IDEAS home Printed from https://ideas.repec.org/a/cai/recosp/reco_712_0225.html
   My bibliography  Save this article

The Nucleolus, the Kernel, and the Bargaining Set: An Update

Author

Listed:
  • Elena Iñarra
  • Roberto Serrano
  • Ken-Ichi Shimomura

Abstract

One of the many important contributions in David Schmeidler?s distinguished career was the introduction of the nucleolus. This paper is an update on the nucleolus and its two related supersolutions, i.e., the kernel and the bargaining set. JEL Codes: C71, C72.

Suggested Citation

  • Elena Iñarra & Roberto Serrano & Ken-Ichi Shimomura, 2020. "The Nucleolus, the Kernel, and the Bargaining Set: An Update," Revue économique, Presses de Sciences-Po, vol. 71(2), pages 225-266.
    • Handle: RePEc:cai:recosp:reco_712_0225
    as

    Download full text from publisher

    File URL: http://www.cairn.info/load_pdf.php?ID_ARTICLE=RECO_712_0225
    Download Restriction: free
    File URL: http://www.cairn.info/revue-economique-2020-2-page-225.htm
    Download Restriction: free
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Vohra, Rajiv, 1991. "An existence theorem for a bargaining set," Journal of Mathematical Economics, Elsevier, vol. 20(1), pages 19-34.
    2. E. Calvo & E. Gutiérrez, 1996. "A prekernel characterization by means of stability properties," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 4(2), pages 257-267, December.
    3. Trudeau, Christian & Vidal-Puga, Juan, 2020. "Clique games: A family of games with coincidence between the nucleolus and the Shapley value," Mathematical Social Sciences, Elsevier, vol. 103(C), pages 8-14.
    4. Guni Orshan & Peter Sudhölter, 2012. "Nonsymmetric variants of the prekernel and the prenucleolus," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(4), pages 809-828, November.
    5. Josep Maria Izquierdo & Carles Rafels, 2018. "The core and the steady bargaining set for convex games," International Journal of Game Theory, Springer;Game Theory Society, vol. 47(1), pages 35-54, March.
    6. Mikel ÁLVAREZ-MOZOS & Lars EHLERS, 2017. "Externalities and the Nucleolus," Cahiers de recherche 08-2017, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    7. Nimrod Megiddo, 1978. "Computational Complexity of the Game Theory Approach to Cost Allocation for a Tree," Mathematics of Operations Research, INFORMS, vol. 3(3), pages 189-196, August.
    8. 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.
    9. 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.
    10. Jackson, Matthew O., 2005. "Allocation rules for network games," Games and Economic Behavior, Elsevier, vol. 51(1), pages 128-154, April.
    11. Chang, Chih, 1991. "Bisection Property of the Kernel," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(1), pages 1-11.
    12. Dov Monderer & Ezra Einy & Diego Moreno, 1998. "The least core, kernel and bargaining sets of large games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 11(3), pages 585-601.
    13. SCHMEIDLER, David, 1969. "The nucleolus of a characteristic function game," LIDAM Reprints CORE 44, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    14. Chih Chang & Chrong-Hsin Lian, 2002. "Some Results On (Pre)Kernel Catchers And The Coincidence Of The Kernel With Prekernel," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 4(03), pages 201-211.
    15. Holger Ingmar Meinhardt, 2014. "The Pre-Kernel as a Tractable Solution for Cooperative Games," Theory and Decision Library C, Springer, edition 127, number 978-3-642-39549-9, March.
    16. Mas-Colell, Andreu, 1989. "An equivalence theorem for a bargaining set," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 129-139, April.
    17. Serrano, Roberto, 1995. "Strategic bargaining, surplus sharing problems and the nucleolus," Journal of Mathematical Economics, Elsevier, vol. 24(4), pages 319-329.
    18. 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.
    19. Maschler, Michael, 1976. "An advantage of the bargaining set over the core," Journal of Economic Theory, Elsevier, vol. 13(2), pages 184-192, October.
    20. Lloyd S. Shapley, 1992. "Kernels of Replicated Market Games," UCLA Economics Working Papers 654, UCLA Department of Economics.
    21. Ron Holzman, 2001. "The comparability of the classical and the Mas-Colell bargaining sets," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(4), pages 543-553.
    22. Hara, Chiaki, 2005. "Bargaining set and anonymous core without the monotonicity assumption," Journal of Mathematical Economics, Elsevier, vol. 41(4-5), pages 545-556, August.
    23. 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.
    24. Robert M. Anderson & Walter Trockel & Lin Zhou, 1997. "Nonconvergence of the Mas-Colell and Zhou Bargaining Sets," Econometrica, Econometric Society, vol. 65(5), pages 1227-1240, September.
    25. Ruiz, Luis M & Valenciano, Federico & Zarzuelo, Jose M, 1996. "The Least Square Prenucleolus and the Least Square Nucleolus. Two Values for TU Games Based on the Excess Vector," International Journal of Game Theory, Springer;Game Theory Society, vol. 25(1), pages 113-134.
    26. Martin J. Osborne & Ariel Rubinstein, 1994. "A Course in Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262650401, December.
    27. Serrano, Roberto, 1997. "Reinterpreting the Kernel," Journal of Economic Theory, Elsevier, vol. 77(1), pages 58-80, November.
    28. 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.
    29. Katsev, Ilya & Yanovskaya, Elena, 2013. "The prenucleolus for games with restricted cooperation," Mathematical Social Sciences, Elsevier, vol. 66(1), pages 56-65.
    30. S.C. Littlechild & G.F. Thompson, 1977. "Aircraft Landing Fees: A Game Theory Approach," Bell Journal of Economics, The RAND Corporation, vol. 8(1), pages 186-204, Spring.
    31. Montero, Maria, 2006. "Noncooperative foundations of the nucleolus in majority games," Games and Economic Behavior, Elsevier, vol. 54(2), pages 380-397, February.
    32. Guni Orshan & Federico Valenciano & José M. Zarzuelo, 2003. "The Bilateral Consistent Prekernel, the Core, and NTU Bankruptcy Problems," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 268-282, May.
    33. Peleg, B, 1986. "On the Reduced Game Property and Its Converse," International Journal of Game Theory, Springer;Game Theory Society, vol. 15(3), pages 187-200.
    34. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, August.
    35. M. Maschler & B. Peleg & L. S. Shapley, 1979. "Geometric Properties of the Kernel, Nucleolus, and Related Solution Concepts," Mathematics of Operations Research, INFORMS, vol. 4(4), pages 303-338, November.
    36. Robert J. Aumann & Ein-Ya Gura & Sergiu Hart & Bezalel Peleg & Hana Shemesh & Shmuel Zamir, 2008. "Michael Maschler: In Memoriam," Discussion Paper Series dp493, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    37. J. Arin & E. Inarra & P. Luquin, 2009. "A noncooperative view on two airport cost sharing rules," Review of Economic Design, Springer;Society for Economic Design, vol. 13(4), pages 361-376, December.
    38. Potters, Jos A M, 1991. "An Axiomatization of the Nucleolus," International Journal of Game Theory, Springer;Game Theory Society, vol. 19(4), pages 365-373.
    39. Maschler, Michael, 1992. "The bargaining set, kernel, and nucleolus," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 18, pages 591-667, Elsevier.
    40. Hu, Cheng-Cheng & Tsay, Min-Hung & Yeh, Chun-Hsien, 2012. "Axiomatic and strategic justifications for the constrained equal benefits rule in the airport problem," Games and Economic Behavior, Elsevier, vol. 75(1), pages 185-197.
    41. Orshan, Gooni & Zarzuelo, Jose M., 2000. "The Bilateral Consistent Prekernel for NTU Games," Games and Economic Behavior, Elsevier, vol. 32(1), pages 67-84, July.
    42. K.I. Shimomura, 1995. "The Bargaining Set and Coalition Formation," Working Papers 95-11, Brown University, Department of Economics.
    43. repec:ehu:ikerla:9314 is not listed on IDEAS
    44. Moldovanu, B, 1990. "Stable Bargained Equilibria for Assignment Games without Side Payments," International Journal of Game Theory, Springer;Game Theory Society, vol. 19(2), pages 171-190.
    45. Hokari, Toru & Kibris, Ozgur, 2003. "Consistency, converse consistency, and aspirations in TU-games," Mathematical Social Sciences, Elsevier, vol. 45(3), pages 313-331, July.
    46. Anna Khmelnitskaya & Peter Sudhölter, 2013. "The prenucleolus and the prekernel for games with communication structures," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(2), pages 285-299, October.
    47. Oriol Tejada & Carles Rafels, 2010. "Symmetrically multilateral-bargained allocations in multi-sided assignment markets," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 249-258, March.
    48. Peleg, B, 1987. "On the Reduced Game Property and Its Converse: A Correction," International Journal of Game Theory, Springer;Game Theory Society, vol. 16(4), pages 290-290.
    49. Serrano, Roberto, 1993. "Non-cooperative Implementation of the Nucleolus: The 3-Player Case," International Journal of Game Theory, Springer;Game Theory Society, vol. 22(4), pages 345-357.
    50. Jeroen Kuipers & Ulrich Faigle & Walter Kern, 2001. "On the computation of the nucleolus of a cooperative game," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(1), pages 79-98.
    51. Zhou Lin, 1994. "A New Bargaining Set of an N-Person Game and Endogenous Coalition Formation," Games and Economic Behavior, Elsevier, vol. 6(3), pages 512-526, May.
    52. L. S. Shapley & Martin Shubik, 1967. "Ownership and the Production Function," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 81(1), pages 88-111.
    53. Youngsub Chun & Nari Park & Duygu Yengin, 2015. "Coincidence of Cooperative Game Theoretic Solutions in the Appointment Problem," School of Economics and Public Policy Working Papers 2015-09, University of Adelaide, School of Economics and Public Policy.
    54. AUMANN, Robert J. & DREZE, Jacques H., 1974. "Cooperative games with coalition structures," LIDAM Reprints CORE 217, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    55. Rodica Brânzei & Elena Iñarra & Stef Tijs & José Zarzuelo, 2006. "A Simple Algorithm for the Nucleolus of Airport Profit Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 34(2), pages 259-272, August.
    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. Holger I. Meinhardt, 2024. "On the Replication of the Pre-kernel and Related Solutions," Computational Economics, Springer;Society for Computational Economics, vol. 64(2), pages 871-946, August.
    2. Ken‐Ichi Shimomura, 2022. "The bargaining set and coalition formation," International Journal of Economic Theory, The International Society for Economic Theory, vol. 18(1), pages 16-37, March.
    3. Pongou, Roland & Tondji, Jean-Baptiste, 2024. "The reciprocity set," Journal of Mathematical Economics, Elsevier, vol. 112(C).
    4. Bossu, Adrien & Durillon, Benoit & Davigny, Arnaud & Barry, Hervé & Robyns, Benoît & Belaïd, Fateh & Saudemont, Christophe, 2024. "Coalitional game-based gain generation and distribution for collective self-consumption in an energy community," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 225(C), pages 129-147.
    5. Yan-An Hwang & Yu-Hsien Liao, 2020. "A Solution Concept and Its Axiomatic Results under Non-Transferable-Utility and Multi-Choice Situations," Mathematics, MDPI, vol. 8(9), pages 1-10, September.

    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. repec:ehu:ikerla:34464 is not listed on IDEAS
    2. Michel Le Breton & Karine Van Der Straeten, 2017. "Alliances Électorales et Gouvernementales : La Contribution de la Théorie des Jeux Coopératifs à la Science Politique," Revue d'économie politique, Dalloz, vol. 127(4), pages 637-736.
    3. Rebelo, S., 1997. "On the Determinant of Economic Growth," RCER Working Papers 443, University of Rochester - Center for Economic Research (RCER).
    4. Serrano, Roberto, 1997. "Reinterpreting the Kernel," Journal of Economic Theory, Elsevier, vol. 77(1), pages 58-80, November.
    5. Serrano, Roberto & Shimomura, Ken-Ichi, 2006. "A comparison of the average prekernel and the prekernel," Mathematical Social Sciences, Elsevier, vol. 52(3), pages 288-301, December.
    6. Roberto Serrano, 2005. "Fifty years of the Nash program, 1953-2003," Investigaciones Economicas, Fundación SEPI, vol. 29(2), pages 219-258, May.
    7. Funaki, Yukihiko & Núñez, Marina, 2024. "Some advances in cooperative game theory: Indivisibilities, externalities and axiomatic approach," Journal of Mathematical Economics, Elsevier, vol. 115(C).
    8. Hu, Cheng-Cheng & Tsay, Min-Hung & Yeh, Chun-Hsien, 2018. "A study of the nucleolus in the nested cost-sharing problem: Axiomatic and strategic perspectives," Games and Economic Behavior, Elsevier, vol. 109(C), pages 82-98.
    9. Guni Orshan & Federico Valenciano & José M. Zarzuelo, 2003. "The Bilateral Consistent Prekernel, the Core, and NTU Bankruptcy Problems," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 268-282, May.
    10. J. Arin & V. Feltkamp & M. Montero, 2015. "A bargaining procedure leading to the serial rule in games with veto players," Annals of Operations Research, Springer, vol. 229(1), pages 41-66, June.
    11. Michel Le Breton & Karine Van der Straeten, 2013. "Alliances électorales entre deux tours de scrutin. Le point de vue de la théorie des jeux coopératifs et une application aux élections régionales de mars 2010," Revue économique, Presses de Sciences-Po, vol. 64(2), pages 173-240.
    12. Serrano, Roberto & Vohra, Rajiv, 2002. "Bargaining and Bargaining Sets," Games and Economic Behavior, Elsevier, vol. 39(2), pages 292-308, May.
    13. Fatma Aslan & Papatya Duman & Walter Trockel, 2019. "Duality for General TU-games Redefined," Working Papers CIE 121, Paderborn University, CIE Center for International Economics.
    14. 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.
    15. Yan-An Hwang, 2006. "Two characterizations of the consistent egalitarian solution and of the core on NTU games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(3), pages 557-568, December.
    16. Sun, Ning & Trockel, Walter & Yang, Zaifu, 2008. "Competitive outcomes and endogenous coalition formation in an n-person game," Journal of Mathematical Economics, Elsevier, vol. 44(7-8), pages 853-860, July.
    17. J. Schouten & B. Dietzenbacher & P. Borm, 2022. "The nucleolus and inheritance of properties in communication situations," Annals of Operations Research, Springer, vol. 318(2), pages 1117-1135, November.
    18. Pierre Dehez, 2017. "On Harsanyi Dividends and Asymmetric Values," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 19(03), pages 1-36, September.
    19. Shin Kishimoto & Naoki Watanabe, 2014. "The Kernel of a Patent Licensing Game," Working Papers e075, Tokyo Center for Economic Research.
    20. Francesc Llerena & Marina Núñez & Carles Rafels, 2015. "An axiomatization of the nucleolus of assignment markets," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(1), pages 1-15, February.
    21. Potters, Jos & Sudholter, Peter, 1999. "Airport problems and consistent allocation rules," Mathematical Social Sciences, Elsevier, vol. 38(1), pages 83-102, July.

    More about this item

    Keywords

    nucleolus; kernel; bargaining set;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

    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:cai:recosp:reco_712_0225. 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: Jean-Baptiste de Vathaire (email available below). General contact details of provider: https://www.cairn.info/revue-economique.htm .

    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.