IDEAS home Printed from https://ideas.repec.org/a/eee/gamebe/v156y2026icp58-63.html

A note on the non-coincidence of the core and the bargaining set in many-to-one assignment markets

Author

Listed:
  • Atay, Ata
  • Núñez, Marina
  • Solymosi, Tamás

Abstract

This paper analyzes the extent to which well-known results on the relationship between the bargaining set, the core, and the kernel in one-to-one assignment games generalize to many-to-one assignment markets, and by extension, many-to-many markets. Using a minimal counterexample, we show that the bargaining set does not necessarily coincide with the core and that the kernel may not be contained within the core. We would like to highlight that the failure of the coincidence between the core and the bargaining set, as observed in the many-to-one assignment game, is quite notable. This is especially true when compared to various other highly structured games, many of which emerge from combinatorial optimization problems, such as the one underlying many-to-one assignment games.

Suggested Citation

  • Atay, Ata & Núñez, Marina & Solymosi, Tamás, 2026. "A note on the non-coincidence of the core and the bargaining set in many-to-one assignment markets," Games and Economic Behavior, Elsevier, vol. 156(C), pages 58-63.
    • Handle: RePEc:eee:gamebe:v:156:y:2026:i:c:p:58-63
    DOI: 10.1016/j.geb.2025.11.011
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0899825625001733
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.geb.2025.11.011?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

    for a different version of it.

    References listed on IDEAS

    as
    1. Atay, Ata & Mauleon, Ana & Vannetelbosch, Vincent, 2021. "A bargaining set for roommate problems," Journal of Mathematical Economics, Elsevier, vol. 94(C).
    2. TamÂs Solymosi, 1999. "On the bargaining set, kernel and core of superadditive games," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(2), pages 229-240.
    3. Atay, Ata & Solymosi, Tamás, 2018. "On bargaining sets of supplier-firm-buyer games," Economics Letters, Elsevier, vol. 167(C), pages 99-103.
    4. Klijn, Flip & Masso, Jordi, 2003. "Weak stability and a bargaining set for the marriage model," Games and Economic Behavior, Elsevier, vol. 42(1), pages 91-100, January.
    5. Tamás Solymosi, 2002. "The bargaining set of four-person balanced games," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(1), pages 1-11.
    6. Sotomayor, Marilda, 2007. "Connecting the cooperative and competitive structures of the multiple-partners assignment game," Journal of Economic Theory, Elsevier, vol. 134(1), pages 155-174, May.
    7. Crawford, Vincent P & Knoer, Elsie Marie, 1981. "Job Matching with Heterogeneous Firms and Workers," Econometrica, Econometric Society, vol. 49(2), pages 437-450, March.
    8. Marilda Sotomayor, 1992. "The Multiple Partners Game," Palgrave Macmillan Books, in: Mukul Majumdar (ed.), Equilibrium and Dynamics, chapter 17, pages 322-354, Palgrave Macmillan.
    9. Morton Davis & Michael Maschler, 1965. "The kernel of a cooperative game," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 12(3), pages 223-259, September.
    10. Maschler, Michael, 1976. "An advantage of the bargaining set over the core," Journal of Economic Theory, Elsevier, vol. 13(2), pages 184-192, October.
    11. Solymosi, T. & Raghavan, T.E.S. & Tijs, S.H., 2003. "Bargaining sets and the core in permutation games," Other publications TiSEM a14f6955-62c4-4bf0-8a54-f, Tilburg University, School of Economics and Management.
    12. Bahel, Eric, 2021. "Hyperadditive games and applications to networks or matching problems," Journal of Economic Theory, Elsevier, vol. 191(C).
    13. Daniel Granot & Frieda Granot, 1992. "On Some Network Flow Games," Mathematics of Operations Research, INFORMS, vol. 17(4), pages 792-841, November.
    14. Daniel Granot, 2010. "The reactive bargaining set for cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 163-170, March.
    15. Kelso, Alexander S, Jr & Crawford, Vincent P, 1982. "Job Matching, Coalition Formation, and Gross Substitutes," Econometrica, Econometric Society, vol. 50(6), pages 1483-1504, November.
    16. Sanchez-Soriano, Joaquin & Lopez, Marco A. & Garcia-Jurado, Ignacio, 2001. "On the core of transportation games," Mathematical Social Sciences, Elsevier, vol. 41(2), pages 215-225, March.
    17. Tamás Solymosi, 2008. "Bargaining sets and the core in partitioning games," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 16(4), pages 425-440, December.
    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. Atay, Ata & Solymosi, Tamás, 2018. "On bargaining sets of supplier-firm-buyer games," Economics Letters, Elsevier, vol. 167(C), pages 99-103.
    2. Ata Atay & Eric Bahel & Tamás Solymosi, 2023. "Matching markets with middlemen under transferable utility," Annals of Operations Research, Springer, vol. 322(2), pages 539-563, March.
    3. Luo, Chenghong & Pérez-Castrillo, David & Sun, Chaoran, 2024. "The equilibrium-value convergence for the multiple-partners game," Journal of Economic Theory, Elsevier, vol. 220(C).
    4. Funaki, Yukihiko & Núñez, Marina, 2024. "Some advances in cooperative game theory: Indivisibilities, externalities and axiomatic approach," Journal of Mathematical Economics, Elsevier, vol. 115(C).
    5. David Pérez-Castrillo & Marilda Sotomayor, 2017. "On the manipulability of competitive equilibrium rules in many-to-many buyer–seller markets," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(4), pages 1137-1161, November.
    6. Eric Bahel & Christian Trudeau, 2018. "Stable cost sharing in production allocation games," Review of Economic Design, Springer;Society for Economic Design, vol. 22(1), pages 25-53, June.
    7. Tamás Solymosi, 2015. "The kernel is in the least core for permutation games," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 23(4), pages 795-809, December.
    8. Sotomayor, Marilda, 2019. "Competition and cooperation in a two-sided matching market with replication," Journal of Economic Theory, Elsevier, vol. 183(C), pages 1030-1056.
    9. Bahel, Eric, 2021. "Hyperadditive games and applications to networks or matching problems," Journal of Economic Theory, Elsevier, vol. 191(C).
    10. Ahmet Alkan & Alparslan Tuncay, 2014. "Pairing Games and Markets," Working Papers 2014.48, Fondazione Eni Enrico Mattei.
    11. Jiang, Zhishan & Tian, Guoqiang, 2013. "Matching with Couples: Stability and Algorithm," MPRA Paper 57936, University Library of Munich, Germany, revised Jul 2014.
    12. Alvin Roth, 2008. "Deferred acceptance algorithms: history, theory, practice, and open questions," International Journal of Game Theory, Springer;Game Theory Society, vol. 36(3), pages 537-569, March.
    13. Bahel, Eric & Trudeau, Christian & Wang, Haoyu, 2026. "Preconvex games," Games and Economic Behavior, Elsevier, vol. 155(C), pages 250-266.
    14. Jesús Getán & Josep Izquierdo & Jesús Montes & Carles Rafels, 2015. "The bargaining set for almost-convex games," Annals of Operations Research, Springer, vol. 225(1), pages 83-89, February.
    15. Josep M. Izquierdo & Carles Rafels, 2010. "On the coincidence between the Shimomuras bargaining sets and the core," Working Papers in Economics 241, Universitat de Barcelona. Espai de Recerca en Economia.
    16. Tamás Solymosi, 2008. "Bargaining sets and the core in partitioning games," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 16(4), pages 425-440, December.
    17. Domènech, Gerard & Núñez, Marina, 2022. "Axioms for the optimal stable rules and fair-division rules in a multiple-partners job market," Games and Economic Behavior, Elsevier, vol. 136(C), pages 469-484.
    18. Pérez-Castrillo, David & Sotomayor, Marilda, 2019. "Comparative statics in the multiple-partners assignment game," Games and Economic Behavior, Elsevier, vol. 114(C), pages 177-192.
    19. Sotomayor, Marilda, 2007. "Connecting the cooperative and competitive structures of the multiple-partners assignment game," Journal of Economic Theory, Elsevier, vol. 134(1), pages 155-174, May.
    20. Inés Macho-Stadler & David Pérez-Castrillo, 2021. "Agency theory meets matching theory," SERIEs: Journal of the Spanish Economic Association, Springer;Spanish Economic Association, vol. 12(1), pages 1-33, March.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory
    • D47 - Microeconomics - - Market Structure, Pricing, and Design - - - Market 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:eee:gamebe:v:156:y:2026:i:c:p:58-63. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/622836 .

    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.