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Characterization of the Average Tree solution and its kernel

Author

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  • Béal, Sylvain
  • Rémila, Eric
  • Solal, Philippe
Abstract
In this article, we study cooperative games with limited cooperation possibilities, represented by a tree on the set of agents. Agents in the game can cooperate if they are connected in the tree. We first derive direct-sum decompositions of the space of TU-games on a fixed tree, and two new basis for these spaces of TU-games. We then focus our attention on the Average (rooted)-Tree solution (see Herings et al. (2008)). We provide a basis for its kernel and a new axiomatic characterization by using the classical axiom for inessential games, and two new axioms of invariance called Invariance with respect to irrelevant coalitions and Weighted addition invariance on bi-partitions. We also solve the inverse problem for the Average (rooted)-Tree solution.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:mateco:v:60:y:2015:i:c:p:159-165
    DOI: 10.1016/j.jmateco.2015.07.001
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    1. 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.
    2. Sylvain Béal & Eric Rémila & Philippe Solal, 2015. "Axioms of invariance for TU-games," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(4), pages 891-902, November.
    3. Debasis Mishra & A. Talman, 2010. "A characterization of the average tree solution for tree games," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 105-111, March.
    4. René Brink & Ilya Katsev & Gerard Laan, 2011. "Axiomatizations of two types of Shapley values for games on union closed systems," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 47(1), pages 175-188, May.
    5. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, August.
    6. 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.
    7. 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.
    8. 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.
    9. René van den Brink, 2009. "Comparable Axiomatizations of the Myerson Value, the Restricted Banzhaf Value, Hierarchical Outcomes and the Average Tree Solution for Cycle-Free Graph Restricted Games," Tinbergen Institute Discussion Papers 09-108/1, Tinbergen Institute.
    10. 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.
    11. van den Brink, René, 2012. "Efficiency and collusion neutrality in cooperative games and networks," Games and Economic Behavior, Elsevier, vol. 76(1), pages 344-348.
    12. Koji Yokote, 2015. "Weak addition invariance and axiomatization of the weighted Shapley value," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(2), pages 275-293, May.
    13. 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.
    14. 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.
    15. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2012. "Weighted component fairness for forest games," Mathematical Social Sciences, Elsevier, vol. 64(2), pages 144-151.
    16. Norman L. Kleinberg & Jeffrey H. Weiss, 1985. "Equivalent N -Person Games and the Null Space of the Shapley Value," Mathematics of Operations Research, INFORMS, vol. 10(2), pages 233-243, May.
    17. Biung-Ghi Ju & Junghum Park, 2012. "Hierarchical Outcomes and Collusion Neutrality," Working Paper Series no87, Institute of Economic Research, Seoul National University.
    18. Evans, Robert A., 1996. "Value, Consistency, and Random Coalition Formation," Games and Economic Behavior, Elsevier, vol. 12(1), pages 68-80, January.
    19. Sylvain Béal & Eric Rémila & Philippe Solal, 2014. "Decomposition of the space of TU-games, Strong Transfer Invariance and the Banzhaf value," Working Papers hal-01377929, HAL.
    20. 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.
    21. Mishra, D. & Talman, A.J.J., 2009. "A Characterization of the Average Tree Solution for Cycle-Free Graph Games," Discussion Paper 2009-17, Tilburg University, Center for Economic Research.
    22. Rene van den Brink & Ilya Katsev & Gerard van der Laan, 2011. "Games on Union Closed Systems," Tinbergen Institute Discussion Papers 11-036/1, Tinbergen Institute.
    23. Sylvain Béal & Eric Rémila & Philippe Solal, 2013. "A Decomposition of the Space of TU-games Using Addition and Transfer Invariance," Working Papers 2013-08, CRESE.
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    Citations

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    Cited by:

    1. Sylvain Béal & André Casajus & Eric Rémila & Philippe Solal, 2021. "Cohesive efficiency in TU-games: axiomatizations of variants of the Shapley value, egalitarian values and their convex combinations," Annals of Operations Research, Springer, vol. 302(1), pages 23-47, July.
    2. Sylvain Béal & Eric Rémila & Philippe Solal, 2015. "Discounted Tree Solutions," Working Papers hal-01377923, HAL.
    3. 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.
    4. Sylvain Béal & André Casajus & Frank Huettner & Eric Rémila & Philippe Solal, 2016. "Characterizations of weighted and equal division values," Theory and Decision, Springer, vol. 80(4), pages 649-667, April.
    5. Özer Selçuk & Takamasa Suzuki, 2023. "Comparable axiomatizations of the average tree solution and the Myerson value," International Journal of Game Theory, Springer;Game Theory Society, vol. 52(2), pages 333-362, June.

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    More about this item

    Keywords

    Average Tree solution; Direct-sum decomposition; Kernel; Weighted addition invariance on bi-partitions; Invariance to irrelevant coalitions; Inverse problem;
    All these keywords.

    JEL classification:

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

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