IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2402.11437.html
   My bibliography  Save this paper

The Assignment Game: New Mechanisms for Equitable Core Imputations

Author

Listed:
  • Vijay V. Vazirani

Abstract

The set of core imputations of the assignment game forms a (non-finite) distributive lattice. So far, efficient algorithms were known for computing only its two extreme imputations; however, each of them maximally favors one side and disfavors the other side of the bipartition, leading to inequitable profit sharing. Another issue is that a sub-coalition consisting of one player (or a set of players from the same side of the bipartition) can make zero profit, therefore a core imputation is not obliged to give them any profit. Hence core imputations make no fairness guarantee at the level of individual agents. This raises the question of computing {\em more equitable core imputations}. In this paper, we give combinatorial (i.e., the mechanism does not invoke an LP-solver) polynomial time mechanisms for computing the leximin and leximax core imputations for the assignment game. These imputations achieve ``fairness'' in different ways: whereas leximin tries to make poor agents more rich, leximax tries to make rich agents less rich. In general, the two imputations are different. Our mechanisms were derived by a suitable adaptation of the classical primal-dual paradigm from combinatorial optimization. The ``engine'' driving them involves recent insights, obtained via complementarity, into core imputations \cite{Va.New-characterizations} and the pristine combinatorial structure of matching. We have identified other natural games which could benefit from our approach.

Suggested Citation

  • Vijay V. Vazirani, 2024. "The Assignment Game: New Mechanisms for Equitable Core Imputations," Papers 2402.11437, arXiv.org.
  • Handle: RePEc:arx:papers:2402.11437
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2402.11437
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Tamás Király & Júlia Pap, 2008. "Total Dual Integrality of Rothblum's Description of the Stable-Marriage Polyhedron," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 283-290, May.
    2. Dutta, Bhaskar & Ray, Debraj, 1989. "A Concept of Egalitarianism under Participation Constraints," Econometrica, Econometric Society, vol. 57(3), pages 615-635, May.
    3. Hervé Moulin, 2019. "Fair Division in the Internet Age," Annual Review of Economics, Annual Reviews, vol. 11(1), pages 407-441, August.
    4. Aumann, Robert J. & Maschler, Michael, 1985. "Game theoretic analysis of a bankruptcy problem from the Talmud," Journal of Economic Theory, Elsevier, vol. 36(2), pages 195-213, August.
    5. Chung-Piaw Teo & Jay Sethuraman, 1998. "The Geometry of Fractional Stable Matchings and Its Applications," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 874-891, November.
    Full references (including those not matched with items on IDEAS)

    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. Peter Knudsen & Lars Østerdal, 2012. "Merging and splitting in cooperative games: some (im)possibility results," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(4), pages 763-774, November.
    2. Hervé Moulin & Jay Sethuraman, 2013. "The Bipartite Rationing Problem," Operations Research, INFORMS, vol. 61(5), pages 1087-1100, October.
    3. Debraj Ray & Rajiv Vohra, 2024. "Nash Bargaining with Coalitional Threats," Working Papers 2024-001, Brown University, Department of Economics.
    4. Michel Le Breton & Juan Moreno-Ternero & Alexei Savvateev & Shlomo Weber, 2013. "Stability and fairness in models with a multiple membership," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(3), pages 673-694, August.
    5. Slikker, Marco & Norde, Henk, 2011. "The monoclus of a coalitional game," Games and Economic Behavior, Elsevier, vol. 71(2), pages 420-435, March.
    6. Helga Habis & P. Herings, 2013. "Stochastic bankruptcy games," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(4), pages 973-988, November.
    7. Koster, Maurice & Boonen, Tim J., 2019. "Constrained stochastic cost allocation," Mathematical Social Sciences, Elsevier, vol. 101(C), pages 20-30.
    8. Endre Bjørndal & Kurt Jörnsten, 2010. "Flow sharing and bankruptcy games," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 11-28, March.
    9. Fatemeh Babaei & Hamidreza Navidi & Stefano Moretti, 2022. "A bankruptcy approach to solve the fixed cost allocation problem in transport systems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(2), pages 332-358, July.
    10. Habis, Helga & Herings, P. Jean-Jacques, 2011. "Transferable utility games with uncertainty," Journal of Economic Theory, Elsevier, vol. 146(5), pages 2126-2139, September.
    11. Keyzer, Michiel & van Wesenbeeck, Cornelia, 2011. "Optimal coalition formation and surplus distribution: Two sides of one coin," European Journal of Operational Research, Elsevier, vol. 215(3), pages 604-615, December.
    12. Thomson, William, 2003. "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey," Mathematical Social Sciences, Elsevier, vol. 45(3), pages 249-297, July.
    13. Flip Klijn & Stef Tijs & Marco Slikker, 2001. "A Dual Egalitarian Solution," Economics Bulletin, AccessEcon, vol. 3(10), pages 1-8.
    14. Duro, Juan Antonio & Giménez-Gómez, José-Manuel & Vilella, Cori, 2020. "The allocation of CO2 emissions as a claims problem," Energy Economics, Elsevier, vol. 86(C).
    15. Giménez-Gómez, José Manuel, 2011. "A way to play bankruptcy problems," Working Papers 2072/169781, Universitat Rovira i Virgili, Department of Economics.
    16. Vijay V. Vazirani, 2023. "LP-Duality Theory and the Cores of Games," Papers 2302.07627, arXiv.org, revised Mar 2023.
    17. José Alcalde & María Marco & José Silva, 2005. "Bankruptcy games and the Ibn Ezra’s proposal," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(1), pages 103-114, July.
    18. Giménez-Gómez, José-Manuel & Osório, Antonio, 2015. "Why and how to differentiate in claims problems? An axiomatic approach," European Journal of Operational Research, Elsevier, vol. 241(3), pages 842-850.
    19. Giménez-Gómez, José-Manuel & Peris, Josep E., 2014. "A proportional approach to claims problems with a guaranteed minimum," European Journal of Operational Research, Elsevier, vol. 232(1), pages 109-116.
    20. Emin Karagözoğlu, 2014. "A noncooperative approach to bankruptcy problems with an endogenous estate," Annals of Operations Research, Springer, vol. 217(1), pages 299-318, June.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:arx:papers:2402.11437. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.