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Competitive division of a mixed manna

Author

Listed:
  • Anna Bogomolnaia

    (National Research University Higher School of Economics)

  • Herve Moulin

    (National Research University Higher School of Economics)

  • Fedor Sandomirskiy

    (National Research University Higher School of Economics)

  • Elena Yanovskaya

    (National Research University Higher School of Economics)

Abstract

A mixed manna contains goods (that everyone likes), bads (that everyone dislikes), as well as items that are goods to some agents, but bads or satiated to others. If all items are goods and utility functions are homothetic, concave (and monotone), the Competitive Equilibrium with Equal Incomes maximizes the Nash product of utilities: hence it is welfarist (determined utility-wise by the feasible set of pro les), single-valued and easy to compute. We generalize the Gale-Eisenberg Theorem to a mixed manna. The Competitive division is still welfarist and related to the product of utilities or disutilities. If the zero utility pro le (before any manna) is Pareto dominated, the competitive pro le is unique and still maximizes the product of utilities. If the zero pro le is unfeasible, the competitive pro les are the critical points of the product of disutilities on the eciency frontier, and multiplicity is pervasive. In particular the task of dividing a mixed manna is either good news for everyone, or bad news for everyone. We re ne our results in the practically important case of linear preferences, where the axiomatic comparison between the division of goods and that of bads is especially sharp. When we divide goods and the manna improves, everyone weakly bene ts under the competitive rule; but no reasonable rule to divide bads can be similarly Resource Monotonic. Also, the much larger set of Non Envious and Ecient divisions of bads can be disconnected so that it will admit no continuous selection.

Suggested Citation

  • Anna Bogomolnaia & Herve Moulin & Fedor Sandomirskiy & Elena Yanovskaya, 2017. "Competitive division of a mixed manna," HSE Working papers WP BRP 158/EC/2017, National Research University Higher School of Economics.
    • Handle: RePEc:hig:wpaper:158/ec/2017
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    8. Ortega, Josué, 2020. "Multi-unit assignment under dichotomous preferences," Mathematical Social Sciences, Elsevier, vol. 103(C), pages 15-24.
    9. Bade, Sophie & Segal-Halevi, Erel, 2023. "Fairness for multi-self agents," Games and Economic Behavior, Elsevier, vol. 141(C), pages 321-336.
    10. Anna Bogomolnaia & Hervé Moulin, 2023. "Guarantees in Fair Division: General or Monotone Preferences," Mathematics of Operations Research, INFORMS, vol. 48(1), pages 160-176, February.
    11. Fedor Sandomirskiy & Erel Segal-Halevi, 2019. "Efficient Fair Division with Minimal Sharing," Papers 1908.01669, arXiv.org, revised Apr 2022.
    12. Samuel Bismuth & Ivan Bliznets & Erel Segal-Halevi, 2019. "Fair Division with Bounded Sharing: Binary and Non-Degenerate Valuations," Papers 1912.00459, arXiv.org, revised Jul 2024.
    13. Van Essen, Matt & Wooders, John, 2021. "Allocating positions fairly: Auctions and Shapley value," Journal of Economic Theory, Elsevier, vol. 196(C).
    14. Fedor Sandomirskiy & Philip Ushchev, 2024. "The geometry of consumer preference aggregation," Papers 2405.06108, arXiv.org.
    15. Erel Segal-Halevi & Shmuel Nitzan, 2019. "Fair cake-cutting among families," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 53(4), pages 709-740, December.
    16. Chambers, Christopher P. & Hayashi, Takashi, 2023. "The structure of representative preference," Journal of Mathematical Economics, Elsevier, vol. 108(C).
    17. Anna Bogomolnaia & Herv'e Moulin, 2024. "Guaranteed shares of benefits and costs," Papers 2406.14198, arXiv.org, revised Nov 2024.
    18. Soroush Ebadian & Dominik Peters & Nisarg Shah, 2022. "How to Fairly Allocate Easy and Difficult Chores," Post-Print hal-03834514, HAL.
    19. Hao Guo & Weidong Li & Bin Deng, 2023. "A Survey on Fair Allocation of Chores," Mathematics, MDPI, vol. 11(16), pages 1-28, August.
    20. Erel Segal-Halevi & Balázs R. Sziklai, 2019. "Monotonicity and competitive equilibrium in cake-cutting," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 68(2), pages 363-401, September.
    21. Simina Br^anzei & Fedor Sandomirskiy, 2019. "Algorithms for Competitive Division of Chores," Papers 1907.01766, arXiv.org, revised Jul 2023.
    22. Marco Dall'Aglio & Camilla Di Luca & Lucia Milone, 2017. "Finding the Pareto optimal equitable allocation of homogeneous divisible goods among three players," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 27(3), pages 35-50.
    23. Vittorio Bil`o & Ioannis Caragiannis & Michele Flammini & Ayumi Igarashi & Gianpiero Monaco & Dominik Peters & Cosimo Vinci & William S. Zwicker, 2018. "Almost Envy-Free Allocations with Connected Bundles," Papers 1808.09406, arXiv.org, revised May 2022.

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    More about this item

    Keywords

    fair division; mixed manna; goods; bads; competitive equilibrium with equal incomes; Nash product; envy-freeness; resource monotonicity; independence of lost bids;
    All these keywords.

    JEL classification:

    • D61 - Microeconomics - - Welfare Economics - - - Allocative Efficiency; Cost-Benefit Analysis
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

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