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The position value and the structures of graphs

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  • Li, Daniel Li
  • Shan, Erfang

Abstract

The position value is an allocation rule based on the Shapley value of the link game from the original communication situation, in which cooperation is restricted by a graph. In the link games, feasible coalitions are connected but their structures are ignored. We introduce structure functions to describe the structures of connected sets, and generalize the link game and the position value to the setting with local structures. We modify an axiomatic characterization for the position value by Slikker to the generalized position value by component efficiency and Balanced link contributions on local structures.

Suggested Citation

  • Li, Daniel Li & Shan, Erfang, 2019. "The position value and the structures of graphs," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 190-197.
    • Handle: RePEc:eee:apmaco:v:356:y:2019:i:c:p:190-197
    DOI: 10.1016/j.amc.2019.03.041
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    References listed on IDEAS

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    1. Marco Slikker, 2005. "A characterization of the position value," International Journal of Game Theory, Springer;Game Theory Society, vol. 33(4), pages 505-514, November.
    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. Borm, P.E.M. & Owen, G. & Tijs, S.H., 1992. "On the position value for communication situations," Other publications TiSEM 5a8473e4-1df7-42df-ad53-f, Tilburg University, School of Economics and Management.
    4. Jackson, Matthew O. & Wolinsky, Asher, 1996. "A Strategic Model of Social and Economic Networks," Journal of Economic Theory, Elsevier, vol. 71(1), pages 44-74, October.
    5. Takumi Kongo, 2010. "Difference between the position value and the Myerson value is due to the existence of coalition structures," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(4), pages 669-675, October.
    6. Yoshio Kamijo & Takumi Kongo, 2010. "Axiomatization of the Shapley value using the balanced cycle contributions property," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(4), pages 563-571, October.
    7. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, August.
    8. Daniel Gómez & Enrique Gonz{'a}lez-Arangüena & Conrado Manuel & Guillermo Owen & Monica Del Pozo, 2004. "A Unified Approach To The Myerson Value And The Position Value," Theory and Decision, Springer, vol. 56(2_2), pages 63-76, February.
    9. van den Nouweland, Anne & Slikker, Marco, 2012. "An axiomatic characterization of the position value for network situations," Mathematical Social Sciences, Elsevier, vol. 64(3), pages 266-271.
    10. 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.
    11. Daniel Gómez & Enrique Gonz{’a}lez-Arangüena & Conrado Manuel & Guillermo Owen & Monica Del Pozo, 2004. "A Unified Approach To The Myerson Value And The Position Value," Theory and Decision, Springer, vol. 56(1), pages 63-76, April.
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    More about this item

    Keywords

    TU game; Graph; Position value; Link game; Graph structure;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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