Nothing Special   »   [go: up one dir, main page]
IDEAS home Printed from https://ideas.repec.org/a/wly/emetrp/v85y2017i6p1847-1871.html
   My bibliography  Save this article

Competitive Division of a Mixed Manna

Author

Listed:
  • Anna Bogomolnaia
  • Hervé Moulin
  • Fedor Sandomirskiy
  • Elena Yanovskaya
Abstract
A mixed manna contains goods (that everyone likes) and 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 homogeneous of degree 1 and concave (and monotone), the competitive division maximizes the Nash product of utilities (Gale–Eisenberg): hence it is welfarist (determined by the set of feasible utility profiles), unique, continuous, and easy to compute. We show that the competitive division of a mixed manna is still welfarist. If the zero utility profile is Pareto dominated, the competitive profile is strictly positive and still uniquely maximizes the product of utilities. If the zero profile is unfeasible (for instance, if all items are bads), the competitive profiles are strictly negative and are the critical points of the product of disutilities on the efficiency frontier. The latter allows for multiple competitive utility profiles, from which no single‐valued selection can be continuous or resource monotonic. Thus the implementation of competitive fairness under linear preferences in interactive platforms like SPLIDDIT will be more difficult when the manna contains bads that overwhelm the goods.

Suggested Citation

  • 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.
    • Handle: RePEc:wly:emetrp:v:85:y:2017:i:6:p:1847-1871
    DOI: 10.3982/ECTA14564
    as

    Download full text from publisher

    File URL: https://doi.org/10.3982/ECTA14564
    Download Restriction: no
    File URL: https://libkey.io/10.3982/ECTA14564?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Eric Budish & Estelle Cantillon, 2012. "The Multi-unit Assignment Problem: Theory and Evidence from Course Allocation at Harvard," American Economic Review, American Economic Association, vol. 102(5), pages 2237-2271, August.
    2. Yinghua He & Antonio Miralles & Marek Pycia & Jianye Yan, 2018. "A Pseudo-Market Approach to Allocation with Priorities," American Economic Journal: Microeconomics, American Economic Association, vol. 10(3), pages 272-314, August.
    3. van den Nouweland, A. & Peleg, B. & Tijs, S., 1996. "Axiomatic characterizations of the Walras correspondence for generalized economies," Journal of Mathematical Economics, Elsevier, vol. 25(3), pages 355-372.
    4. Segal-Halevi, Erel & Sziklai, Balázs R., 2018. "Resource-monotonicity and population-monotonicity in connected cake-cutting," Mathematical Social Sciences, Elsevier, vol. 95(C), pages 19-30.
    5. Hurwicz, Leonid, 1979. "On allocations attainable through Nash equilibria," Journal of Economic Theory, Elsevier, vol. 21(1), pages 140-165, August.
    6. Eric Maskin, 1999. "Nash Equilibrium and Welfare Optimality," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 66(1), pages 23-38.
    7. 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.
    8. Jacob Gyntelberg & Frank Hansen, 2004. "Expected Utility Theory with “Small Worlds”," FRU Working Papers 2004/04, University of Copenhagen. Department of Economics. Finance Research Unit.
    9. Shafer, Wayne & Sonnenschein, Hugo, 1975. "Equilibrium in abstract economies without ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 2(3), pages 345-348, December.
    10. Azrieli, Yaron & Shmaya, Eran, 2014. "Rental harmony with roommates," Journal of Economic Theory, Elsevier, vol. 153(C), pages 128-137.
    11. Marco Mariotti & Antonio Villar, 2005. "The Nash rationing problem," International Journal of Game Theory, Springer;Game Theory Society, vol. 33(3), pages 367-377, September.
    12. Pappa, Evi, 2004. "Do the ECB and the fed really need to cooperate? Optimal monetary policy in a two-country world," Journal of Monetary Economics, Elsevier, vol. 51(4), pages 753-779, May.
    13. Eric Budish, 2011. "The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes," Journal of Political Economy, University of Chicago Press, vol. 119(6), pages 1061-1103.
    14. Shafer, Wayne & Sonnenschein, Hugo, 1993. "Market demand and excess demand functions," Handbook of Mathematical Economics, in: K. J. Arrow & M.D. Intriligator (ed.), Handbook of Mathematical Economics, edition 4, volume 2, chapter 14, pages 671-693, Elsevier.
    15. Nakamura, Ken-ichi & Nakamura, Kenichi, 1993. "日本的雇用慣行と特殊人的資本再考 : 新たな理論へ向けての覚書 = The Japanese Employment Relation and Specific Human Capital Reconsidered : A Note for New Approach," 商学討究 (Shogaku Tokyu), Otaru University of Commerce, vol. 43(3/4), pages 369-385.
    16. Sven de Vries & Rakesh V. Vohra, 2003. "Combinatorial Auctions: A Survey," INFORMS Journal on Computing, INFORMS, vol. 15(3), pages 284-309, August.
    17. Varian, Hal R., 1974. "Equity, envy, and efficiency," Journal of Economic Theory, Elsevier, vol. 9(1), pages 63-91, September.
    18. Ioannis Caragiannis & David Kurokawa & Herve Moulin & Ariel D. Procaccia & Nisarg Shah & Junxing Wang, 2016. "The Unreasonable Fairness of Maximum Nash Welfare," Working Papers 2016_08, Business School - Economics, University of Glasgow.
    19. John Winsor Pratt & Richard Jay Zeckhauser, 1990. "The Fair and Efficient Division of the Winsor Family Silver," Management Science, INFORMS, vol. 36(11), pages 1293-1301, November.
    20. Moulin, Herve & Thomson, William, 1988. "Can everyone benefit from growth? : Two difficulties," Journal of Mathematical Economics, Elsevier, vol. 17(4), pages 339-345, September.
    21. GEVERS, Louis, 1986. "Walrasian social choice: some simple axiomatic approaches," LIDAM Reprints CORE 708, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    22. Herrero, Maria Jose, 1989. "The nash program: Non-convex bargaining problems," Journal of Economic Theory, Elsevier, vol. 49(2), pages 266-277, December.
    23. Balazs Sziklai & Erel Segal-Halevi, 2015. "Resource-monotonicity and Population-monotonicity in Cake-cutting," CERS-IE WORKING PAPERS 1552, Institute of Economics, Centre for Economic and Regional Studies.
    24. Nagahisa, Ryo-ichi, 1991. "A local independence condition for characterization of Walrasian allocations rule," Journal of Economic Theory, Elsevier, vol. 54(1), pages 106-123, June.
    25. 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.
    26. Chipman, John S., 1974. "Homothetic preferences and aggregation," Journal of Economic Theory, Elsevier, vol. 8(1), pages 26-38, May.
    27. Elisha A. Pazner & David Schmeidler, 1978. "Egalitarian Equivalent Allocations: A New Concept of Economic Equity," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 92(4), pages 671-687.
    28. N/A, 1977. "CMEA—A Quinquennial Balance-Sheet," Foreign Trade Review, , vol. 12(2), pages 296-317, July.
    29. Serrano, Roberto & Shimomura, Ken-Ichi, 1998. "Beyond Nash Bargaining Theory: The Nash Set," Journal of Economic Theory, Elsevier, vol. 83(2), pages 286-307, December.
    30. Jain, Kamal & Vazirani, Vijay V., 2010. "Eisenberg-Gale markets: Algorithms and game-theoretic properties," Games and Economic Behavior, Elsevier, vol. 70(1), pages 84-106, September.
    31. Hylland, Aanund & Zeckhauser, Richard, 1979. "The Efficient Allocation of Individuals to Positions," Journal of Political Economy, University of Chicago Press, vol. 87(2), pages 293-314, April.
    32. Peter Cramton & Yoav Shoham & Richard Steinberg (ed.), 2006. "Combinatorial Auctions," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262033429, April.
    33. Rosalinda T. Cuizon, 1973. "Bargaining Behavior Under Price Leadership," Philippine Review of Economics, University of the Philippines School of Economics and Philippine Economic Society, vol. 10(1), pages 99-125, June.
    34. Thomson, William, 1988. "A study of choice correspondences in economies with a variable number of agents," Journal of Economic Theory, Elsevier, vol. 46(2), pages 237-254, December.
    35. Fliaster, Alexander & Golly, Tanja, 2014. "Innovation in small and medium-sized companies: Knowledge integration mechanisms and the role of top managers’ networks," management revue. Socio-economic Studies, Rainer Hampp Verlag, vol. 25(2), pages 125-147.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Mariotti, Marco & Wen, Quan, 2021. "A noncooperative foundation of the competitive divisions for bads," Journal of Economic Theory, Elsevier, vol. 194(C).
    2. Erel Segal-Halevi & Warut Suksompong, 2023. "Cutting a Cake Fairly for Groups Revisited," Papers 2301.09061, arXiv.org.
    3. Anna Bogomolnaia & Hervé Moulin & Fedor Sandomirskiy & Elena Yanovskaia, 2019. "Dividing bads under additive utilities," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 52(3), pages 395-417, March.
    4. Kondratev, Aleksei Y. & Nesterov, Alexander S., 2022. "Minimal envy and popular matchings," European Journal of Operational Research, Elsevier, vol. 296(3), pages 776-787.
    5. Anna Bogomolnaia & Hervé Moulin & Fedor Sandomirskiy, 2022. "On the Fair Division of a Random Object," Management Science, INFORMS, vol. 68(2), pages 1174-1194, February.
    6. Miralles, Antonio & Pycia, Marek, 2021. "Foundations of pseudomarkets: Walrasian equilibria for discrete resources," Journal of Economic Theory, Elsevier, vol. 196(C).
    7. Dall’Aglio, Marco, 2023. "Fair division of goods in the shadow of market values," European Journal of Operational Research, Elsevier, vol. 307(2), pages 785-801.
    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. Sandomirskiy, Fedor & Ushchev, Philip, 2024. "The geometry of consumer preference aggregation," CEPR Discussion Papers 19100, C.E.P.R. Discussion Papers.
    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.
    18. Sandomirskiy, Fedor & Ushchev, Philip, 2024. "The geometry of consumer preference aggregation," CEPR Discussion Papers 19100, C.E.P.R. Discussion Papers.
    19. Soroush Ebadian & Dominik Peters & Nisarg Shah, 2022. "How to Fairly Allocate Easy and Difficult Chores," Post-Print hal-03834514, HAL.
    20. Hao Guo & Weidong Li & Bin Deng, 2023. "A Survey on Fair Allocation of Chores," Mathematics, MDPI, vol. 11(16), pages 1-28, August.
    21. 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.
    22. Simina Br^anzei & Fedor Sandomirskiy, 2019. "Algorithms for Competitive Division of Chores," Papers 1907.01766, arXiv.org, revised Jul 2023.
    23. 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.
    24. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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.
    2. Anna Bogomolnaia & Hervé Moulin & Fedor Sandomirskiy & Elena Yanovskaia, 2019. "Dividing bads under additive utilities," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 52(3), pages 395-417, March.
    3. Ortega, Josué, 2020. "Multi-unit assignment under dichotomous preferences," Mathematical Social Sciences, Elsevier, vol. 103(C), pages 15-24.
    4. Thomson, William, 2011. "Chapter Twenty-One - Fair Allocation Rules," Handbook of Social Choice and Welfare, in: K. J. Arrow & A. K. Sen & K. Suzumura (ed.), Handbook of Social Choice and Welfare, edition 1, volume 2, chapter 21, pages 393-506, Elsevier.
    5. Rebelo, S., 1997. "On the Determinant of Economic Growth," RCER Working Papers 443, University of Rochester - Center for Economic Research (RCER).
    6. Sakai, Toyotaka, 2007. "Fairness and implementability in allocation of indivisible objects with monetary compensations," Journal of Mathematical Economics, Elsevier, vol. 43(5), pages 549-563, June.
    7. Fedor Sandomirskiy & Erel Segal-Halevi, 2019. "Efficient Fair Division with Minimal Sharing," Papers 1908.01669, arXiv.org, revised Apr 2022.
    8. Fleurbaey, Marc & Maniquet, Francois, 1996. "Fair allocation with unequal production skills: The No Envy approach to compensation," Mathematical Social Sciences, Elsevier, vol. 32(1), pages 71-93, August.
    9. Christopher P. Chambers & Takashi Hayashi, 2017. "Resource allocation with partial responsibilities for initial endowments," International Journal of Economic Theory, The International Society for Economic Theory, vol. 13(4), pages 355-368, December.
    10. Scott Duke Kominers & Alexander Teytelboym & Vincent P Crawford, 2017. "An invitation to market design," Oxford Review of Economic Policy, Oxford University Press and Oxford Review of Economic Policy Limited, vol. 33(4), pages 541-571.
    11. Miralles, Antonio & Pycia, Marek, 2021. "Foundations of pseudomarkets: Walrasian equilibria for discrete resources," Journal of Economic Theory, Elsevier, vol. 196(C).
    12. Biung†Ghi Ju & Juan D. Moreno†Ternero, 2017. "Fair Allocation Of Disputed Properties," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 58(4), pages 1279-1301, November.
    13. Kojima, Fuhito, 2013. "Efficient resource allocation under multi-unit demand," Games and Economic Behavior, Elsevier, vol. 82(C), pages 1-14.
    14. Serrano, Roberto & Volij, Oscar, 1998. "Axiomatizations of neoclassical concepts for economies," Journal of Mathematical Economics, Elsevier, vol. 30(1), pages 87-108, August.
    15. Maniquet, Francois, 1998. "An equal right solution to the compensation-responsibility dilemma," Mathematical Social Sciences, Elsevier, vol. 35(2), pages 185-202, March.
    16. He, Yinghua & Li, Sanxi & Yan, Jianye, 2015. "Evaluating assignment without transfers: A market perspective," Economics Letters, Elsevier, vol. 133(C), pages 40-44.
    17. 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.
    18. Fleurbaey, Marc, 2007. "Two criteria for social decisions," Journal of Economic Theory, Elsevier, vol. 134(1), pages 421-447, May.
    19. Jin Li & Jingyi Xue, 2013. "Egalitarian division under Leontief Preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 54(3), pages 597-622, November.
    20. Hadi Hosseini & Zhiyi Huang & Ayumi Igarashi & Nisarg Shah, 2022. "Class Fairness in Online Matching," Papers 2203.03751, arXiv.org.

    More about this item

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:emetrp:v:85:y:2017:i:6:p:1847-1871. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://edirc.repec.org/data/essssea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.