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Values on regular games under Kirchhoff’s laws

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  • Fabien Lange

    ()
    (Budapest Tech)

  • Michel Grabisch

    ()
    (Centre d’Economie de la Sorbonne)

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 leading to applications to games with communication graphs, 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 Shapley value proprosed 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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File URL: http://uni-obuda.hu/users/vecseya/RePEc/pkk/wpaper/0807.pdf
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Bibliographic Info

Paper provided by Óbuda University, Keleti Faculty of Business and Management in its series Working Paper Series with number 0807.

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Length: 23 pages
Date of creation: 2006
Date of revision: Nov 2008
Handle: RePEc:pkk:wpaper:0807

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

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References

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  1. Pradeep Dubey & Robert J. Weber, 1977. "Probabilistic Values for Games," Cowles Foundation Discussion Papers 471, Cowles Foundation for Research in Economics, Yale University.
  2. René van den Brink & Gerard van der Laan & Vitaly Pruzhansky, 2004. "Harsanyi Power Solutions for Graph-restricted Games," Tinbergen Institute Discussion Papers 04-095/1, Tinbergen Institute.
  3. 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.
  4. Michel Grabisch & Fabien Lange, 2007. "Games on lattices, multichoice games and the shapley value: a new approach," Computational Statistics, Springer, vol. 65(1), pages 153-167, February.
  5. 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.
  6. 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.
  7. 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.
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Citations

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Cited by:
  1. repec:hal:cesptp:hal-00281598 is not listed on IDEAS
  2. Richard Baron & Sylvain Béal & Eric Rémila & Philippe Solal, 2011. "Average tree solutions and the distribution of Harsanyi dividends," International Journal of Game Theory, Springer, vol. 40(2), pages 331-349, May.
  3. Michel Grabisch, 2010. "Ensuring the boundedness of the core of games with restricted cooperation," Documents de travail du Centre d'Economie de la Sorbonne 10093, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
  4. Grabisch, Michel & Sudhölter, Peter, 2014. "The positive core for games with precedence constraints," Discussion Papers of Business and Economics 8/2014, Department of Business and Economics, University of Southern Denmark.
  5. Michel Grabisch & Peter Sudhölter, 2014. "The positive core for games with precedence constraints," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01020282, HAL.
  6. Honda, Aoi & Grabisch, Michel, 2008. "An axiomatization of entropy of capacities on set systems," European Journal of Operational Research, Elsevier, vol. 190(2), pages 526-538, October.
  7. repec:hal:cesptp:hal-00803233 is not listed on IDEAS
  8. Selcuk, O. & Talman, A.J.J., 2013. "Games With General Coalitional Structure," Discussion Paper 2013-002, Tilburg University, Center for Economic Research.
  9. Baron, Richard & Béal, Sylvain & Remila, Eric & Solal, Philippe, 2008. "Average tree solutions for graph games," MPRA Paper 10189, University Library of Munich, Germany.

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