IDEAS home Printed from https://ideas.repec.org/a/spr/jogath/v52y2023i1d10.1007_s00182-022-00814-3.html
   My bibliography  Save this article

(In)efficiency and equitability of equilibrium outcomes in a family of bargaining games

Author

Listed:
  • Emin Karagözoğlu

    (Bilkent University)

  • Kerim Keskin

    (ADA University)

  • Çağrı Sağlam

    (Bilkent University)

Abstract

We construct a parametric family of (modified) divide-the-dollar games: when there is excess demand, some portion of the dollar may disappear and the remaining portion is distributed in a bankruptcy problem. In two extremes, this game family captures the standard divide-the-dollar game of Nash (Econometrica 21:128–140, 1953) (when the whole dollar vanishes) and the game studied in Ashlagi et al. (Math Soc Sci 63:228–233, 2012) (when the whole dollar remains) as special cases. We first show that in all interior members of our game family, all Nash equilibria are inefficient under the proportional rule if there are ‘too many’ players in the game. Moreover, in any interior member of the game family, the inefficiency increases as the number of players increases, and the whole surplus vanishes as the number of players goes to infinity. On the other hand, we show that any bankruptcy rule that satisfies certain normatively appealing axioms induces a unique and efficient Nash equilibrium in which everyone demands and receives an equal share of the dollar. The constrained equal awards rule is one such rule.

Suggested Citation

  • Emin Karagözoğlu & Kerim Keskin & Çağrı Sağlam, 2023. "(In)efficiency and equitability of equilibrium outcomes in a family of bargaining games," International Journal of Game Theory, Springer;Game Theory Society, vol. 52(1), pages 175-193, March.
    • Handle: RePEc:spr:jogath:v:52:y:2023:i:1:d:10.1007_s00182-022-00814-3
    DOI: 10.1007/s00182-022-00814-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00182-022-00814-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00182-022-00814-3?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Tore Ellingsen & Topi Miettinen, 2008. "Commitment and Conflict in Bilateral Bargaining," American Economic Review, American Economic Association, vol. 98(4), pages 1629-1635, September.
    2. Young H. P., 1993. "An Evolutionary Model of Bargaining," Journal of Economic Theory, Elsevier, vol. 59(1), pages 145-168, February.
    3. Moulin, Herve, 2002. "Axiomatic cost and surplus sharing," 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 1, chapter 6, pages 289-357, Elsevier.
    4. Ashlagi, Itai & Karagözoğlu, Emin & Klaus, Bettina, 2012. "A non-cooperative support for equal division in estate division problems," Mathematical Social Sciences, Elsevier, vol. 63(3), pages 228-233.
    5. 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.
    6. Deepal Basak & Joyee Deb, 2020. "Gambling over Public Opinion," American Economic Review, American Economic Association, vol. 110(11), pages 3492-3521, November.
    7. Chun, Youngsub, 1989. "A noncooperative justification for egalitarian surplus sharing," Mathematical Social Sciences, Elsevier, vol. 17(3), pages 245-261, June.
    8. Helga Habis & P. Herings, 2013. "Stochastic bankruptcy games," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(4), pages 973-988, November.
    9. Herrero, Carmen & Villar, Antonio, 2001. "The three musketeers: four classical solutions to bankruptcy problems," Mathematical Social Sciences, Elsevier, vol. 42(3), pages 307-328, November.
    10. Stovall, John E., 2014. "Collective rationality and monotone path division rules," Journal of Economic Theory, Elsevier, vol. 154(C), pages 1-24.
    11. Nejat Anbarci, 2001. "Divide-the-Dollar Game Revisited," Theory and Decision, Springer, vol. 50(4), pages 295-303, June.
    12. Makoto Hagiwara & Shunsuke Hanato, 2021. "A strategic justification of the constrained equal awards rule through a procedurally fair multilateral bargaining game," Theory and Decision, Springer, vol. 90(2), pages 233-243, March.
    13. Shiran Rachmilevitch, 2017. "Punishing greediness in divide-the-dollar games," Theory and Decision, Springer, vol. 82(3), pages 341-351, March.
    14. Van Huyck, John & Battalio, Raymond & Mathur, Sondip & Van Huyck, Patsy & Ortmann, Andreas, 1995. "On the Origin of Convention: Evidence from Symmetric Bargaining Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 24(2), pages 187-212.
    15. Hervé Moulin, 2000. "Priority Rules and Other Asymmetric Rationing Methods," Econometrica, Econometric Society, vol. 68(3), pages 643-684, May.
    16. Esat Cetemen & Emin Karagözoğlu, 2014. "Implementing equal division with an ultimatum threat," Theory and Decision, Springer, vol. 77(2), pages 223-236, August.
    17. Andreozzi, Luciano, 2010. "An evolutionary theory of social justice: Choosing the right game," European Journal of Political Economy, Elsevier, vol. 26(3), pages 320-329, September.
    18. Thomson, William, 2015. "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: An update," Mathematical Social Sciences, Elsevier, vol. 74(C), pages 41-59.
    19. Tsay, Min-Hung & Yeh, Chun-Hsien, 2019. "Relations among the central rules in bankruptcy problems: A strategic perspective," Games and Economic Behavior, Elsevier, vol. 113(C), pages 515-532.
    20. Emin Karagözoğlu & Shiran Rachmilevitch, 2018. "Implementing egalitarianism in a class of Nash demand games," Theory and Decision, Springer, vol. 85(3), pages 495-508, October.
    21. Carlsson, Hans, 1991. "A Bargaining Model Where Parties Make Errors," Econometrica, Econometric Society, vol. 59(5), pages 1487-1496, September.
    22. repec:cup:cbooks:9780521555838 is not listed on IDEAS
    23. Leininger, W. & Linhart, P. B. & Radner, R., 1989. "Equilibria of the sealed-bid mechanism for bargaining with incomplete information," Journal of Economic Theory, Elsevier, vol. 48(1), pages 63-106, June.
    24. Bolton, Gary E., 1997. "The rationality of splitting equally," Journal of Economic Behavior & Organization, Elsevier, vol. 32(3), pages 365-381, March.
    25. Dutta, Rohan, 2012. "Bargaining with revoking costs," Games and Economic Behavior, Elsevier, vol. 74(1), pages 144-153.
    26. Dilip Abreu & David Pearce, 2015. "A Dynamic Reinterpretation of Nash Bargaining With Endogenous Threats," Econometrica, Econometric Society, vol. 83(4), pages 1641-1655, July.
    27. Nejat Anbarci & Kang Rong & Jaideep Roy, 2019. "Random-settlement arbitration and the generalized Nash solution: one-shot and infinite-horizon cases," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 68(1), pages 21-52, July.
    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. Shiran Rachmilevitch, 2020. "Rewarding moderate behavior in a dynamic Nash Demand Game," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(2), pages 639-650, June.
    2. Schouten, Jop, 2022. "Cooperation, allocation and strategy in interactive decision-making," Other publications TiSEM d5d41448-8033-4f6b-8ec0-c, Tilburg University, School of Economics and Management.
    3. 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.
    4. Jingyi Xue, 2018. "Fair division with uncertain needs," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 51(1), pages 105-136, June.
    5. Shiran Rachmilevitch, 2022. "Reasonable Nash demand games," Theory and Decision, Springer, vol. 93(2), pages 319-330, September.
    6. Andrea Gallice, 2019. "Bankruptcy problems with reference-dependent preferences," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(1), pages 311-336, March.
    7. Thomson, William, 2015. "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: An update," Mathematical Social Sciences, Elsevier, vol. 74(C), pages 41-59.
    8. Jaume García-Segarra & Miguel Ginés-Vilar, 2023. "Additive adjudication of conflicting claims," International Journal of Game Theory, Springer;Game Theory Society, vol. 52(1), pages 93-116, March.
    9. Carmen Herrero & Juan Moreno-Ternero & Giovanni Ponti, 2010. "On the adjudication of conflicting claims: an experimental study," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 34(1), pages 145-179, January.
    10. Martínez, Ricardo & Moreno-Ternero, Juan D., 2022. "Compensation and sacrifice in the probabilistic rationing of indivisible units," European Journal of Operational Research, Elsevier, vol. 302(2), pages 740-751.
    11. Koster, Maurice & Boonen, Tim J., 2019. "Constrained stochastic cost allocation," Mathematical Social Sciences, Elsevier, vol. 101(C), pages 20-30.
    12. José-Manuel Giménez-Gómez & M. Marco-Gil, 2014. "A new approach for bounding awards in bankruptcy problems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 43(2), pages 447-469, August.
    13. Sinan Ertemel & Rajnish Kumar, 2018. "Proportional rules for state contingent claims," International Journal of Game Theory, Springer;Game Theory Society, vol. 47(1), pages 229-246, March.
    14. Karol Flores-Szwagrzak & Jaume García-Segarra & Miguel Ginés-Vilar, 2020. "Priority and proportionality in bankruptcy," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 54(4), pages 559-579, April.
    15. Emin Karagözoğlu & Shiran Rachmilevitch, 2018. "Implementing egalitarianism in a class of Nash demand games," Theory and Decision, Springer, vol. 85(3), pages 495-508, October.
    16. Flores-Szwagrzak, Karol, 2015. "Priority classes and weighted constrained equal awards rules for the claims problem," Journal of Economic Theory, Elsevier, vol. 160(C), pages 36-55.
    17. Esat Cetemen & Emin Karagözoğlu, 2014. "Implementing equal division with an ultimatum threat," Theory and Decision, Springer, vol. 77(2), pages 223-236, August.
    18. Long, Yan & Sethuraman, Jay & Xue, Jingyi, 2021. "Equal-quantile rules in resource allocation with uncertain needs," Journal of Economic Theory, Elsevier, vol. 197(C).
    19. Doudou Gong & Genjiu Xu & Xuanzhu Jin & Loyimee Gogoi, 2022. "A sequential partition method for non-cooperative games of bankruptcy problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(2), pages 359-379, July.
    20. Erik Ansink & Hans-Peter Weikard, 2012. "Sequential sharing rules for river sharing problems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 38(2), pages 187-210, February.

    More about this item

    Keywords

    Bankruptcy problem; Bargaining; Constrained equal awards rule; Divide-the-dollar game; Efficiency; Equal division; Proportional rule;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
    • D74 - Microeconomics - - Analysis of Collective Decision-Making - - - Conflict; Conflict Resolution; Alliances; Revolutions

    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:spr:jogath:v:52:y:2023:i:1:d:10.1007_s00182-022-00814-3. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.