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The extended and generalized Shapley value: Simultaneous consideration of coalitional externalities and coalitional structure

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  • McQuillin, Ben

Abstract

The Shapley value assigns, to each game that is adequately represented by its characteristic function, an outcome for each player. An elaboration on the Shapley value that assigns, to characteristic function games, a "partition function" outcome is broadly established and accepted, but elaborations to encompass games with externalities (represented by partition functions) are not. Here, I show that simultaneous consideration of the two elaborations ("generalization" and "extension") obtains a unique Shapley-type value for games in partition function form. The key requirement is that the "Extended, Generalized Shapley Value" (EGSV) should be "recursive": the EGSV of any game should be the EGSV of itself. This requirement forces us to ignore all but the payoffs to bilateral partitions. The EGSV can be conceptualized as the ex ante value of a process of successive bilateral amalgamations. Previous Shapley value extensions, if generalized, are not recursive; indeed, they iterate to the EGSV.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Economic Theory.

Volume (Year): 144 (2009)
Issue (Month): 2 (March)
Pages: 696-721

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Handle: RePEc:eee:jetheo:v:144:y:2009:i:2:p:696-721

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Web page: http://www.elsevier.com/locate/inca/622869

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Keywords: Coalition structure Externalities Partition function games Recursion Shapley value;

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  1. Martin Shubik, 1961. "Incentives, Decentralized Control, the Assignment of Joint Costs and Internal Pricing," Cowles Foundation Discussion Papers 112, Cowles Foundation for Research in Economics, Yale University.
  2. Debraj Ray & Rajiv Vohra, 1998. "A Theory of Endogenous Coalition Structures," Working Papers 98-1, Brown University, Department of Economics, revised Jan 1998.
  3. Geoffroy de Clippel & Roberto Serrano, 2007. "Marginal contributions and externalities in the value," Working Papers 2007-04, Instituto Madrileño de Estudios Avanzados (IMDEA) Ciencias Sociales.
  4. Bloch, Francis, 1996. "Sequential Formation of Coalitions in Games with Externalities and Fixed Payoff Division," Games and Economic Behavior, Elsevier, vol. 14(1), pages 90-123, May.
  5. Salant, Stephen W & Switzer, Sheldon & Reynolds, Robert J, 1983. "Losses from Horizontal Merger: The Effects of an Exogenous Change in Industry Structure on Cournot-Nash Equilibrium," The Quarterly Journal of Economics, MIT Press, vol. 98(2), pages 185-99, May.
  6. Macho-Stadler, Ines & Perez-Castrillo, David & Wettstein, David, 2007. "Sharing the surplus: An extension of the Shapley value for environments with externalities," Journal of Economic Theory, Elsevier, vol. 135(1), pages 339-356, July.
  7. Hart, Sergiu & Kurz, Mordecai, 1983. "Endogenous Formation of Coalitions," Econometrica, Econometric Society, vol. 51(4), pages 1047-64, July.
  8. Pham Do, K.H. & Norde, H.W., 2002. "The Shapley Value for Partition Function Form Games," Discussion Paper 2002-4, Tilburg University, Center for Economic Research.
  9. Gul, Faruk, 1989. "Bargaining Foundations of Shapley Value," Econometrica, Econometric Society, vol. 57(1), pages 81-95, January.
  10. Bolger, E M, 1989. "A Set of Axioms for a Value for Partition Function Games," International Journal of Game Theory, Springer, vol. 18(1), pages 37-44.
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Cited by:
  1. Inés Macho-Stadler & David Pérez-Castrillo & David Wettstein, 2008. "Dividends and Weighted Values in Games with Externalities," UFAE and IAE Working Papers 758.08, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
  2. Beard, Rodney & Mallawaarachchi, Thilak, 2011. "Are international environmental agreements stable ex-post?," MPRA Paper 34303, University Library of Munich, Germany.
  3. repec:pdn:wpaper:73 is not listed on IDEAS
  4. Effrosyni Diamantoudi & Inés Macho-Stadler & David Pérez-Castrillo & Licun Xue, 2011. "Sharing the surplus in games with externalities within and across issues," UFAE and IAE Working Papers 880.11, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).

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