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Average tree solutions and the distribution of Harsanyi dividends

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  • Richard Baron

    ()

  • Sylvain Béal

    ()

  • Eric Rémila

    ()

  • Philippe Solal

    ()

Abstract

We consider communication situations games being the combination of a TU-game and a communication graph. We study the average tree (AT) solutions introduced by Herings \sl et al. [9] and [10]. The AT solutions are defined with respect to a set, say T, of rooted spanning trees of the communication graph. We characterize these solutions by efficiency, linearity and an axiom of T-hierarchy. Then we prove the following results. Firstly, the AT solution with respect to T is a Harsanyi solution if and only if T is a subset of the set of trees introduced in [10]. Secondly, the latter set is constructed by the classical DFS algorithm and the associated AT solution coincides with the Shapley value when the communication graph is complete. Thirdly, the AT solution with respect to trees constructed by the other classical algorithm BFS yields the equal surplus division when the communication graph is complete.

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Bibliographic Info

Article provided by Springer in its journal International Journal of Game Theory.

Volume (Year): 40 (2011)
Issue (Month): 2 (May)
Pages: 331-349

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Handle: RePEc:spr:jogath:v:40:y:2011:i:2:p:331-349

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Related research

Keywords: Communication situations; Average tree solutions; Harsanyi solutions; DFS ; BFS ; Shapley value; Equal surplus division;

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References

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  1. Fabien Lange & Michel Grabisch, 2006. "Values on regular games under Kirchhoff’s laws," Working Paper Series 0807, Óbuda University, Keleti Faculty of Business and Management, revised Nov 2008.
  2. René Brink & Gerard Laan & Valeri Vasil’ev, 2007. "Component efficient solutions in line-graph games with applications," Economic Theory, Springer, vol. 33(2), pages 349-364, November.
  3. René van den Brink, 2006. "On Hierarchies and Communication," Tinbergen Institute Discussion Papers 06-056/1, Tinbergen Institute.
  4. Herings, P.J.J. & Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 2010. "The average tree solution for cooperative games with communication structure," Open Access publications from Tilburg University urn:nbn:nl:ui:12-3736837, Tilburg University.
  5. Rene van den Brink & Ilya Katsev & Gerard van der Laan, 2009. "Axiomatizations of Two Types of Shapley Values for Games on Union Closed Systems," Tinbergen Institute Discussion Papers 09-064/1, Tinbergen Institute.
  6. HERINGS, P. Jean-Jacques & van der LAAN, Gerard & TALMAN, Dolf, . "The average tree solution for cycle-free graph games," CORE Discussion Papers RP -2155, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  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. Faigle, U & Kern, W, 1992. "The Shapley Value for Cooperative Games under Precedence Constraints," International Journal of Game Theory, Springer, vol. 21(3), pages 249-66.
  9. 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.
  10. Borm, P.E.M. & Owen, G. & Tijs, S.H., 1992. "On the position value for communication situations," Open Access publications from Tilburg University urn:nbn:nl:ui:12-154855, Tilburg University.
  11. Gilles, R.P. & Owen, G. & Brink, J.R. van den, 1991. "Games with permission structures: The conjunctive approach," Discussion Paper 1991-14, Tilburg University, Center for Economic Research.
  12. Ren� 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.
  13. Gabrielle Demange, 2004. "On group stability in hierarchies and networks," Post-Print halshs-00581662, HAL.
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Cited by:
  1. repec:hal:cesptp:hal-00803233 is not listed on IDEAS
  2. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2009. "Weighted Component Fairness for Forest Games," MPRA Paper 17455, University Library of Munich, Germany.
  3. Huseynov, T. & Talman, A.J.J., 2012. "The Communication Tree Value for TU-games with Graph Communication," Discussion Paper 2012-095, Tilburg University, Center for Economic Research.
  4. Sylvain Béal & Eric Rémila & Philippe Solal, 2012. "Compensations in the Shapley value and the compensation solutions for graph games," International Journal of Game Theory, Springer, vol. 41(1), pages 157-178, February.
  5. Suzuki, T. & Talman, A.J.J., 2011. "Solution Concepts for Cooperative Games with Circular Communication Structure," Discussion Paper 2011-100, Tilburg University, Center for Economic Research.
  6. Napel, Stefan & Nohn, Andreas & Alonso-Meijide, José Maria, 2012. "Monotonicity of power in weighted voting games with restricted communication," Mathematical Social Sciences, Elsevier, vol. 64(3), pages 247-257.

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