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Reduced-Form Allocations for Multiple Indivisible Objects under Constraints

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  • Xu Lang
  • Zaifu Yang

Abstract

We examine the implementation of reduced-form allocation rules that assign multiple indivisible objects to many agents, with incomplete information and distributional constraints across objects and agents. To obtain implementability results, we adopt a lift-and-project approach, which reduces the problem to a problem of enumerating finite generators of a projection cone. We study geometric and combinatorial properties of the projection cone and provide a total unimodularity condition that leads to several characterization results including those on hierarchies and bihierarchies. Our results have applications in matching markets with constraints where agents may have ordinal or cardinal preferences.

Suggested Citation

  • Xu Lang & Zaifu Yang, 2021. "Reduced-Form Allocations for Multiple Indivisible Objects under Constraints," Discussion Papers 21/04, Department of Economics, University of York.
    • Handle: RePEc:yor:yorken:21/04
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    References listed on IDEAS

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    1. Börgers, Tilman & Postl, Peter, 2009. "Efficient compromising," Journal of Economic Theory, Elsevier, vol. 144(5), pages 2057-2076, September.
    2. Crawford, Vincent P & Knoer, Elsie Marie, 1981. "Job Matching with Heterogeneous Firms and Workers," Econometrica, Econometric Society, vol. 49(2), pages 437-450, March.
    3. Border, Kim C, 1991. "Implementation of Reduced Form Auctions: A Geometric Approach," Econometrica, Econometric Society, vol. 59(4), pages 1175-1187, July.
    4. Tayfun Sönmez & M. Utku Ünver, 2010. "Course Bidding At Business Schools," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 51(1), pages 99-123, February.
    5. Ehlers, Lars & Hafalir, Isa E. & Yenmez, M. Bumin & Yildirim, Muhammed A., 2014. "School choice with controlled choice constraints: Hard bounds versus soft bounds," Journal of Economic Theory, Elsevier, vol. 153(C), pages 648-683.
    6. Myerson, Roger B. & Satterthwaite, Mark A., 1983. "Efficient mechanisms for bilateral trading," Journal of Economic Theory, Elsevier, vol. 29(2), pages 265-281, April.
    7. Gabrielle Demange, 2004. "On Group Stability in Hierarchies and Networks," Journal of Political Economy, University of Chicago Press, vol. 112(4), pages 754-778, August.
    8. 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.
    9. 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.
    10. 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.
    11. Roger B. Myerson, 1981. "Optimal Auction Design," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 58-73, February.
    12. Eric Budish & Yeon-Koo Che & Fuhito Kojima & Paul Milgrom, 2013. "Designing Random Allocation Mechanisms: Theory and Applications," American Economic Review, American Economic Association, vol. 103(2), pages 585-623, April.
    13. Yuichiro Kamada & Fuhito Kojima, 2015. "Efficient Matching under Distributional Constraints: Theory and Applications," American Economic Review, American Economic Association, vol. 105(1), pages 67-99, January.
    14. 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.
    15. Bogomolnaia, Anna & Moulin, Herve, 2001. "A New Solution to the Random Assignment Problem," Journal of Economic Theory, Elsevier, vol. 100(2), pages 295-328, October.
    16. Yeon‐Koo Che & Jinwoo Kim & Konrad Mierendorff, 2013. "Generalized Reduced‐Form Auctions: A Network‐Flow Approach," Econometrica, Econometric Society, vol. 81(6), pages 2487-2520, November.
    17. Maskin, Eric S & Riley, John G, 1984. "Optimal Auctions with Risk Averse Buyers," Econometrica, Econometric Society, vol. 52(6), pages 1473-1518, November.
    18. 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.
    19. Matthews, Steven A, 1984. "On the Implementability of Reduced Form Auctions," Econometrica, Econometric Society, vol. 52(6), pages 1519-1522, November.
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    Citations

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    Cited by:

    1. Debasis Mishra & Xu Lang, 2022. "Symmetric reduced form voting," Discussion Papers 22-03, Indian Statistical Institute, Delhi.
    2. Lang, Xu & Mishra, Debasis, 2024. "Symmetric reduced form voting," Theoretical Economics, Econometric Society, vol. 19(2), May.
    3. Xu Lang, 2022. "Reduced-form budget allocation with multiple public alternatives," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 59(2), pages 335-359, August.
    4. Xu Lang & Debasis Mishra, 2022. "Symmetric reduced form voting," Papers 2207.09253, arXiv.org, revised Apr 2023.
    5. Xu Lang & Zaifu Yang, 2023. "Reduced-Form Allocations for Multiple Indivisible Objects under Constraints," Discussion Papers 23/02, Department of Economics, University of York.

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    More about this item

    Keywords

    Implementation; Reduced-form rules; Indivisible goods; Distributional constraints; Total unimodularity; Incomplete information.;
    All these keywords.

    JEL classification:

    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools

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