IDEAS home Printed from https://ideas.repec.org/a/spr/jogath/v53y2024i3d10.1007_s00182-024-00893-4.html
   My bibliography  Save this article

Second-order productivity, second-order payoffs, and the Banzhaf value

Author

Listed:
  • André Casajus

    (HHL Leipzig Graduate School of Management
    Dr. Hops Craft Beer Bar)

  • Rodrigue Tido Takeng

    (Université de Caen Normandie, CREM, UMR 6211)

Abstract

First, we suggest and discuss second-order versions of properties for solutions for TU games used to characterize the Banzhaf value, in particular, of standardness for two-player games, of the dummy player property, and of 2-efficiency. Then, we provide a number of characterizations of the Banzhaf value invoking the following properties: (i) [second-order standardness for two-player games or the second-order dummy player property] and 2-efficiency, (ii) standardness for one-player games, standardness for two-player games, and second-order 2-efficiency, (iii) standardness for one-player games, [second-order standardness for two-player games or the second-order dummy player property], and second-order 2-efficiency. These characterizations also work within the classes of simple games, of superadditive games, and of simple superadditive games.

Suggested Citation

  • André Casajus & Rodrigue Tido Takeng, 2024. "Second-order productivity, second-order payoffs, and the Banzhaf value," International Journal of Game Theory, Springer;Game Theory Society, vol. 53(3), pages 989-1004, September.
    • Handle: RePEc:spr:jogath:v:53:y:2024:i:3:d:10.1007_s00182-024-00893-4
    DOI: 10.1007/s00182-024-00893-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00182-024-00893-4
    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/s00182-024-00893-4?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. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer;Game Theory Society, vol. 17(2), pages 89-99.
    2. Besner, Manfred, 2022. "Disjointly productive players and the Shapley value," Games and Economic Behavior, Elsevier, vol. 133(C), pages 109-114.
    3. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(3), pages 261-281.
    4. Haimanko, Ori, 2018. "The axiom of equivalence to individual power and the Banzhaf index," Games and Economic Behavior, Elsevier, vol. 108(C), pages 391-400.
    5. André Casajus, 2011. "Marginality, differential marginality, and the Banzhaf value," Theory and Decision, Springer, vol. 71(3), pages 365-372, September.
    6. Pradeep Dubey & Lloyd S. Shapley, 1979. "Mathematical Properties of the Banzhaf Power Index," Mathematics of Operations Research, INFORMS, vol. 4(2), pages 99-131, May.
    7. 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).
    8. André Casajus, 2012. "Amalgamating players, symmetry, and the Banzhaf value," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(3), pages 497-515, August.
    9. Guillermo Owen, 1975. "Multilinear extensions and the banzhaf value," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 22(4), pages 741-750, December.
    10. Guillermo Owen, 1972. "Multilinear Extensions of Games," Management Science, INFORMS, vol. 18(5-Part-2), pages 64-79, January.
    11. Feltkamp, Vincent, 1995. "Alternative Axiomatic Characterizations of the Shapley and Banzhaf Values," International Journal of Game Theory, Springer;Game Theory Society, vol. 24(2), pages 179-186.
    12. Casajus, André & Huettner, Frank, 2018. "Decomposition of solutions and the Shapley value," Games and Economic Behavior, Elsevier, vol. 108(C), pages 37-48.
    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. André Casajus & Frank Huettner, 2019. "The Coleman–Shapley index: being decisive within the coalition of the interested," Public Choice, Springer, vol. 181(3), pages 275-289, December.
    2. Conrado M. Manuel & Daniel Martín, 2021. "A Monotonic Weighted Banzhaf Value for Voting Games," Mathematics, MDPI, vol. 9(12), pages 1-23, June.
    3. Josep Freixas & Montserrat Pons, 2021. "An Appropriate Way to Extend the Banzhaf Index for Multiple Levels of Approval," Group Decision and Negotiation, Springer, vol. 30(2), pages 447-462, April.
    4. Gerard van der Laan & René van den Brink, 2002. "A Banzhaf share function for cooperative games in coalition structure," Theory and Decision, Springer, vol. 53(1), pages 61-86, August.
    5. André Casajus, 2014. "Collusion, quarrel, and the Banzhaf value," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(1), pages 1-11, February.
    6. Ori Haimanko, 2020. "Generalized Coleman-Shapley indices and total-power monotonicity," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(1), pages 299-320, March.
    7. McQuillin, Ben & Sugden, Robert, 2018. "Balanced externalities and the Shapley value," Games and Economic Behavior, Elsevier, vol. 108(C), pages 81-92.
    8. Ori Haimanko, 2019. "Composition independence in compound games: a characterization of the Banzhaf power index and the Banzhaf value," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(3), pages 755-768, September.
    9. En-Cheng Chi & Yu-Hsien Liao, 2021. "Sustainable Usability Distribution Mechanisms under Multi-Attribute Sports Management Schemes," Sustainability, MDPI, vol. 13(3), pages 1-16, February.
    10. Yu-Hsien Liao, 2023. "Power Indices under Specific Multicriteria Status," Games, MDPI, vol. 14(4), pages 1-10, June.
    11. Haimanko, Ori, 2018. "The axiom of equivalence to individual power and the Banzhaf index," Games and Economic Behavior, Elsevier, vol. 108(C), pages 391-400.
    12. van den Brink, Rene & van der Laan, Gerard, 2005. "A class of consistent share functions for games in coalition structure," Games and Economic Behavior, Elsevier, vol. 51(1), pages 193-212, April.
    13. Sylvain Béal & Eric Rémila & Philippe Solal, 2014. "Decomposition of the space of TU-games, Strong Transfer Invariance and the Banzhaf value," Working Papers 2014-05, CRESE.
    14. Nicolas Andjiga & Sebastien Courtin, 2015. "Coalition configurations and share functions," Annals of Operations Research, Springer, vol. 225(1), pages 3-25, February.
    15. Bendel, Dan & Haviv, Moshe, 2018. "Cooperation and sharing costs in a tandem queueing network," European Journal of Operational Research, Elsevier, vol. 271(3), pages 926-933.
    16. Rafael Amer & José Miguel Giménez, 2007. "Technical note: Characterization of binomial semivalues through delegation games," Naval Research Logistics (NRL), John Wiley & Sons, vol. 54(6), pages 702-708, September.
    17. Annick Laruelle & Federico Valenciano, 2001. "Shapley-Shubik and Banzhaf Indices Revisited," Mathematics of Operations Research, INFORMS, vol. 26(1), pages 89-104, February.
    18. Carreras, Francesc & Freixas, Josep & Puente, Maria Albina, 2003. "Semivalues as power indices," European Journal of Operational Research, Elsevier, vol. 149(3), pages 676-687, September.
    19. Besner, Manfred, 2021. "Disjointly productive players and the Shapley value," MPRA Paper 108241, University Library of Munich, Germany.
    20. Kamijo, Yoshio & Kongo, Takumi, 2012. "Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value, and the Banzhaf value," European Journal of Operational Research, Elsevier, vol. 216(3), pages 638-646.

    More about this item

    Keywords

    TU game; Banzhaf value; Second-order marginal contributions; Second-order payoffs; Amalgamation;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D60 - Microeconomics - - Welfare Economics - - - General

    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:53:y:2024:i:3:d:10.1007_s00182-024-00893-4. 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.