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

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McQuillin, Ben

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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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Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 12049.

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Date of creation: 11 Nov 2008
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Handle: RePEc:pra:mprapa:12049

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

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Find related papers by JEL classification:
D62 - Microeconomics - - Welfare Economics - - - Externalities
C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. 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. [Downloadable!] (restricted)
  2. Gul, Faruk, 1989. "Bargaining Foundations of Shapley Value," Econometrica, Econometric Society, vol. 57(1), pages 81-95, January. [Downloadable!] (restricted)
  3. 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. [Downloadable!] (restricted)
  4. Ray, Debraj & Vohra, Rajiv, 1999. "A Theory of Endogenous Coalition Structures," Games and Economic Behavior, Elsevier, vol. 26(2), pages 286-336, January. [Downloadable!] (restricted)
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  5. Hart, Sergiu & Kurz, Mordecai, 1983. "Endogenous Formation of Coalitions," Econometrica, Econometric Society, vol. 51(4), pages 1047-64, July. [Downloadable!] (restricted)
  6. 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. [Downloadable!]
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  7. 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.
  8. Kim Hang Pham Do & Henk Norde, 2007. "The Shapley Value For Partition Function Form Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 9(02), pages 353-360. [Downloadable!] (restricted)
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  9. Martin Shubik, 1961. "Incentives, Decentralized Control, the Assignment of Joint Costs and Internal Pricing," Cowles Foundation Discussion Papers 112, Cowles Foundation, Yale University. [Downloadable!]
  10. 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. [Downloadable!] (restricted)
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  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). [Downloadable!]
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