IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i15p2705-d876615.html
   My bibliography  Save this article

Equilibrium in a Bargaining Game of Two Sellers and Two Buyers

Author

Listed:
  • Jiawei Li

    (School of Computer Science, University of Nottingham Ningbo China, Ningbo 315100, China)

  • Tianxiang Cui

    (School of Computer Science, University of Nottingham Ningbo China, Ningbo 315100, China)

  • Graham Kendall

    (School of Computer Science, University of Nottingham UK, Nottingham NG7 2RD, UK)

Abstract

The uniqueness of equilibrium in bargaining games with three or more players is a problem preventing bargaining theory from general real world applications. We study the uniqueness of bargaining equilibrium in a bargaining game of two sellers and two buyers, which has instances in real-world markets. Each seller (or buyer) wants to reach an agreement with a buyer (or seller) on the division of a pie in the bargaining game. A seller and a buyer will receive their agreed divisions if they can reach an agreement. Otherwise, they receive nothing. The bargaining game includes a finite number of rounds. In each round, a player can propose an offer or accept an offer. Each player has a constant discounting factor. Under the assumption of complete information, we prove that the equilibrium of this bargaining game is the same division of two pies. The ratio of division as a function of the discount factors of all players is also deduced. The result can be extended to a bargaining game of n -sellers and n -buyers, which reveals the relevance of bargaining equilibrium to the general equilibrium of a market.

Suggested Citation

  • Jiawei Li & Tianxiang Cui & Graham Kendall, 2022. "Equilibrium in a Bargaining Game of Two Sellers and Two Buyers," Mathematics, MDPI, vol. 10(15), pages 1-9, July.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2705-:d:876615
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/15/2705/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/15/2705/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Rubinstein, Ariel, 1982. "Perfect Equilibrium in a Bargaining Model," Econometrica, Econometric Society, vol. 50(1), pages 97-109, January.
    2. Baron, David P. & Ferejohn, John A., 1989. "Bargaining in Legislatures," American Political Science Review, Cambridge University Press, vol. 83(4), pages 1181-1206, December.
    3. Ken Binmore & Ariel Rubinstein & Asher Wolinsky, 1986. "The Nash Bargaining Solution in Economic Modelling," RAND Journal of Economics, The RAND Corporation, vol. 17(2), pages 176-188, Summer.
    4. Silvana Krasteva & Huseyin Yildirim, 2012. "Payoff uncertainty, bargaining power, and the strategic sequencing of bilateral negotiations," RAND Journal of Economics, RAND Corporation, vol. 43(3), pages 514-536, September.
    5. Merlo, Antonio & Wilson, Charles A, 1995. "A Stochastic Model of Sequential Bargaining with Complete Information," Econometrica, Econometric Society, vol. 63(2), pages 371-399, March.
    6. Juan J. Vidal-Puga, 2004. "Bargaining with commitments," International Journal of Game Theory, Springer;Game Theory Society, vol. 33(1), pages 129-144, January.
    7. Kultti, Klaus & Vartiainen, Hannu, 2008. "Bargaining with many players: A limit result," Economics Letters, Elsevier, vol. 101(3), pages 249-252, December.
    8. Chae, Suchan & Yang, Jeong-Ae, 1988. "The unique perfect equilibrium of an n-person bargaining game," Economics Letters, Elsevier, vol. 28(3), pages 221-223.
    9. Eraslan, Hulya & Evdokimov, Kirill S., 2019. "Legislative and Multilateral Bargaining," Working Papers 19-007, Rice University, Department of Economics.
    10. Matthew Backus & Thomas Blakee & Brad Larsen & Steven Tadelis, 2020. "Sequential Bargaining in the Field: Evidence from Millions of Online Bargaining Interactions," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 135(3), pages 1319-1361.
    11. Asheim, Geir B., 1992. "A unique solution to n-person sequential bargaining," Games and Economic Behavior, Elsevier, vol. 4(2), pages 169-181, April.
    12. Winter, Eyal, 1994. "The Demand Commitment Bargaining and Snowballing Cooperation," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(2), pages 255-273, March.
    13. Vincent J. Vannetelbosch, 1999. "Rationalizability and equilibrium in N-person sequential bargaining," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 14(2), pages 353-371.
    14. Allan Collard-Wexler & Gautam Gowrisankaran & Robin S. Lee, 2019. ""Nash-in-Nash" Bargaining: A Microfoundation for Applied Work," Journal of Political Economy, University of Chicago Press, vol. 127(1), pages 163-195.
    15. Kalandrakis, Tasos, 2015. "Computation of equilibrium values in the Baron and Ferejohn bargaining model," Games and Economic Behavior, Elsevier, vol. 94(C), pages 29-38.
    16. (*), Y. Stephen Chiu & Ani Dasgupta, 1998. "On implementation via demand commitment games," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(2), pages 161-189.
    17. Silvana Krasteva & Huseyin Yildirim, 2019. "Strategic Ignorance in Sequential Procurement," American Economic Journal: Microeconomics, American Economic Association, vol. 11(2), pages 131-172, May.
    18. John Sutton, 1986. "Non-Cooperative Bargaining Theory: An Introduction," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 53(5), pages 709-724.
    19. Yang, Jeong-Ae, 1992. "Another n-person bargaining game with a unique perfect equilibrium," Economics Letters, Elsevier, vol. 38(3), pages 275-277, March.
    20. Kalyan Chatterjee & Hamid Sabourian, 2000. "Multiperson Bargaining and Strategic Complexity," Econometrica, Econometric Society, vol. 68(6), pages 1491-1510, November.
    21. Hülya Eraslan & Kirill S. Evdokimov, 2019. "Legislative and Multilateral Bargaining," Annual Review of Economics, Annual Reviews, vol. 11(1), pages 443-472, August.
    22. Cai, Hongbin, 2000. "Delay in Multilateral Bargaining under Complete Information," Journal of Economic Theory, Elsevier, vol. 93(2), pages 260-276, August.
    23. Montero, Maria, 2017. "Proportional Payoffs in Legislative Bargaining with Weighted Voting: A Characterization," Quarterly Journal of Political Science, now publishers, vol. 12(3), pages 325-346, October.
    24. Eraslan, Hulya, 2002. "Uniqueness of Stationary Equilibrium Payoffs in the Baron-Ferejohn Model," Journal of Economic Theory, Elsevier, vol. 103(1), pages 11-30, March.
    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. Xuefeng Min & Yang Yu & Rui Zhao & Xin Xiong & Sen Du & Qin Mou & Wenjie Xu, 2023. "System Dynamics-Based Game Theoretical Analysis for Watershed Eco-Compensation Design: A Case Study in Tuojiang River, Sichuan Province, China," Land, MDPI, vol. 12(9), pages 1-16, August.
    2. Adrian Marius Deaconu & Daniel Tudor Cotfas & Petru Adrian Cotfas, 2023. "Advanced Optimization Methods and Applications," Mathematics, MDPI, vol. 11(9), pages 1-7, May.

    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. Torstensson, Pär, 2005. "An n-person Rubinstein bargaining game," Working Papers 2005:39, Lund University, Department of Economics.
    2. Harstad, Bård, 2023. "Pledge-and-review bargaining," Journal of Economic Theory, Elsevier, vol. 207(C).
    3. Andrzej Baranski & Ernesto Reuben, 2023. "Competing for Proposal Rights: Theory and Experimental Evidence," Working Papers 20220085, New York University Abu Dhabi, Department of Social Science, revised Mar 2023.
    4. P. Jean-Jacques Herings & Harold Houba, 2022. "Costless delay in negotiations," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 74(1), pages 69-93, July.
    5. Suh, Sang-Chul & Wen, Quan, 2006. "Multi-agent bilateral bargaining and the Nash bargaining solution," Journal of Mathematical Economics, Elsevier, vol. 42(1), pages 61-73, February.
    6. Uyanik, Metin & Yengin, Duygu, 2023. "Expropriation power in private dealings: Quota rule in collective sales," Games and Economic Behavior, Elsevier, vol. 141(C), pages 548-580.
    7. Sang-Chul Suh & Quan Wen, 2009. "A multi-agent bilateral bargaining model with endogenous protocol," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 40(2), pages 203-226, August.
    8. Johannes Münster & Markus Reisinger, 2021. "Sequencing Bilateral Negotiations with Externalities," ECONtribute Discussion Papers Series 096, University of Bonn and University of Cologne, Germany.
    9. Sang-Chul Suh & Quan Wen, 2003. "Multi-Agent Bilateral Bargaining with Endogenous Protocol," Vanderbilt University Department of Economics Working Papers 0305, Vanderbilt University Department of Economics.
    10. Daniel Cardona-Coll, 2003. "Bargaining and Strategic Demand Commitment," Theory and Decision, Springer, vol. 54(4), pages 357-374, June.
    11. Li, Shuwen & Houser, Daniel, 2022. "Stochastic bargaining in the lab," Journal of Economic Behavior & Organization, Elsevier, vol. 200(C), pages 687-715.
    12. Kazuo Yamaguchi, 2022. "Spatial bargaining in rectilinear facility location problem," Theory and Decision, Springer, vol. 93(1), pages 69-104, July.
    13. Ray, Debraj & Vohra, Rajiv, 2015. "Coalition Formation," Handbook of Game Theory with Economic Applications,, Elsevier.
    14. Cho, Seok-ju & Duggan, John, 2009. "Bargaining foundations of the median voter theorem," Journal of Economic Theory, Elsevier, vol. 144(2), pages 851-868, March.
    15. Miller, Luis & Montero, Maria & Vanberg, Christoph, 2018. "Legislative bargaining with heterogeneous disagreement values: Theory and experiments," Games and Economic Behavior, Elsevier, vol. 107(C), pages 60-92.
    16. Miettinen, Topi & Vanberg, Christoph, 2020. "Commitment and Conflict in Multilateral Bargaining," Working Papers 0679, University of Heidelberg, Department of Economics.
    17. Klaus Kultti & Hannu Vartiainen, 2010. "Multilateral non-cooperative bargaining in a general utility space," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(4), pages 677-689, October.
    18. Gersbach, Hans & Britz, Volker, 2018. "Open Rule Legislative Bargaining," CEPR Discussion Papers 12966, C.E.P.R. Discussion Papers.
    19. Andrzej Baranski & Caleb A. Cox, 2023. "Communication in multilateral bargaining with joint production," Experimental Economics, Springer;Economic Science Association, vol. 26(1), pages 55-77, March.
    20. Alfredo Valencia-Toledo & Juan Vidal-Puga, 2020. "A sequential bargaining protocol for land rental arrangements," Review of Economic Design, Springer;Society for Economic Design, vol. 24(1), pages 65-99, June.

    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:gam:jmathe:v:10:y:2022:i:15:p:2705-:d:876615. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.