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Characterizations of three linear values for TU games by associated consistency: simple proofs using the Jordan normal form

Author

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  • Sylvain Béal

    (CRESE - Centre de REcherches sur les Stratégies Economiques (UR 3190) - UFC - Université de Franche-Comté - UBFC - Université Bourgogne Franche-Comté [COMUE])

  • Eric Rémila

    (GATE Lyon Saint-Étienne - Groupe d'Analyse et de Théorie Economique Lyon - Saint-Etienne - ENS de Lyon - École normale supérieure de Lyon - UL2 - Université Lumière - Lyon 2 - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon - UJM - Université Jean Monnet - Saint-Étienne - CNRS - Centre National de la Recherche Scientifique)

  • Philippe Solal

    (GATE Lyon Saint-Étienne - Groupe d'Analyse et de Théorie Economique Lyon - Saint-Etienne - ENS de Lyon - École normale supérieure de Lyon - UL2 - Université Lumière - Lyon 2 - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon - UJM - Université Jean Monnet - Saint-Étienne - CNRS - Centre National de la Recherche Scientifique)

Abstract

This article studies values for cooperative games with transferable utility. Numerous such values can be characterized by axioms of Ψ ε-associated consistency, which require that a value is invariant under some parametrized linear transformation Ψ ε on the vector space of cooperative games with transferable utility. Xu et al. (2008, 2009, Linear Algebra Appl.), Xu et al. (2013, Linear Algebra Appl.), Hamiache (2010, Int. Game Theory Rev.) and more recently Xu et al. (2015, Linear Algebra Appl.) 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, 2015. "Characterizations of three linear values for TU games by associated consistency: simple proofs using the Jordan normal form," Working Papers hal-01376909, HAL.
    • Handle: RePEc:hal:wpaper:hal-01376909
    Note: View the original document on HAL open archive server: https://hal.science/hal-01376909
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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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    3. Nowak Andrzej S. & Radzik Tadeusz, 1994. "The Shapley Value for n-Person Games in Generalized Characteristic Function Form," Games and Economic Behavior, Elsevier, vol. 6(1), pages 150-161, January.
    4. Gérard Hamiache, 2001. "Associated consistency and Shapley value," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(2), pages 279-289.
    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).
    6. Gérard Hamiache, 2012. "A Matrix Approach to TU Games with Coalition and Communication Structures," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 38(1), pages 85-100, January.
    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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    Citations

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    Cited by:

    1. Wenna Wang, 2021. "Bilateral associated game: Gain and loss in revaluation," PLOS ONE, Public Library of Science, vol. 16(7), pages 1-12, July.
    2. Sylvain Béal & Mihai Manea & Eric Rémila & Phillippe Solal, 2018. "Games With Identical Shapley Values," Working Papers 2018-03, CRESE.
    3. 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.
    4. 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.
    5. 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.
    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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