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The average tree solution for cooperative games with communication structure

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Author Info

  • Herings, P.J.J.

    (Tilburg University)

  • Laan, G. van der
  • Talman, A.J.J.

    (Tilburg University)

  • Yang, Z.F.

    (Tilburg University)

Abstract

We study cooperative games with communication structure, represented by an undirectedgraph. Players in the game are able to cooperate only if they can form a network in the graph. A single-valued solution, the average tree solution, is proposed for this class ofgames. Given the graph structure we define a collection of spanning trees, where eachspanning tree specifies a particular way by which players communicate and determines a payoff vector of marginal contributions of all the players. The average tree solution is defined to be the average of all these payoff vectors. It is shown that if a game has acomplete communication structure, then the proposed solution coincides with the Shapleyvalue, and that if the game has a cycle-free communication structure, it is the solutionproposed by Herings, van der Laan and Talman (2008). We introduce the notion of linkconvexity, under which the game is shown to have a non-empty core and the average tree solution lies in the core. In general, link-convexity is weaker than convexity. For games with a cycle-free communication structure, link-convexity is even weaker than super-additivity.

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Bibliographic Info

Paper provided by Tilburg University in its series Open Access publications from Tilburg University with number urn:nbn:nl:ui:12-3736837.

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Date of creation: 2010
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Publication status: Published in Games and Economic Behavior (2010) v.68, p.626-633
Handle: RePEc:ner:tilbur:urn:nbn:nl:ui:12-3736837

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Web page: http://www.tilburguniversity.edu/

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References

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  1. Baron, Richard & Béal, Sylvain & Remila, Eric & Solal, Philippe, 2008. "Average tree solutions for graph games," MPRA Paper 10189, University Library of Munich, Germany.
  2. Mamoru Kaneko & Myrna Holtz Wooders, 1982. "Cores of Partitioning Games," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 620, Cowles Foundation for Research in Economics, Yale University.
  3. 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.
  4. René van den Brink & Gerard van der Laan & Vitaly Pruzhansky, 2004. "Harsanyi Power Solutions for Graph-restricted Games," Tinbergen Institute Discussion Papers, Tinbergen Institute 04-095/1, Tinbergen Institute.
  5. Le Breton,Michel & Owen,Guillermo & Weber,Shlomo, 1991. "Strongly balanced cooperative games," Discussion Paper Serie A, University of Bonn, Germany 338, University of Bonn, Germany.
  6. Gabrielle Demange, 2004. "On Group Stability in Hierarchies and Networks," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 112(4), pages 754-778, August.
  7. repec:dgr:uvatin:2004095 is not listed on IDEAS
  8. Marco Slikker, 2005. "A characterization of the position value," International Journal of Game Theory, Springer, Springer, vol. 33(4), pages 505-514, November.
  9. Talman, A.J.J. & Yamamoto, Y., 2007. "Games With Limited Communication Structure," Discussion Paper, Tilburg University, Center for Economic Research 2007-19, Tilburg University, Center for Economic Research.
  10. Demange, G., 1991. "Intermediate Preferences and Stable Coalition Structures," DELTA Working Papers, DELTA (Ecole normale supérieure) 91-16, DELTA (Ecole normale supérieure).
  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. Talman, A.J.J. & Yamamoto, Y., 2008. "Average tree solution and subcore for acyclic graph games," Open Access publications from Tilburg University, Tilburg University urn:nbn:nl:ui:12-3129818, Tilburg University.
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Citations

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Cited by:
  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, Springer, vol. 40(2), pages 331-349, May.
  2. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2012. "Weighted component fairness for forest games," Mathematical Social Sciences, Elsevier, Elsevier, vol. 64(2), pages 144-151.
  3. Rene van den Brink & Gerard van der Laan & Nigel Moes, 2012. "A Strategic Implementation of the Average Tree Solution for Cycle-Free Graph Games," Tinbergen Institute Discussion Papers, Tinbergen Institute 12-050/1, Tinbergen Institute.
  4. S. Ryuo & K. Sato & Y. Yamamoto, 2012. "Parameterized fairness axioms on cycle-free graph games," Journal of Global Optimization, Springer, Springer, vol. 52(3), pages 487-497, March.
  5. Khmelnitskaya, Anna B. & Sudhölter, Peter, 2011. "The prenucleolus for games with communication structures," Discussion Papers of Business and Economics, Department of Business and Economics, University of Southern Denmark 10/2011, Department of Business and Economics, University of Southern Denmark.
  6. Albizuri, M. Josune & Sudhölter, Peter, 2014. "On the Core of Games with Communication Structures," Discussion Papers of Business and Economics, Department of Business and Economics, University of Southern Denmark 6/2014, Department of Business and Economics, University of Southern Denmark.
  7. 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.
  8. 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.
  9. repec:dgr:uvatin:2012050 is not listed on IDEAS
  10. Koshevoy, Gleb & Talman, Dolf, 2014. "Solution concepts for games with general coalitional structure," Mathematical Social Sciences, Elsevier, Elsevier, vol. 68(C), pages 19-30.
  11. repec:pdn:wpaper:73 is not listed on IDEAS
  12. Khmelnitskaya, A. & Selcuk, O. & Talman, A.J.J., 2012. "The Average Covering Tree Value for Directed Graph Games," Discussion Paper, Tilburg University, Center for Economic Research 2012-037, Tilburg University, Center for Economic Research.
  13. 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.
  14. Baron, Richard & Béal, Sylvain & Remila, Eric & Solal, Philippe, 2008. "Average tree solutions for graph games," MPRA Paper 10189, University Library of Munich, Germany.
  15. Koshevoy, G.A. & Suzuki, T. & Talman, A.J.J., 2013. "Solutions For Games With General Coalitional Structure And Choice Sets," Discussion Paper, Tilburg University, Center for Economic Research 2013-012, Tilburg University, Center for Economic Research.
  16. Koshevoy, G.A. & Talman, A.J.J., 2011. "Solution Concepts for Games with General Coalitional Structure (Replaces CentER DP 2011-025)," Discussion Paper, Tilburg University, Center for Economic Research 2011-119, Tilburg University, Center for Economic Research.
  17. 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.
  18. Selçuk, Özer & Suzuki, Takamasa & Talman, Dolf, 2013. "Equivalence and axiomatization of solutions for cooperative games with circular communication structure," Economics Letters, Elsevier, Elsevier, vol. 121(3), pages 428-431.

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