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Average tree solutions and the distribution of Harsanyi dividends

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

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

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 17909.

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Date of creation: 04 Sep 2009
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Handle: RePEc:pra:mprapa:17909

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Keywords: Communication situations ; average tree solution ; Harsanyi solutions ; DFS ; BFS} ; Shapley value ; equal surplus division;

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  1. Fabien Lange & Michel Grabisch, 2009. "Values on regular games under Kirchhoff's laws," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers), HAL halshs-00496553, HAL.
  2. 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, Tinbergen Institute 09-108/1, Tinbergen Institute.
  3. Gilles, R.P. & Owen, G. & Brink, J.R. van den, 1991. "Games with permission structures: The conjunctive approach," Discussion Paper, Tilburg University, Center for Economic Research 1991-14, Tilburg University, Center for Economic Research.
  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, Elsevier, vol. 68(2), pages 626-633, March.
  5. René Brink & Ilya Katsev & Gerard Laan, 2011. "Axiomatizations of two types of Shapley values for games on union closed systems," Economic Theory, Springer, Springer, vol. 47(1), pages 175-188, May.
  6. Gabrielle Demange, 2004. "On group stability in hierarchies and networks," Post-Print, HAL halshs-00581662, HAL.
  7. Mishra, D. & Talman, A.J.J., 2010. "A characterization of the average tree solution for cycle-free graph games," Open Access publications from Tilburg University, Tilburg University urn:nbn:nl:ui:12-3736838, Tilburg University.
  8. Faigle, U & Kern, W, 1992. "The Shapley Value for Cooperative Games under Precedence Constraints," International Journal of Game Theory, Springer, Springer, vol. 21(3), pages 249-66.
  9. Herings, P.J.J. & Laan, G. van der & Talman, A.J.J., 2008. "The average tree solution for cycle-free graph games," Open Access publications from Tilburg University, Tilburg University urn:nbn:nl:ui:12-377604, Tilburg University.
  10. René Brink, 2012. "On hierarchies and communication," Social Choice and Welfare, Springer, Springer, vol. 39(4), pages 721-735, October.
  11. Borm, P.E.M. & Owen, G. & Tijs, S.H., 1992. "On the position value for communication situations," Open Access publications from Tilburg University, Tilburg University urn:nbn:nl:ui:12-154855, Tilburg University.
  12. René Brink & Gerard Laan & Valeri Vasil’ev, 2007. "Component efficient solutions in line-graph games with applications," Economic Theory, Springer, Springer, vol. 33(2), pages 349-364, November.
  13. van den Brink, Rene, 2007. "Null or nullifying players: The difference between the Shapley value and equal division solutions," Journal of Economic Theory, Elsevier, Elsevier, vol. 136(1), pages 767-775, September.
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Citations

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Cited by:
  1. Napel, Stefan & Nohn, Andreas & Alonso-Meijide, José Maria, 2012. "Monotonicity of power in weighted voting games with restricted communication," Mathematical Social Sciences, Elsevier, Elsevier, vol. 64(3), pages 247-257.
  2. 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.
  3. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2012. "Weighted component fairness for forest games," Mathematical Social Sciences, Elsevier, Elsevier, vol. 64(2), pages 144-151.
  4. repec:hal:cesptp:hal-00803233 is not listed on IDEAS
  5. Huseynov, T. & Talman, A.J.J., 2012. "The Communication Tree Value for TU-games with Graph Communication," Discussion Paper, Tilburg University, Center for Economic Research 2012-095, Tilburg University, Center for Economic Research.
  6. Suzuki, T. & Talman, A.J.J., 2011. "Solution Concepts for Cooperative Games with Circular Communication Structure," Discussion Paper, Tilburg University, Center for Economic Research 2011-100, Tilburg University, Center for Economic Research.

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