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A general equivalence theorem for allocation of indivisible objects

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  • Carroll, Gabriel

Abstract

We consider situations in which n indivisible objects are to be allocated to n agents. A number of recent papers studying such allocation problems have shown various interesting equivalences between randomized mechanisms based on trading and randomized mechanisms based on serial dictatorship. We prove a very general equivalence theorem from which many previous equivalence results immediately follow, and we give several new applications. Our general result sheds some light on why these equivalences hold by presenting the existing serial-dictatorship-based mechanisms as randomizations of a general mechanism which we call serial dictatorship in groups. Our proof technique also streamlines the bijective methods used in previous proofs, showing that it is enough to assemble a bijection from smaller pieces, without needing to construct the pieces explicitly.

Suggested Citation

  • Carroll, Gabriel, 2014. "A general equivalence theorem for allocation of indivisible objects," Journal of Mathematical Economics, Elsevier, vol. 51(C), pages 163-177.
    • Handle: RePEc:eee:mateco:v:51:y:2014:i:c:p:163-177
    DOI: 10.1016/j.jmateco.2013.12.006
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    2. Ivan Balbuzanov & Maciej H. Kotowski, 2019. "Endowments, Exclusion, and Exchange," Econometrica, Econometric Society, vol. 87(5), pages 1663-1692, September.
    3. Sophie Bade, 2020. "Random Serial Dictatorship: The One and Only," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 353-368, February.
    4. Tamura, Yuki & Hosseini, Hadi, 2022. "The crawler: Three equivalence results for object (re)allocation problems when preferences are single-peaked," Journal of Economic Theory, Elsevier, vol. 203(C).
    5. Nick Arnosti, 2023. "Lottery Design for School Choice," Management Science, INFORMS, vol. 69(1), pages 244-259, January.
    6. Che, Yeon-Koo & Tercieux, Olivier, 2018. "Payoff equivalence of efficient mechanisms in large matching markets," Theoretical Economics, Econometric Society, vol. 13(1), January.
    7. Marek Pycia & Peter Troyan, 2023. "A Theory of Simplicity in Games and Mechanism Design," Econometrica, Econometric Society, vol. 91(4), pages 1495-1526, July.
    8. Marek Pycia & M. Utku Ünver, 2022. "Outside options in neutral allocation of discrete resources," Review of Economic Design, Springer;Society for Economic Design, vol. 26(4), pages 581-604, December.
    9. Ashlagi, Itai & Nikzad, Afshin, 2020. "What matters in school choice tie-breaking? How competition guides design," Journal of Economic Theory, Elsevier, vol. 190(C).
    10. Altuntaş, Açelya & Phan, William, 2022. "Trading probabilities along cycles," Journal of Mathematical Economics, Elsevier, vol. 100(C).
    11. Afacan, Mustafa Oğuz & Hu, Gaoji & Li, Jiangtao, 2024. "Housing markets since Shapley and Scarf," Journal of Mathematical Economics, Elsevier, vol. 111(C).
    12. Ekici, Özgün, 2020. "Random mechanisms for house allocation with existing tenants," Journal of Mathematical Economics, Elsevier, vol. 89(C), pages 53-65.
    13. Ashlagi, Itai & Nikzad, Afshin & Romm, Assaf, 2019. "Assigning more students to their top choices: A comparison of tie-breaking rules," Games and Economic Behavior, Elsevier, vol. 115(C), pages 167-187.
    14. Morrill, Thayer & Roth, Alvin E., 2024. "Top trading cycles," Journal of Mathematical Economics, Elsevier, vol. 112(C).
    15. Imamura, Kenzo & Kawase, Yasushi, 2024. "Efficient matching under general constraints," Games and Economic Behavior, Elsevier, vol. 145(C), pages 197-207.
    16. Nesterov, Alexander S., 2017. "Fairness and efficiency in strategy-proof object allocation mechanisms," Journal of Economic Theory, Elsevier, vol. 170(C), pages 145-168.
    17. Sophie Bade, 2016. "Pareto-optimal matching allocation mechanisms for boundedly rational agents," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 47(3), pages 501-510, October.
    18. Tayfun Sönmez & M. Utku Ünver, 2024. "Matching under Non-transferable Utility: Theory," Boston College Working Papers in Economics 1068, Boston College Department of Economics.
    19. Xinghua Long & Rodrigo A. Velez, 2021. "Balanced House Allocation," Papers 2109.01992, arXiv.org.

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