IDEAS home Printed from https://ideas.repec.org/p/hal/cesptp/halshs-01659804.html
   My bibliography  Save this paper

Convexity of Graph-Restricted Games Induced by Minimum Partitions

Author

Listed:
  • Alexandre Skoda

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

Abstract

We consider restricted games on weighted graphs associated with minimum partitions. We replace in the classical definition of Myerson restricted game the connected components of any subgraph by the subcomponents corresponding to a minimum partition. This minimum partition Pmin is induced by the deletion of the minimum weight edges. We provide five necessary conditions on the graph edge-weights to have inheritance of convexity from the underlying game to the restricted game associated with Pmin. Then, we establish that these conditions are also sufficient for a weaker condition, called F-convexity, obtained by restriction of convexity to connected subsets. Moreover, we prove that inheritance of convexity for Myerson restricted game associated with a given graph G is equivalent to inheritance of F-convexity for the Pmin-restricted game associated with a particular weighted graph G' built from G by adding a dominating vertex, and with only two different edge-weights. Then, we prove that G is cycle-complete if and only if a specific condition on adjacent cycles is satisfied on G'.

Suggested Citation

  • Alexandre Skoda, 2017. "Convexity of Graph-Restricted Games Induced by Minimum Partitions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01659804, HAL.
    • Handle: RePEc:hal:cesptp:halshs-01659804
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-01659804
    as

    Download full text from publisher

    File URL: https://shs.hal.science/halshs-01659804/document
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Annals of Operations Research, Springer, vol. 204(1), pages 33-64, April.
    2. 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.
    3. M. Grabisch & A. Skoda, 2012. "Games induced by the partitioning of a graph," Annals of Operations Research, Springer, vol. 201(1), pages 229-249, December.
    4. repec:hal:pseose:hal-00830291 is not listed on IDEAS
    5. Faigle, U. & Grabisch, M. & Heyne, M., 2010. "Monge extensions of cooperation and communication structures," European Journal of Operational Research, Elsevier, vol. 206(1), pages 104-110, October.
    6. EDMONDS, Jack & GILES, Rick, 1977. "A min-max relation for submodular functions on graphs," LIDAM Reprints CORE 301, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. repec:hal:pseose:hal-00803233 is not listed on IDEAS
    8. Alexandre Skoda, 2017. "Inheritance of Convexity for the Pmin-Restricted Game," Documents de travail du Centre d'Economie de la Sorbonne 17051, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    9. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, August.
    10. Alexandre Skoda, 2017. "Inheritance of convexity for partition restricted games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01487381, HAL.
    11. 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.
    12. Alexandre Skoda, 2017. "Inheritance of convexity for partition restricted games," Post-Print halshs-01487381, HAL.
    13. van den Nouweland, Anne & Borm, Peter, 1991. "On the Convexity of Communication Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 19(4), pages 421-430.
    14. E. Algaba & J.M. Bilbao & J.J. López, 2001. "A unified approach to restricted games," Theory and Decision, Springer, vol. 50(4), pages 333-345, June.
    15. Alexandre Skoda, 2016. "Complexity of inheritance of F-convexity for restricted games induced by minimum partitions," Post-Print halshs-01382502, HAL.
    16. Alexandre Skoda, 2016. "Complexity of inheritance of F-convexity for restricted games induced by minimum partitions," Documents de travail du Centre d'Economie de la Sorbonne 16055, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    17. Alexandre Skoda, 2016. "Convexity of Network Restricted Games Induced by Minimum Partitions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01305005, HAL.
    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. Alexandre Skoda, 2019. "Convexity of graph-restricted games induced by minimum partitions," Post-Print halshs-01617023, HAL.
    2. Alexandre Skoda, 2017. "Convexity of Graph-Restricted Games Induced by Minimum Partitions," Post-Print halshs-01659804, HAL.
    3. Alexandre Skoda, 2019. "Convexity of graph-restricted games induced by minimum partitions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01617023, HAL.
    4. Alexandre Skoda, 2017. "Convexity of Graph-Restricted Games Induced by Minimum Partitions," Documents de travail du Centre d'Economie de la Sorbonne 17049, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    5. Alexandre Skoda, 2020. "Inheritance of Convexity for the P˜min-Restricted Game," Post-Print halshs-02967120, HAL.
    6. Alexandre Skoda, 2020. "Inheritance of Convexity for the P˜min-Restricted Game," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-02967120, HAL.
    7. Alexandre Skoda, 2020. "Inheritance of Convexity for the P˜min-Restricted Game," Documents de travail du Centre d'Economie de la Sorbonne 20020, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    8. Alexandre Skoda, 2016. "Inheritance of Convexity for Partition Restricted," Documents de travail du Centre d'Economie de la Sorbonne 16040, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    9. Alexandre Skoda, 2016. "Inheritance of Convexity for Partition Restricted Games," Post-Print halshs-01318105, HAL.
    10. Alexandre Skoda, 2016. "Inheritance of Convexity for Partition Restricted Games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01318105, HAL.
    11. Alexandre Skoda, 2017. "Inheritance of Convexity for the P min-Restricted Game," Post-Print halshs-01660670, HAL.
    12. Alexandre Skoda, 2017. "Inheritance of Convexity for the P min-Restricted Game," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01660670, HAL.
    13. Alexandre Skoda, 2017. "Inheritance of Convexity for the Pmin-Restricted Game," Documents de travail du Centre d'Economie de la Sorbonne 17051, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    14. 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.
    15. Alexandre Skoda, 2016. "Complexity of inheritance of F-convexity for restricted games induced by minimum partitions," Post-Print halshs-01382502, HAL.
    16. Alexandre Skoda, 2016. "Complexity of inheritance of F-convexity for restricted games induced by minimum partitions," Documents de travail du Centre d'Economie de la Sorbonne 16055, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    17. A. Skoda, 2021. "Inheritance of convexity for the $$\mathcal {P}_{\min }$$ P min -restricted game," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(1), pages 1-32, February.
    18. 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.
    19. Alexandre Skoda, 2016. "Convexity of Network Restricted Games Induced by Minimum Partitions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01305005, HAL.
    20. Alexandre Skoda, 2016. "Convexity of Network Restricted Games Induced by Minimum Partitions," Post-Print hal-01305005, HAL.

    More about this item

    Keywords

    communication networks; cooperative game; convex game; restricted game; partitions;
    All these keywords.

    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:hal:cesptp:halshs-01659804. 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: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    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.