IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v340y2024i2d10.1007_s10479-024-06082-6.html
   My bibliography  Save this article

Positivity and convexity in incomplete cooperative games

Author

Listed:
  • Martin Černý

    (Charles University)

  • Jan Bok

    (Charles University)

  • David Hartman

    (Charles University
    Czech Academy of Sciences)

  • Milan Hladík

    (Charles University)

Abstract

Incomplete cooperative games generalize the classical model of cooperative games by omitting the values of some of the coalitions. This allows for incorporating uncertainty into the model and studying the underlying games and possible payoff distributions based only on the partial information. In this paper, we conduct a systematic investigation of incomplete games, focusing on two important classes: positive and convex games. Regarding positivity, we generalize previous results from a special class of minimal incomplete games to a general setting. We characterize the non-extendability to a positive game by the existence of a certificate and provide a description of the set of positive extensions using its extreme games. These results also enable the construction of explicit formulas for several classes of incomplete games with special structures. The second part deals with convexity. We begin with the case of non-negative, minimal incomplete games. We establish the connection between incomplete games and the problem of completing partial functions and, consequently, provide a characterization of extendability and a full description of the set of symmetric convex extensions. This set serves as an approximation of the set of convex extensions.

Suggested Citation

  • Martin Černý & Jan Bok & David Hartman & Milan Hladík, 2024. "Positivity and convexity in incomplete cooperative games," Annals of Operations Research, Springer, vol. 340(2), pages 785-809, September.
    • Handle: RePEc:spr:annopr:v:340:y:2024:i:2:d:10.1007_s10479-024-06082-6
    DOI: 10.1007/s10479-024-06082-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-024-06082-6
    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/s10479-024-06082-6?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. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, August.
    2. Robert P. Gilles, 2010. "The Cooperative Game Theory of Networks and Hierarchies," Theory and Decision Library C, Springer, number 978-3-642-05282-8, December.
    3. Aymeric Lardon, 2017. "Endogenous interval games in oligopolies and the cores," Annals of Operations Research, Springer, vol. 248(1), pages 345-363, January.
    4. Satoshi Masuya & Masahiro Inuiguchi, 2016. "A fundamental study for partially defined cooperative games," Fuzzy Optimization and Decision Making, Springer, vol. 15(3), pages 281-306, September.
    5. Michel Grabisch, 2016. "Set Functions, Games and Capacities in Decision Making," Theory and Decision Library C, Springer, number 978-3-319-30690-2, December.
    6. O. Palancı & S. Alparslan Gök & G. Weber, 2014. "Cooperative games under bubbly uncertainty," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 80(2), pages 129-137, October.
    7. Elisenda Molina & Juan Tejada & Tom Weiss, 2022. "Some game theoretic marketing attribution models," Annals of Operations Research, Springer, vol. 318(2), pages 1043-1075, November.
    8. S. Alparslan-Gök & Silvia Miquel & Stef Tijs, 2009. "Cooperation under interval uncertainty," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(1), pages 99-109, March.
    9. E. Algaba & J. M. Bilbao & P. Borm & J. J. López, 2000. "The position value for union stable systems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 52(2), pages 221-236, November.
    10. M. Josune Albizuri & Satoshi Masuya & José M. Zarzuelo, 2022. "Characterization of a value for games under restricted cooperation," Annals of Operations Research, Springer, vol. 318(2), pages 773-785, November.
    11. Lloyd S. Shapley, 1967. "On balanced sets and cores," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 14(4), pages 453-460.
    12. Willson, Stephen J, 1993. "A Value for Partially Defined Cooperative Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(4), pages 371-384.
    13. Hashem Omrani & Khatereh Shafaat & Arash Alizadeh, 2019. "Integrated data envelopment analysis and cooperative game for evaluating energy efficiency of transportation sector: a case of Iran," Annals of Operations Research, Springer, vol. 274(1), pages 471-499, March.
    14. repec:spr:thdchp:978-3-319-30690-2_2 is not listed on IDEAS
    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. Jan Bok & Martin Černý, 2024. "1-convex extensions of incomplete cooperative games and the average value," Theory and Decision, Springer, vol. 96(2), pages 239-268, March.
    2. Rene van den Brink & Ilya Katsev & Gerard van der Laan, 2023. "Properties of Solutions for Games on Union-Closed Systems," Mathematics, MDPI, vol. 11(4), pages 1-16, February.
    3. 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.
    4. Satoshi Masuya, 2023. "Two Approaches to Estimate the Shapley Value for Convex Partially Defined Games," Mathematics, MDPI, vol. 12(1), pages 1-15, December.
    5. Martin Cerny & Michel Grabisch, 2023. "Player-centered incomplete cooperative games," Documents de travail du Centre d'Economie de la Sorbonne 23006, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    6. Encarnacion Algaba & Rene van den Brink, 2019. "The Shapley Value and Games with Hierarchies," Tinbergen Institute Discussion Papers 19-064/II, Tinbergen Institute.
    7. M. Josune Albizuri & Satoshi Masuya & José M. Zarzuelo, 2022. "Characterization of a value for games under restricted cooperation," Annals of Operations Research, Springer, vol. 318(2), pages 773-785, November.
    8. Alexandre Skoda, 2016. "Convexity of Network Restricted Games Induced by Minimum Partitions," Documents de travail du Centre d'Economie de la Sorbonne 16019, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    9. Sridhar Mandyam & Usha Sridhar, 2017. "DON and Shapley Value for Allocation among Cooperating Agents in a Network: Conditions for Equivalence," Studies in Microeconomics, , vol. 5(2), pages 143-161, December.
    10. Alexandre Skoda, 2019. "Convexity of graph-restricted games induced by minimum partitions," Post-Print halshs-01617023, HAL.
    11. Tobias Hiller, 2021. "Hierarchy and the size of a firm," International Review of Economics, Springer;Happiness Economics and Interpersonal Relations (HEIRS), vol. 68(3), pages 389-404, September.
    12. E. Algaba & J. Bilbao & R. Brink, 2015. "Harsanyi power solutions for games on union stable systems," Annals of Operations Research, Springer, vol. 225(1), pages 27-44, February.
    13. Daniel Li Li & Erfang Shan, 2024. "A new value for communication situations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(2), pages 535-551, October.
    14. Alexandre Skoda, 2017. "Inheritance of Convexity for the P min-Restricted Game," Post-Print halshs-01660670, HAL.
    15. Takashi Ui & Hiroyuki Kojima & Atsushi Kajii, 2011. "The Myerson value for complete coalition structures," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 74(3), pages 427-443, December.
    16. André Casajus, 2007. "The position value is the Myerson value, in a sense," International Journal of Game Theory, Springer;Game Theory Society, vol. 36(1), pages 47-55, September.
    17. Encarnaciön Algaba & Sylvain Béal & Eric Rémila & Phillippe Solal, 2018. "Harsanyi power solutions for cooperative games on voting structures," Working Papers 2018-05, CRESE.
    18. Erfang Shan & Guang Zhang & Xiaokang Shan, 2018. "The degree value for games with communication structure," International Journal of Game Theory, Springer;Game Theory Society, vol. 47(3), pages 857-871, September.
    19. Demange, Gabrielle, 2024. "Stable outcomes in simple cooperative games," Journal of Mathematical Economics, Elsevier, vol. 111(C).
    20. Alexandre Skoda, 2016. "Complexity of inheritance of F-convexity for restricted games induced by minimum partitions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01382502, HAL.

    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:annopr:v:340:y:2024:i:2:d:10.1007_s10479-024-06082-6. 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.