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Guaranteed size ratio of ordinally efficient and envy-free mechanisms in the assignment problem

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  • Huang, Chao
  • Tian, Guoqiang

Abstract

In the assignment problem where agents can stay unassigned, the size of the assignment is an important consideration for designers. Bogomolnaia and Moulin (2015) show that there is a tension between size and fairness: the guaranteed size ratio of any envy-free mechanism is at most rm, which converges decreasingly to 1−1e≈63.2% as the maximum size increases. They then ask whether rm is also the guaranteed size ratio for any ordinally efficient and envy-free mechanism. We study this issue and show that the lower bound of the guaranteed size ratio of ordinally efficient and envy-free mechanisms converges to 12 as the maximum size increases, which means that almost half of the maximum size is wasted at the lower bound. Moreover, the exact lower bound is m+12m when the maximum size m is odd.

Suggested Citation

  • Huang, Chao & Tian, Guoqiang, 2017. "Guaranteed size ratio of ordinally efficient and envy-free mechanisms in the assignment problem," Games and Economic Behavior, Elsevier, vol. 105(C), pages 1-8.
    • Handle: RePEc:eee:gamebe:v:105:y:2017:i:c:p:1-8
    DOI: 10.1016/j.geb.2017.07.001
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    References listed on IDEAS

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    1. Bogomolnaia, Anna, 2015. "Random assignment: Redefining the serial rule," Journal of Economic Theory, Elsevier, vol. 158(PA), pages 308-318.
    2. Bogomolnaia, Anna & Heo, Eun Jeong, 2012. "Probabilistic assignment of objects: Characterizing the serial rule," Journal of Economic Theory, Elsevier, vol. 147(5), pages 2072-2082.
    3. Bogomolnaia, Anna & Moulin, Herve, 2015. "Size versus fairness in the assignment problem," Games and Economic Behavior, Elsevier, vol. 90(C), pages 119-127.
    4. Hervé Crès & Hervé Moulin, 2001. "Scheduling with Opting Out: Improving upon Random Priority," Operations Research, INFORMS, vol. 49(4), pages 565-577, August.
    5. 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.
    6. Hashimoto, Tadashi & Hirata, Daisuke & Kesten, Onur & Kurino, Morimitsu & Unver, Utku, 2014. "Two axiomatic approaches to the probabilistic serial mechanism," Theoretical Economics, Econometric Society, vol. 9(1), January.
    7. Yeon-Koo Che & Fuhito Kojima, 2010. "Asymptotic Equivalence of Probabilistic Serial and Random Priority Mechanisms," Econometrica, Econometric Society, vol. 78(5), pages 1625-1672, September.
    8. Erdil, Aytek, 2014. "Strategy-proof stochastic assignment," Journal of Economic Theory, Elsevier, vol. 151(C), pages 146-162.
    9. 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.
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    Cited by:

    1. Zhang, Jun, 2023. "Strategy-proof allocation with outside option," Games and Economic Behavior, Elsevier, vol. 137(C), pages 50-67.

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    More about this item

    Keywords

    Size of assignment; Ordinal efficiency; Envy-freeness;
    All these keywords.

    JEL classification:

    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
    • D70 - Microeconomics - - Analysis of Collective Decision-Making - - - General

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