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Dividing Goods and Bads Under Additive Utilities

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

When utilities are additive, we uncovered in our previous paper (Bogomolnaia et al. "Dividing Goods or Bads under Additive Utilities") many similarities but also surprising differences in the behavior of the familiar Competitive rule (with equal incomes), when we divide (private) goods or bads. The rule picks in both cases the critical points of the product of utilities (or disutilities) on the efficiency frontier, but there is only one such point if we share goods, while there can be exponentially many in the case of bads. We extend this analysis to the fair division of mixed items: each item can be viewed by some participants as a good and by others as a bad, with corresponding positive or negative marginal utilities. We find that the division of mixed items boils down, normatively as well as computationally, to a variant of an all goods problem, or of an all bads problem: in particular the task of dividing the non disposable items must be either good news for everyone, or bad news for everyone. If at least one feasible utility profile is positive, the Competitive rule picks the unique maximum of the product of (positive) utilities. If no feasible utility profile is positive, this rule picks all critical points of the product of disutilities on the efficient frontier

Suggested Citation

  • Anna Bogomolnaia & Herve Moulin & Fedor Sandomirskiy & Elena Yanovskaya, 2016. "Dividing Goods and Bads Under Additive Utilities," HSE Working papers WP BRP 153/EC/2016, National Research University Higher School of Economics.
    • Handle: RePEc:hig:wpaper:153/ec/2016
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    File URL: https://wp.hse.ru/data/2016/10/27/1110471280/153EC2016.pdf
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    References listed on IDEAS

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    1. Anna Bogomolnaia & Herve Moulin & Fedor Sandomirskiy & Elena Yanovskaya, 2016. "Dividing Goods or Bads Under Additive Utilities," HSE Working papers WP BRP 147/EC/2016, National Research University Higher School of Economics.
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    Cited by:

    1. 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.
    2. Vittorio Bilò & Ioannis Caragiannis & Michele Flammini & Ayumi Igarashi & Gianpiero Monaco & Dominik Peters & Cosimo Vinci & William Zwicker, 2021. "Almost Envy-Free Allocations with Connected Bundles," Post-Print hal-03834506, HAL.
    3. Pavel Konyukhovskiy & Victoria Holodkova & Aleksander Titov, 2019. "Modeling Competition between Countries in the Development of Arctic Resources," Resources, MDPI, vol. 8(1), pages 1-17, March.
    4. Anna Bogomolnaia & Hervé Moulin & Fedor Sandomirskiy & Elena Yanovskaya, 2017. "Competitive Division of a Mixed Manna," Econometrica, Econometric Society, vol. 85(6), pages 1847-1871, November.
    5. Yves Sprumont, 2020. "Nash welfarism and the distributive implications of informational constraints," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 8(1), pages 49-64, April.
    6. Misha Gavrilovich & Victoria Kreps, 2016. "Games with Incomplete Information on One Side as Games with Incomplete Information on Both Sides and Asymmetric Computational Resources," HSE Working papers WP BRP 154/EC/2016, National Research University Higher School of Economics.
    7. 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.
    8. Fedor Sandomirskiy & Erel Segal-Halevi, 2019. "Efficient Fair Division with Minimal Sharing," Papers 1908.01669, arXiv.org, revised Apr 2022.

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    1. 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.
    2. Pavel Konyukhovskiy & Victoria Holodkova & Aleksander Titov, 2019. "Modeling Competition between Countries in the Development of Arctic Resources," Resources, MDPI, vol. 8(1), pages 1-17, March.
    3. Misha Gavrilovich & Victoria Kreps, 2016. "Games with Incomplete Information on One Side as Games with Incomplete Information on Both Sides and Asymmetric Computational Resources," HSE Working papers WP BRP 154/EC/2016, National Research University Higher School of Economics.
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    5. 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.
    6. Fedor Sandomirskiy & Erel Segal-Halevi, 2019. "Efficient Fair Division with Minimal Sharing," Papers 1908.01669, arXiv.org, revised Apr 2022.
    7. Vittorio Bilò & Ioannis Caragiannis & Michele Flammini & Ayumi Igarashi & Gianpiero Monaco & Dominik Peters & Cosimo Vinci & William Zwicker, 2021. "Almost Envy-Free Allocations with Connected Bundles," Post-Print hal-03834506, HAL.
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    More about this item

    Keywords

    fair division of goods and bads; competitive equilibrium with equal incomes; Nash product; envy-freeness;
    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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