Values on regular games under Kirchhoff's laws

Fabien Lange () and Michel Grabisch
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Fabien Lange: CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique

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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.

Keywords: regular values; Kirchhoff's laws; probabilistic efficient values; Shapley value; regular games; Regular set systems; lois de Kirchhoff; jeu régulier; Système d'ensemble; valeur de Shapley; valeur probabiliste; axiome d'anonymat; axiome d'efficacité (search for similar items in EconPapers)
Date: 2006-12
Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00130449v1
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Published in 2006

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Related works:
Journal Article: Values on regular games under Kirchhoff's laws (2009)
Working Paper: Values on regular games under Kirchhoff's laws (2009)
Working Paper: Values on regular games under Kirchhoff's laws (2009)
Working Paper: Values on regular games under Kirchhoff’s laws (2008)
Working Paper: Values on regular games under Kirchhoff's laws (2006)
Working Paper: Values on regular games under Kirchhoff's laws (2006)
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