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Values on regular games under Kirchhoff's laws

  • Fabien Lange

    ()

    (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris I - Panthéon-Sorbonne)

  • Michel Grabisch

    ()

    (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris I - Panthéon-Sorbonne)

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.

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Date of creation: Dec 2006
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Handle: RePEc:hal:cesptp:halshs-00130449
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  1. Hwang, Yan-An & Liao, Yu-Hsien, 2008. "Potential approach and characterizations of a Shapley value in multi-choice games," Mathematical Social Sciences, Elsevier, vol. 56(3), pages 321-335, November.
  2. Pradeep Dubey & Abraham Neyman & Robert J. Weber, 1979. "Value Theory without Efficiency," Cowles Foundation Discussion Papers 513, Cowles Foundation for Research in Economics, Yale University.
  3. Calvo, Emilio & Lasaga, Javier & van den Nouweland, Anne, 1999. "Values of games with probabilistic graphs," Mathematical Social Sciences, Elsevier, vol. 37(1), pages 79-95, January.
  4. Robert J. Weber, 1977. "Probabilistic Values for Games," Cowles Foundation Discussion Papers 471R, Cowles Foundation for Research in Economics, Yale University.
  5. 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.
  6. Michel Grabisch & Fabien Lange, 2007. "Games on lattices, multichoice games and the Shapley value: a new approach," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00178916, HAL.
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