IDEAS home Printed from https://ideas.repec.org/a/spr/operea/v23y2023i1d10.1007_s12351-023-00742-4.html
   My bibliography  Save this article

Efficiently computing the Shapley value of connectivity games in low-treewidth graphs

Author

Listed:
  • Tom C. van der Zanden

    (Maastricht University)

  • Hans L. Bodlaender

    (Utrecht University)

  • Herbert J. M. Hamers

    (Tilburg University)

Abstract

The Shapley value is the solution concept in cooperative game theory that is most used in both theoretical and practical settings. Unfortunately, in general, computing the Shapley value is computationally intractable. This paper focuses on computing the Shapley value of (weighted) connectivity games. For these connectivity games, which are defined on an underlying (weighted) graph, computing the Shapley value is $$\#\textsf {P}$$ # P -hard, and thus (likely) intractable even for graphs with a moderate number of vertices. We present an algorithm that can efficiently compute the Shapley value if the underlying graph has bounded treewidth. Next, we apply our algorithm to several real-world (covert) networks. We show that our algorithm can quickly compute exact Shapley values for these networks, whereas in prior work these values could only be approximated using a heuristic method. Finally, it is demonstrated that our algorithm can also efficiently compute the Shapley value time for several larger (artificial) benchmark graphs from the PACE 2018 challenge.

Suggested Citation

  • Tom C. van der Zanden & Hans L. Bodlaender & Herbert J. M. Hamers, 2023. "Efficiently computing the Shapley value of connectivity games in low-treewidth graphs," Operational Research, Springer, vol. 23(1), pages 1-23, March.
    • Handle: RePEc:spr:operea:v:23:y:2023:i:1:d:10.1007_s12351-023-00742-4
    DOI: 10.1007/s12351-023-00742-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s12351-023-00742-4
    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/s12351-023-00742-4?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. Stefano Moretti & Fioravante Patrone, 2008. "Rejoinder on: Transversality of the Shapley value," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 16(1), pages 60-61, July.
    2. Stefano Moretti & Fioravante Patrone, 2008. "Transversality of the Shapley value," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 16(1), pages 1-41, July.
    3. Stefano Moretti & Fioravante Patrone & Stefano Bonassi, 2007. "The class of microarray games and the relevance index for genes," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 15(2), pages 256-280, December.
    4. S. C. Littlechild & G. Owen, 1973. "A Simple Expression for the Shapley Value in a Special Case," Management Science, INFORMS, vol. 20(3), pages 370-372, November.
    5. Lindelauf, R.H.A. & Hamers, H.J.M. & Husslage, B.G.M., 2013. "Cooperative game theoretic centrality analysis of terrorist networks: The cases of Jemaah Islamiyah and Al Qaeda," European Journal of Operational Research, Elsevier, vol. 229(1), pages 230-238.
    6. Hisao Tamaki, 2019. "Positive-instance driven dynamic programming for treewidth," Journal of Combinatorial Optimization, Springer, vol. 37(4), pages 1283-1311, May.
    7. Rafael Amer & José Miguel Giménez, 2004. "A connectivity game for graphs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 60(3), pages 453-470, December.
    8. Faigle, U & Kern, W, 1992. "The Shapley Value for Cooperative Games under Precedence Constraints," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(3), pages 249-266.
    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. Hamers, Herbert & Husslage, Bart & Lindelauf, R. & Campen, Tjeerd, 2016. "A New Approximation Method for the Shapley Value Applied to the WTC 9/11 Terrorist Attack," Other publications TiSEM 8a67b416-1091-4efe-a1a6-7, Tilburg University, School of Economics and Management.
    2. Hamers, Herbert & Husslage, Bart & Lindelauf, R. & Campen, Tjeerd, 2016. "A New Approximation Method for the Shapley Value Applied to the WTC 9/11 Terrorist Attack," Discussion Paper 2016-042, Tilburg University, Center for Economic Research.
    3. van den Brink, René & González-Arangüena, Enrique & Manuel, Conrado & del Pozo, Mónica, 2014. "Order monotonic solutions for generalized characteristic functions," European Journal of Operational Research, Elsevier, vol. 238(3), pages 786-796.
    4. Giulia Bernardi & Roberto Lucchetti, 2015. "Generating Semivalues via Unanimity Games," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 1051-1062, September.
    5. Béal, Sylvain & Moyouwou, Issofa & Rémila, Eric & Solal, Philippe, 2020. "Cooperative games on intersection closed systems and the Shapley value," Mathematical Social Sciences, Elsevier, vol. 104(C), pages 15-22.
    6. Béal, Sylvain & Ferrières, Sylvain & Rémila, Eric & Solal, Philippe, 2018. "The proportional Shapley value and applications," Games and Economic Behavior, Elsevier, vol. 108(C), pages 93-112.
    7. Encarnacion Algaba & Rene van den Brink, 2021. "Networks, Communication and Hierarchy: Applications to Cooperative Games," Tinbergen Institute Discussion Papers 21-019/IV, Tinbergen Institute.
    8. García-Martínez, Jose A. & Mayor-Serra, Antonio J. & Meca, Ana, 2023. "Efficient effort equilibrium in cooperation with pairwise cost reduction," Omega, Elsevier, vol. 121(C).
    9. Giulia Cesari & Roberto Lucchetti & Stefano Moretti, 2017. "Generalized additive games," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(4), pages 919-939, November.
    10. Stefano Moretti & Henk Norde, 2022. "Some new results on generalized additive games," International Journal of Game Theory, Springer;Game Theory Society, vol. 51(1), pages 87-118, March.
    11. Arroyo, Federico, 2024. "Cost Allocation in Vehicle Routing Problems with Time Windows," Junior Management Science (JUMS), Junior Management Science e. V., vol. 9(1), pages 1241-1268.
    12. Hougaard, Jens Leth & Moreno-Ternero, Juan D. & Tvede, Mich & Østerdal, Lars Peter, 2017. "Sharing the proceeds from a hierarchical venture," Games and Economic Behavior, Elsevier, vol. 102(C), pages 98-110.
    13. Kouladoum, Jean-Claude, 2019. "Décision du mariage des ménages tchadiens et Caractéristiques socio-économiques [Marriage decision of Chadian households and socio-economic characteristics]," MPRA Paper 91590, University Library of Munich, Germany.
    14. René van den Brink, 2017. "Games with a Permission Structure: a survey on generalizations and applications," Tinbergen Institute Discussion Papers 17-016/II, Tinbergen Institute.
    15. Perea, Federico & Puerto, Justo & Fernández, Francisco R., 2012. "Avoiding unfairness of Owen allocations in linear production processes," European Journal of Operational Research, Elsevier, vol. 220(1), pages 125-131.
    16. M. Fiestras-Janeiro & Ignacio García-Jurado & Manuel Mosquera, 2011. "Cooperative games and cost allocation problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(1), pages 1-22, July.
    17. Stefano Moretti & Fioravante Patrone, 2008. "Transversality of the Shapley value," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 16(1), pages 1-41, July.
    18. Roberto Roson & Franz Hubert, 2015. "Bargaining Power and Value Sharing in Distribution Networks: A Cooperative Game Theory Approach," Networks and Spatial Economics, Springer, vol. 15(1), pages 71-87, March.
    19. René Brink & P. Herings & Gerard Laan & A. Talman, 2015. "The Average Tree permission value for games with a permission tree," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 58(1), pages 99-123, January.
    20. 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.

    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:operea:v:23:y:2023:i:1:d:10.1007_s12351-023-00742-4. 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.