IDEAS home Printed from https://ideas.repec.org/p/has/discpr/1914.html
   My bibliography  Save this paper

Weighted nucleoli and dually essential coalitions (extended version)

Author

Listed:
  • Tamás Solymosi

    (Institute of Economics, Centre for Economic and Regional Studies, Hungarian Academy of Sciences and Department of Operations Research and Actuarial Sciences, Corvinus University of Budapest)

Abstract

We study linearly weighted versions of the least core and the (pre)nucleolus and investigate the reduction possibilities in their computation. We slightly extend some well-known related results and establish their counterparts by using the dual game. Our main results imply, for example, that if the core of the game is not empty, all dually inessential coalitions (which can be weakly minorized by a partition in the dual game) can be ignored when we compute the per-capita least core and the per-capita (pre)nucleolus from the dual game. This could lead to the design of polynomial time algorithms for the per-capita (and other monotone nondecreasingly weighted versions of the) least core and the (pre)nucleolus in specific classes of balanced games with polynomial many dually essential coalitions.

Suggested Citation

  • Tamás Solymosi, 2019. "Weighted nucleoli and dually essential coalitions (extended version)," CERS-IE WORKING PAPERS 1914, Institute of Economics, Centre for Economic and Regional Studies.
  • Handle: RePEc:has:discpr:1914
    as

    Download full text from publisher

    File URL: https://www.mtakti.hu/wp-content/uploads/2019/05/MTDP1914.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Jos A. M. Potters & Stef H. Tijs, 1992. "The Nucleolus of a Matrix Game and Other Nucleoli," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 164-174, February.
    2. 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.
    3. Oishi, Takayuki & Nakayama, Mikio & Hokari, Toru & Funaki, Yukihiko, 2016. "Duality and anti-duality in TU games applied to solutions, axioms, and axiomatizations," Journal of Mathematical Economics, Elsevier, vol. 63(C), pages 44-53.
    4. Mirjam Groote Schaarsberg & Peter Borm & Herbert Hamers & Hans Reijnierse, 2013. "Game theoretic analysis of maximum cooperative purchasing situations," Naval Research Logistics (NRL), John Wiley & Sons, vol. 60(8), pages 607-624, December.
    5. Nguyen, Tri-Dung & Thomas, Lyn, 2016. "Finding the nucleoli of large cooperative games," European Journal of Operational Research, Elsevier, vol. 248(3), pages 1078-1092.
    6. Rodica Brânzei & Tamás Solymosi & Stef Tijs, 2005. "Strongly essential coalitions and the nucleolus of peer group games," International Journal of Game Theory, Springer;Game Theory Society, vol. 33(3), pages 447-460, September.
    7. René Brink & Ilya Katsev & Gerard Laan, 2011. "A polynomial time algorithm for computing the nucleolus for a class of disjunctive games with a permission structure," International Journal of Game Theory, Springer;Game Theory Society, vol. 40(3), pages 591-616, August.
    8. Maschler, M. & Potters, J.A.M. & Tijs, S.H., 1992. "The general nucleolus and the reduced game property," Other publications TiSEM ab187dab-1b5b-40c3-a673-8, Tilburg University, School of Economics and Management.
    9. Mitsuo Suzuki & Mikio Nakayama, 1976. "The Cost Assignment of the Cooperative Water Resource Development: A Game Theoretical Approach," Management Science, INFORMS, vol. 22(10), pages 1081-1086, June.
    10. Kleppe, J., 2010. "Modelling interactive behaviour, and solution concepts," Other publications TiSEM b9b96884-5761-48f0-9ee4-4, Tilburg University, School of Economics and Management.
    11. Potters, J.A.M. & Tijs, S.H., 1992. "The nucleolus of a matrix game and other nucleoli," Other publications TiSEM ae3402e7-bd19-494b-b0a1-f, Tilburg University, School of Economics and Management.
    12. 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).
    13. Maschler, M & Potters, J A M & Tijs, S H, 1992. "The General Nucleolus and the Reduced Game Property," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(1), pages 85-106.
    14. Solymosi, Tamas & Raghavan, Tirukkannamangai E S, 1994. "An Algorithm for Finding the Nucleolus of Asignment Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(2), pages 119-143.
    15. Derks, Jean & Peters, Hans, 1998. "Orderings, excess functions, and the nucleolus," Mathematical Social Sciences, Elsevier, vol. 36(2), pages 175-182, September.
    16. John Kleppe & Hans Reijnierse & Peter Sudhölter, 2016. "Axiomatizations of symmetrically weighted solutions," Annals of Operations Research, Springer, vol. 243(1), pages 37-53, August.
    17. 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.
    18. Takayuki Oishi & Mikio Nakayama, 2009. "Anti‐Dual Of Economic Coalitional Tu Games," The Japanese Economic Review, Japanese Economic Association, vol. 60(4), pages 560-566, 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. Tamás Solymosi, 2019. "Weighted nucleoli and dually essential coalitions," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(4), pages 1087-1109, December.
    2. Márton Benedek & Jörg Fliege & Tri-Dung Nguyen, 2020. "Finding and verifying the nucleolus of cooperative games," CERS-IE WORKING PAPERS 2021, Institute of Economics, Centre for Economic and Regional Studies.
    3. Tamas Solymosi & Balazs Sziklai, 2015. "Universal Characterization Sets for the Nucleolus in Balanced Games," CERS-IE WORKING PAPERS 1512, Institute of Economics, Centre for Economic and Regional Studies.
    4. Michael Maschler, 2004. "Encouraging a Coalition Formation," Discussion Paper Series dp392, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    5. Eijffinger, S.C.W. & Gruijters, A.P.D., 1989. "On the effectiveness of daily interventions by the Deutsche Bundesbank and the federal reserve system in the U.S. Dollar-Deutsche Mark exchange market," Other publications TiSEM cd65eff1-5f9e-4262-8f38-b, Tilburg University, School of Economics and Management.
    6. Pedro Calleja & Francesc Llerena & Peter Sudhölter, 2020. "Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 1056-1068, August.
    7. Rodica Brânzei & Tamás Solymosi & Stef Tijs, 2005. "Strongly essential coalitions and the nucleolus of peer group games," International Journal of Game Theory, Springer;Game Theory Society, vol. 33(3), pages 447-460, September.
    8. Vijay V. Vazirani, 2022. "Insights into the Core of Matching Games via Complementarity," Papers 2202.00619, arXiv.org, revised May 2022.
    9. René Brink, 2017. "Games with a permission structure - A survey on generalizations and applications," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 1-33, April.
    10. Eijffinger, Sylvester & van Rixtel, Adrian, 1992. "The Japanese financial system and monetary policy: a descriptive review," Japan and the World Economy, Elsevier, vol. 4(4), pages 291-309, December.
    11. Xiaotie Deng & Qizhi Fang & Xiaoxun Sun, 2009. "Finding nucleolus of flow game," Journal of Combinatorial Optimization, Springer, vol. 18(1), pages 64-86, July.
    12. Verbeek, M.J.C.M. & Nijman, T.E., 1993. "Incomplete panels and selection bias : A survey," Other publications TiSEM 08061352-957b-4f56-b303-9, Tilburg University, School of Economics and Management.
    13. Karl Wärneryd, 1993. "Anarchy, Uncertainty, And The Emergence Of Property Rights," Economics and Politics, Wiley Blackwell, vol. 5(1), pages 1-14, March.
    14. Ogryczak, Wlodzimierz, 1997. "On the lexicographic minimax approach to location problems," European Journal of Operational Research, Elsevier, vol. 100(3), pages 566-585, August.
    15. Vazirani, Vijay V., 2022. "The general graph matching game: Approximate core," Games and Economic Behavior, Elsevier, vol. 132(C), pages 478-486.
    16. Reijnierse, Hans & Potters, Jos, 1998. "The -Nucleolus of TU-Games," Games and Economic Behavior, Elsevier, vol. 24(1-2), pages 77-96, July.
    17. Walter Kern & Daniël Paulusma, 2003. "Matching Games: The Least Core and the Nucleolus," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 294-308, May.
    18. Luo, Chunlin & Zhou, Xiaoyang & Lev, Benjamin, 2022. "Core, shapley value, nucleolus and nash bargaining solution: A Survey of recent developments and applications in operations management," Omega, Elsevier, vol. 110(C).
    19. Daniel Granot & Jeroen Kuipers & Sunil Chopra, 2002. "Cost Allocation for a Tree Network with Heterogeneous Customers," Mathematics of Operations Research, INFORMS, vol. 27(4), pages 647-661, November.
    20. Quant, Marieke & Borm, Peter & Hendrickx, Ruud & Zwikker, Peter, 2006. "Compromise solutions based on bankruptcy," Mathematical Social Sciences, Elsevier, vol. 51(3), pages 247-256, May.

    More about this item

    Keywords

    nucleolus; least core; weighted nucleoli; efficient computation; cooperative game;
    All these keywords.

    JEL classification:

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

    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:has:discpr:1914. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: https://edirc.repec.org/data/iehashu.html .

    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: Nora Horvath (email available below). General contact details of provider: https://edirc.repec.org/data/iehashu.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.