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Solutions For Games With General Coalitional Structure And Choice Sets

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  • Koshevoy, G.A.
  • Suzuki, T.
  • Talman, A.J.J.

    (Tilburg University, Center for Economic Research)

Abstract

In this paper we introduce the concept of quasi-building set that may underlie the coalitional structure of a cooperative game with restricted communication between the players. Each feasible coalition, including the set of all players, contains a nonempty subset called the choice set of the coalition. Only players that are in the choice set of a coalition are able to join to feasible subcoalitions to form the coalition and to obtain a marginal contribution. We demonstrate that all restricted communication systems that have been studied in the literature take the form of a quasi-building set for an appropriate set system and choice set. Every quasi-building set determines a nonempty collection of maximal strictly nested sets and each such set induces a rooted tree satisfying that every node of the tree is a player that is in the choice set of the feasible coalition that consists of himself and all his successors in the tree. Each tree corresponds to a marginal vector of the underlying game at which each player gets as payo his marginal contribution when he joins his successors in the tree. As solution concept of a quasi-building set game we propose the average marginal vector (AMV) value, being the average of the marginal vectors that correspond to the trees induced by all maximal strictly nested sets of the quasi-building set. Properties of this solution are also studied. To establish core stability we introduce appropriate convexity conditions of the game with respect to the underlying quasi-building set. For some speci cations of quasi-building sets, the AMV-value coincides with solutions known in the literature, for example, for building set games the solution coincides with the gravity center solution and the Shapley value recently de ned for this class. For graph games it therefore di ers from the well-known Myerson value. For a full communication system the solution coincides with the classical Shapley value.

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Bibliographic Info

Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 2013-012.

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Date of creation: 2013
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Handle: RePEc:dgr:kubcen:2013012

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Web page: http://center.uvt.nl

Related research

Keywords: Set system; nested set; rooted tree; chain; core; convexity; marginal vector; Shapley value;

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References

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  1. repec:hal:cesptp:hal-00625336 is not listed on IDEAS
  2. Khmelnitskaya, A. & Selcuk, O. & Talman, A.J.J., 2012. "The Average Covering Tree Value for Directed Graph Games," Discussion Paper 2012-037, Tilburg University, Center for Economic Research.
  3. P. Jean-Jacques Herings & Gerard van der Laan & Dolf Talman & Zaifu Yang, 2008. "The Average Tree Solution for Cooperative Games with Communication Structure," Tinbergen Institute Discussion Papers 08-083/1, Tinbergen Institute.
  4. Bilbao, J. M. & Lebron, E. & Jimenez, N., 1999. "The core of games on convex geometries," European Journal of Operational Research, Elsevier, vol. 119(2), pages 365-372, December.
  5. Faigle, U. & Grabisch, M. & Heyne, M., 2010. "Monge extensions of cooperation and communication structures," European Journal of Operational Research, Elsevier, vol. 206(1), pages 104-110, October.
  6. Algaba, A. & Bilbao, J.M. & Brink, J.R. van den & Jiménez-Losada, A., 2000. "Cooperative Games on Antimatroids," Discussion Paper 2000-124, Tilburg University, Center for Economic Research.
  7. 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.
  8. Bilbao, J.M. & Ordóñez, M., 2009. "Axiomatizations of the Shapley value for games on augmenting systems," European Journal of Operational Research, Elsevier, vol. 196(3), pages 1008-1014, August.
  9. Derks, Jean J M & Gilles, Robert P, 1995. "Hierarchical Organization Structures and Constraints on Coalition Formation," International Journal of Game Theory, Springer, vol. 24(2), pages 147-63.
  10. Koshevoy, G.A. & Talman, A.J.J., 2011. "Solution Concepts for Games with General Coalitional Structure (Replaces CentER DP 2011-025)," Discussion Paper 2011-119, Tilburg University, Center for Economic Research.
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