IDEAS home Printed from https://ideas.repec.org/p/pra/mprapa/20955.html
   My bibliography  Save this paper

Compensations in the Shapley value and the compensation solutions for graph games

Author

Listed:
  • Béal, Sylvain
  • Rémila, Eric
  • Solal, Philippe

Abstract

We consider an alternative expression of the Shapley value that reveals a system of compensations: each player receives an equal share of the worth of each coalition he belongs to, and has to compensate an equal share of the worth of any coalition he does not belong to. We give an interpretation in terms of formation of the grand coalition according to an ordering of the players and define the corresponding compensation vector. Then, we generalize this idea to cooperative games with a communication graph. Firstly, we consider cooperative games with a forest (cycle-free graph). We extend the compensation vector by considering all rooted spanning trees of the forest (see Demange 2004) instead of orderings of the players. The associated allocation rule, called the compensation solution, is characterized by component efficiency and relative fairness. The latter axiom takes into account the relative position of a player with respect to his component. Secondly, we consider cooperative games with arbitrary graphs and construct rooted spanning trees by using the classical algorithms DFS and BFS. If the graph is complete, we show that the compensation solutions associated with DFS and BFS coincide with the Shapley value and the equal surplus division respectively.

Suggested Citation

  • Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2010. "Compensations in the Shapley value and the compensation solutions for graph games," MPRA Paper 20955, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:20955
    as

    Download full text from publisher

    File URL: https://mpra.ub.uni-muenchen.de/20955/1/MPRA_paper_20955.pdf
    File Function: original version
    Download Restriction: no
    ---><---

    Other versions of this item:

    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. Gabrielle Demange, 2004. "On Group Stability in Hierarchies and Networks," Journal of Political Economy, University of Chicago Press, vol. 112(4), pages 754-778, August.
    3. Pierre Dehez & Daniela Tellone, 2013. "Data Games: Sharing Public Goods with Exclusion," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 15(4), pages 654-673, August.
    4. Herings, P.J.J. & van der Laan, G. & Talman, A.J.J. & Yang, Z., 2010. "The average tree solution for cooperative games with communication structure," Games and Economic Behavior, Elsevier, vol. 68(2), pages 626-633, March.
    5. Ernst Fehr & Klaus M. Schmidt, 1999. "A Theory of Fairness, Competition, and Cooperation," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 114(3), pages 817-868.
    6. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2010. "Rooted-tree solutions for tree games," European Journal of Operational Research, Elsevier, vol. 203(2), pages 404-408, June.
    7. van den Brink, Rene, 2007. "Null or nullifying players: The difference between the Shapley value and equal division solutions," Journal of Economic Theory, Elsevier, vol. 136(1), pages 767-775, September.
    8. René Brink & Gerard Laan & Valeri Vasil’ev, 2007. "Component efficient solutions in line-graph games with applications," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 33(2), pages 349-364, November.
    9. Herings, P. Jean Jacques & van der Laan, Gerard & Talman, Dolf, 2008. "The average tree solution for cycle-free graph games," Games and Economic Behavior, Elsevier, vol. 62(1), pages 77-92, January.
    10. Richard Baron & Sylvain Béal & Eric Rémila & Philippe Solal, 2011. "Average tree solutions and the distribution of Harsanyi dividends," International Journal of Game Theory, Springer;Game Theory Society, vol. 40(2), pages 331-349, May.
    11. Aadland, David & Kolpin, Van, 2004. "Environmental determinants of cost sharing," Journal of Economic Behavior & Organization, Elsevier, vol. 53(4), pages 495-511, April.
    12. Evans, Robert A., 1996. "Value, Consistency, and Random Coalition Formation," Games and Economic Behavior, Elsevier, vol. 12(1), pages 68-80, January.
    13. Anna Khmelnitskaya, 2010. "Values for rooted-tree and sink-tree digraph games and sharing a river," Theory and Decision, Springer, vol. 69(4), pages 657-669, October.
    14. Teck-Hua Ho & Xuanming Su, 2009. "Peer-Induced Fairness in Games," American Economic Review, American Economic Association, vol. 99(5), pages 2022-2049, December.
    15. Norman L. Kleinberg & Jeffrey H. Weiss, 1986. "The Orthogonal Decomposition of Games and an Averaging Formula for the Shapley Value," Mathematics of Operations Research, INFORMS, vol. 11(1), pages 117-124, February.
    16. Aadland, David & Kolpin, Van, 1998. "Shared irrigation costs: An empirical and axiomatic analysis," Mathematical Social Sciences, Elsevier, vol. 35(2), pages 203-218, March.
    17. Ruiz, Luis M. & Valenciano, Federico & Zarzuelo, Jose M., 1998. "The Family of Least Square Values for Transferable Utility Games," Games and Economic Behavior, Elsevier, vol. 24(1-2), pages 109-130, July.
    18. Aadland, David & Kolpin, Van, 2004. "Erratum to "Environmental determinants of cost sharing"," Journal of Economic Behavior & Organization, Elsevier, vol. 55(1), pages 105-121, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Sylvain Béal & André Casajus & Frank Huettner, 2018. "Efficient extensions of communication values," Annals of Operations Research, Springer, vol. 264(1), pages 41-56, May.
    2. Sylvain Béal & Anna Khmelnitskaya & Philippe Solal, 2018. "Two-step values for games with two-level communication structure," Journal of Combinatorial Optimization, Springer, vol. 35(2), pages 563-587, February.
    3. Ramón Flores & Elisenda Molina & Juan Tejada, 2014. "Pyramidal values," Annals of Operations Research, Springer, vol. 217(1), pages 233-252, June.
    4. de Clippel, Geoffroy, 2018. "Membership separability: A new axiomatization of the Shapley value," Games and Economic Behavior, Elsevier, vol. 108(C), pages 125-129.
    5. Sylvain Béal & Eric Rémila & Philippe Solal, 2022. "Allocation rules for cooperative games with restricted communication and a priori unions based on the Myerson value and the average tree solution," Journal of Combinatorial Optimization, Springer, vol. 43(4), pages 818-849, May.

    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. Richard Baron & Sylvain Béal & Eric Rémila & Philippe Solal, 2011. "Average tree solutions and the distribution of Harsanyi dividends," International Journal of Game Theory, Springer;Game Theory Society, vol. 40(2), pages 331-349, May.
    2. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2015. "Characterization of the Average Tree solution and its kernel," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 159-165.
    3. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2012. "Weighted component fairness for forest games," Mathematical Social Sciences, Elsevier, vol. 64(2), pages 144-151.
    4. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Annals of Operations Research, Springer, vol. 204(1), pages 33-64, April.
    5. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2012. "The sequential equal surplus division for sharing a river," MPRA Paper 37346, University Library of Munich, Germany.
    6. Sylvain Béal & Amandine Ghintran & Eric Rémila & Philippe Solal, 2015. "The sequential equal surplus division for rooted forest games and an application to sharing a river with bifurcations," Theory and Decision, Springer, vol. 79(2), pages 251-283, September.
    7. repec:hal:pseose:hal-00803233 is not listed on IDEAS
    8. 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.
    9. Khmelnitskaya, Anna & Talman, Dolf, 2014. "Tree, web and average web values for cycle-free directed graph games," European Journal of Operational Research, Elsevier, vol. 235(1), pages 233-246.
    10. Sylvain Béal & Eric Rémila & Philippe Solal, 2022. "Allocation rules for cooperative games with restricted communication and a priori unions based on the Myerson value and the average tree solution," Journal of Combinatorial Optimization, Springer, vol. 43(4), pages 818-849, May.
    11. Sylvain Béal & Amandine Ghintran & Eric Rémila & Philippe Solal, 2012. "The Sequential Equal Surplus Division for Sharing International Rivers with Bifurcations," Working Papers 2012-02, CRESE.
    12. S. Béal & A. Lardon & E. Rémila & P. Solal, 2012. "The average tree solution for multi-choice forest games," Annals of Operations Research, Springer, vol. 196(1), pages 27-51, July.
    13. Anna Khmelnitskaya & Özer Selçuk & Dolf Talman, 2020. "The average covering tree value for directed graph games," Journal of Combinatorial Optimization, Springer, vol. 39(2), pages 315-333, February.
    14. Huseynov, T. & Talman, A.J.J., 2012. "The Communication Tree Value for TU-games with Graph Communication," Other publications TiSEM 6ba97d87-1ac6-4af7-a981-a, Tilburg University, School of Economics and Management.
    15. van den Brink, René & van der Laan, Gerard & Moes, Nigel, 2012. "Fair agreements for sharing international rivers with multiple springs and externalities," Journal of Environmental Economics and Management, Elsevier, vol. 63(3), pages 388-403.
    16. László Á. Kóczy, 2018. "Partition Function Form Games," Theory and Decision Library C, Springer, number 978-3-319-69841-0, September.
    17. Khmelnitskaya, A. & Talman, A.J.J., 2010. "Tree-Type Values for Cycle-Free Directed Graph Games," Discussion Paper 2010-113, Tilburg University, Center for Economic Research.
    18. Sylvain Béal & Eric Rémila & Philippe Solal, 2015. "Discounted Tree Solutions," Working Papers hal-01377923, HAL.
    19. van den Brink, René & van der Laan, Gerard & Moes, Nigel, 2013. "A strategic implementation of the Average Tree solution for cycle-free graph games," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2737-2748.
    20. Khmelnitskaya, A. & Talman, A.J.J., 2011. "Two solution concepts for TU games with cycle-free directed cooperation structures," Other publications TiSEM df61db11-45de-42b3-91e6-9, Tilburg University, School of Economics and Management.
    21. Wu, Hao & van den Brink, René & Estévez-Fernández, Arantza, 2024. "Highway toll allocation," Transportation Research Part B: Methodological, Elsevier, vol. 180(C).

    More about this item

    Keywords

    Shapley value ; compensations ; relative fairness ; compensation solution ; DFS ; BFS ; equal surplus division;
    All these keywords.

    JEL classification:

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

    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:pra:mprapa:20955. 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: Joachim Winter (email available below). General contact details of provider: https://edirc.repec.org/data/vfmunde.html .

    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.