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On linear transformations of intersections

Author

Listed:
  • Alexey Kushnir
  • Shuo Liu

Abstract

For any linear transformation and two convex closed sets, we provide necessary and sufficient conditions for when the transformation of the intersection of the sets coincides with the intersection of their images. We also identify analogous conditions for non-convex sets, general transformations, and multiple sets. We demonstrate the usefulness of our results via an application to the economics literature of mechanism design.

Suggested Citation

  • Alexey Kushnir & Shuo Liu, 2017. "On linear transformations of intersections," ECON - Working Papers 255, Department of Economics - University of Zurich.
  • Handle: RePEc:zur:econwp:255
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    File URL: https://www.zora.uzh.ch/id/eprint/137636/1/econwp255.pdf
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    References listed on IDEAS

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    1. Alvin E. Roth, 2009. "What Have We Learned from Market Design?," Innovation Policy and the Economy, University of Chicago Press, vol. 9(1), pages 79-112.
    2. Myerson, Roger B, 1979. "Incentive Compatibility and the Bargaining Problem," Econometrica, Econometric Society, vol. 47(1), pages 61-73, January.
    3. Alex Gershkov & Jacob K. Goeree & Alexey Kushnir & Benny Moldovanu & Xianwen Shi, 2013. "On the Equivalence of Bayesian and Dominant Strategy Implementation," Econometrica, Econometric Society, vol. 81(1), pages 197-220, January.
    4. Laffont, Jean-Jacques & Maskin, Eric, 1980. "A Differential Approach to Dominant Strategy Mechanisms," Econometrica, Econometric Society, vol. 48(6), pages 1507-1520, September.
    5. 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.
    6. Milgrom,Paul, 2004. "Putting Auction Theory to Work," Cambridge Books, Cambridge University Press, number 9780521536721, October.
    7. 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.
    8. 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.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Alexey Kushnir & Shuo Liu, 2019. "On the equivalence of Bayesian and dominant strategy implementation for environments with nonlinear utilities," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 67(3), pages 617-644, April.

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    More about this item

    Keywords

    Linear transformation; convex closed set; intersection; directional convexity; mechanism design; dominant-strategy implementation; Bayesian implementation;
    All these keywords.

    JEL classification:

    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

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