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Values on regular games under Kirchhoff's laws

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Author Info
Fabien Lange () (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Panthéon-Sorbonne - Paris I)
Michel Grabisch () (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Panthéon-Sorbonne - Paris I)

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Abstract

In cooperative game theory, the Shapley value is a central notion defining a rational way to share the total worth of a game among players. In this paper, we address a general framework, namely regular set systems, where the set of feasible coalitions forms a poset where all maximal chains have the same length. We first show that previous definitions and axiomatizations of the Shaphey value proposed by Faigle and Kern and Bilbao and Edelman still work. our main contribution is then to propose a new axiomatization avoiding the hierarchical strength axiom of Faigle and Kern, and considering a new way to define the symmetry among players. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the classical efficiency axiom correspond actually to the two Kirchhoff's laws in the resistor circuit associated to the Hasse diagram of feasible coalitions. We finally work out a weak form of the monotonicity axiom which is satisfied by the proposed value.

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Paper provided by HAL in its series Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) with number halshs-00130449_v1.

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Date of creation: Dec 2006
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Handle: RePEc:hal:cesptp:halshs-00130449_v1

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Related research
Keywords: Regular set systems; regular games; Shapley value; probabilistic efficient values; regular values; Kirchhoff's laws.;

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References listed on IDEAS
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  1. Pradeep Dubey & Abraham Neyman & Robert J. Weber, 1979. "Value Theory without Efficiency," Cowles Foundation Discussion Papers 513, Cowles Foundation, Yale University. [Downloadable!]
  2. Robert J. Weber, 1977. "Probabilistic Values for Games," Cowles Foundation Discussion Papers 471R, Cowles Foundation, Yale University. [Downloadable!]
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  3. 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.
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Cited by:
(explanations, 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. Baron, Richard & Béal, Sylvain & Remila, Eric & Solal, Philippe, 2008. "Average tree solutions for graph games," MPRA Paper 10189, University Library of Munich, Germany. [Downloadable!]
  2. Aoi Honda & Michel Grabisch, 2008. "An axiomatization of entropy of capacities on set systems," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00281598_v1, HAL. [Downloadable!]
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