro M. Manelli & Daniel R. Vincent, 2010. "Bayesian and Dominant‐Strategy Implementation in the Independent Private‐Values Model," Econometrica, Econometric Society, vol. 78(6), pages 1905-1938, November.
    7. Roger B. Myerson, 1981. "Optimal Auction Design," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 58-73, February.
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    Cited by:

    1. Alexey Kushnir, 2013. "On the equivalence between Bayesian and dominant strategy implementation: the case of correlated types," ECON - Working Papers 129, Department of Economics - University of Zurich.
    2. Elizabeth Baldwin & Paul Klemperer, 2015. "Understanding Preferences: “Demand Types”, and the Existence of Equilibrium with Indivisibilities," Economics Papers 2015-W10, Economics Group, Nuffield College, University of Oxford.
    3. Kushnir, Alexey, 2015. "On sufficiency of dominant strategy implementation in environments with correlated types," Economics Letters, Elsevier, vol. 133(C), pages 4-6.
    4. Alexey Kushnir & Shuo Liu, 2015. "On the equivalence of bayesian and dominant strategy implementation: the case of non-linear utilities," ECON - Working Papers 212, Department of Economics - University of Zurich.
    5. Alexey Kushnir & Shuo Liu, 2017. "On linear transformations of intersections," ECON - Working Papers 255, Department of Economics - University of Zurich.

    More about this item

    Keywords

    Mechanism design; convex set; support function; duality; majorization; ironing; Hotelling's lemma; reduced-from implementation; BIC-DIC equivalence; concave objectives; interdependent values; second-best mechanisms;

    JEL classification:

    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions

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