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Characterizations of Three Linear Values for TU Games by Associated Consistency: Simple Proofs Using the Jordan Normal Form

Author

Listed:
  • Sylvain Béal

    (Université de Franche-Comté, CRESE EA3190, Universite Bourgogne Franche-Comté, F-25000 Besançon, France)

  • Eric Rémila

    (��Universite Lyon, UJM Saint-Etienne, CNRS, GATE L-SE UMR 5824, F-42023 Saint-Etienne, France‡IXXI, 46, allée d’Italie, 69007 Lyon, France)

  • Philippe Solal

    (�Université de Saint-Etienne, CNRS UMR 5824 GATE Lyon Saint-Etienne, France)

Abstract

This paper studies values for cooperative games with transferable utility. Numerous such values can be characterized by axioms of Ψε-associated consistency, which require that a value is invariant under some parametrized linear transformation Ψε on the vector space of cooperative games with transferable utility. Xu et al. [(2008) Linear Algebr. Appl. 428, 1571–1586; (2009) Linear Algebr. Appl. 430, 2896–2897] Xu et al. [(2013) Linear Algebr. Appl. 439, 2205–2215], Hamiache [(2010) Int. Game Theor. Rev. 12, 175–187] and more recently Xu et al. [(2015) Linear Algebr. Appl. 471, 224–240] follow this approach by using a matrix analysis. The main drawback of these articles is the heaviness of the proofs to show that the matrix expression of the linear transformations is diagonalizable. By contrast, we provide quick proofs by relying on the Jordan normal form of the previous matrix.

Suggested Citation

  • Sylvain Béal & Eric Rémila & Philippe Solal, 2016. "Characterizations of Three Linear Values for TU Games by Associated Consistency: Simple Proofs Using the Jordan Normal Form," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 18(01), pages 1-21, March.
    • Handle: RePEc:wsi:igtrxx:v:18:y:2016:i:01:n:s0219198916500031
    DOI: 10.1142/S0219198916500031
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    References listed on IDEAS

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    1. Moulin, Herve, 1985. "The separability axiom and equal-sharing methods," Journal of Economic Theory, Elsevier, vol. 36(1), pages 120-148, June.
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    5. AUMANN, Robert J. & DREZE, Jacques H., 1974. "Cooperative games with coalition structures," LIDAM Reprints CORE 217, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    7. Gérard Hamiache, 2010. "A Matrix Approach To The Associated Consistency With An Application To The Shapley Value," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 12(02), pages 175-187.
    8. Yan-An Hwang, 2006. "Associated consistency and equal allocation of nonseparable costs," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 28(3), pages 709-719, August.
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    Cited by:

    1. Dongshuang Hou & Aymeric Lardon & Panfei Sun & Theo Driessen, 2018. "Compromise for the Per Capita Complaint: An Optimization Characterization of Two Equalitarian Values," GREDEG Working Papers 2018-13, Groupe de REcherche en Droit, Economie, Gestion (GREDEG CNRS), Université Côte d'Azur, France.
    2. Norman L. Kleinberg, 2018. "A note on associated consistency and linear, symmetric values," International Journal of Game Theory, Springer;Game Theory Society, vol. 47(3), pages 913-925, September.
    3. Florian Navarro, 2019. "Necessary players, Myerson fairness and the equal treatment of equals," Annals of Operations Research, Springer, vol. 280(1), pages 111-119, September.
    4. Wenna Wang, 2021. "Bilateral associated game: Gain and loss in revaluation," PLOS ONE, Public Library of Science, vol. 16(7), pages 1-12, July.
    5. Sylvain Béal & Mihai Manea & Eric Rémila & Phillippe Solal, 2018. "Games With Identical Shapley Values," Working Papers 2018-03, CRESE.
    6. Dongshuang Hou & Aymeric Lardon & Panfei Sun & Theo Driessen, 2018. "Compromise for the per Capita Complaint: an optimization CharaCterization of two equalitarian values," Working Papers halshs-01931224, HAL.

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    More about this item

    Keywords

    Associated consistency; Jordan normal form; Shapley value; center of imputation set; equal allocation of nonseparable costs;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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