IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v77y2013i1p65-99.html
   My bibliography  Save this article

Two-step coalition values for multichoice games

Author

Listed:
  • Michael Jones

    ()

  • Jennifer Wilson

    ()

Abstract

We introduce and compare several coalition values for multichoice games. Albizuri defined coalition structures and an extension of the Owen coalition value for multichoice games using the average marginal contribution of a player over a set of orderings of the player’s representatives. Following an approach used for cooperative games, we introduce a set of nested or two-step coalition values on multichoice games which measure the value of each coalition and then divide this among the players in the coalition using either a Shapley or Banzhaf value at each step. We show that when a Shapley value is used in both steps, the resulting coalition value coincides with that of Albizuri. We axiomatize the three new coalition values and show that each set of axioms, including that of Albizuri, is independent. Further we show how the multilinear extension can be used to compute the coalition values. We conclude with a brief discussion about the applicability of the different values. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Michael Jones & Jennifer Wilson, 2013. "Two-step coalition values for multichoice games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(1), pages 65-99, February.
  • Handle: RePEc:spr:mathme:v:77:y:2013:i:1:p:65-99
    DOI: 10.1007/s00186-012-0415-4
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00186-012-0415-4
    Download Restriction: Access to full text is restricted to subscribers.

    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. repec:spr:compst:v:66:y:2007:i:2:p:255-261 is not listed on IDEAS
    2. Winter, Eyal, 1992. "The consistency and potential for values of games with coalition structure," Games and Economic Behavior, Elsevier, vol. 4(1), pages 132-144, January.
    3. M. Fiestras-Janeiro & Ignacio García-Jurado & Manuel Mosquera, 2011. "Rejoinder on: 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 33-34, July.
    4. Philip Straffin, 1977. "Homogeneity, independence, and power indices," Public Choice, Springer, vol. 30(1), pages 107-118, June.
    5. Michel Grabisch & Fabien Lange, 2007. "Games on lattices, multichoice games and the shapley value: a new approach," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(1), pages 153-167, February.
    6. Owen, Guillermo & Winter, Eyal, 1992. "The multilinear extension and the coalition structure value," Games and Economic Behavior, Elsevier, vol. 4(4), pages 582-587, October.
    7. Amer, Rafael & Carreras, Francese & Gimenez, Jose Miguel, 2002. "The modified Banzhaf value for games with coalition structure: an axiomatic characterization," Mathematical Social Sciences, Elsevier, vol. 43(1), pages 45-54, January.
    8. repec:spr:compst:v:65:y:2007:i:1:p:153-167 is not listed on IDEAS
    9. 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.
    10. J. Alonso-Meijide & F. Carreras & M. Fiestras-Janeiro, 2005. "The Multilinear Extension and the Symmetric Coalition Banzhaf Value," Theory and Decision, Springer, vol. 59(2), pages 111-126, September.
    11. Vito Fragnelli & Anna Iandolino, 2004. "A cost allocation problem in urban solid wastes collection and disposal," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 59(3), pages 447-463, July.
    12. José Alonso-Meijide & M. Fiestras-Janeiro, 2002. "Modification of the Banzhaf Value for Games with a Coalition Structure," Annals of Operations Research, Springer, vol. 109(1), pages 213-227, January.
    13. repec:spr:compst:v:72:y:2010:i:1:p:145-169 is not listed on IDEAS
    14. M. Albizuri & Jesus Aurrekoetxea, 2006. "Coalition Configurations and the Banzhaf Index," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 26(3), pages 571-596, June.
    15. Hart, Sergiu & Kurz, Mordecai, 1983. "Endogenous Formation of Coalitions," Econometrica, Econometric Society, vol. 51(4), pages 1047-1064, July.
    16. repec:hal:journl:halshs-00178916 is not listed on IDEAS
    17. Guillermo Owen, 1972. "Multilinear Extensions of Games," Management Science, INFORMS, vol. 18(5-Part-2), pages 64-79, January.
    18. Derks, Jean & Peters, Hans, 1993. "A Shapley Value for Games with Restricted Coalitions," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(4), pages 351-360.
    19. Amer, Rafael & Giménez, José Miguel, 2008. "A general procedure to compute mixed modified semivalues for cooperative games with structure of coalition blocks," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 269-282, September.
    20. Hsiao Chih-Ru & Raghavan T. E. S., 1993. "Shapley Value for Multichoice Cooperative Games, I," Games and Economic Behavior, Elsevier, vol. 5(2), pages 240-256, April.
    21. M. Albizuri, 2009. "The multichoice coalition value," Annals of Operations Research, Springer, vol. 172(1), pages 363-374, November.
    22. Michael Jones & Jennifer Wilson, 2010. "Multilinear extensions and values for multichoice games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 72(1), pages 145-169, August.
    23. Vazquez-Brage, M. & van den Nouweland, A. & Garcia-Jurado, I., 1997. "Owen's coalitional value and aircraft landing fees," Mathematical Social Sciences, Elsevier, vol. 34(3), pages 273-286, October.
    24. Anna Khmelnitskaya & Elena Yanovskaya, 2007. "Owen coalitional value without additivity axiom," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(2), pages 255-261, October.
    25. repec:wsi:igtrxx:v:11:y:2009:i:02:n:s0219198909002261 is not listed on IDEAS
    Full references (including those not matched with items on IDEAS)

    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:mathme:v:77:y:2013:i:1:p:65-99. 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: (Sonal Shukla) or (Rebekah McClure). General contact details of provider: http://www.springer.com .

    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 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.

    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.