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Necessary players and values

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Abstract

In this paper we introduce the \(\Gamma \) value, a new value for cooperative games with transferable utility. We also provide an axiomatic characterization of the \(\Gamma \) value based on a property concerning the so-called necessary players. A necessary player of a game is one without which the characteristic function is zero. We illustrate the performance of the \(\Gamma \) value in a particular cost allocation problem that arises when the owners of the apartments in a building plan to install an elevator and share its installation cost; in the resulting example we compare the proposals of the \(\Gamma \) value, the equal division value and the Shapley value in two different scenarios. In addition, we propose an extension of the \(\Gamma \) value for cooperative games with transferable utility and with a coalition structure. Finally, we provide axiomatic characterizations of the coalitional \(\Gamma \) value and of the Owen and Banzhaf-Owen values using alternative properties concerning necessary players.

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Notes

  1. Notice that \(\frac{v(N)}{n}+\frac{1}{2^{n-1}}\sum _{S\subseteq N, i\in S} v^z(S)= \frac{v(N)}{n}+\frac{1}{2^{n-1}}\sum _{S\subset N, i\in S} v(S)\).

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Acknowledgements

This work has been supported by the ERDF, the MINECO/AEI grants MTM2017-87197-C3-1-P, MTM2017-87197-C3-3-P, and by the Xunta de Galicia (Grupos de Referencia Competitiva ED431C-2016-015, ED431C-2017-38 and ED431C-2020-14, and Centro Singular de Investigación de Galicia ED431G 2019/01). We also acknowledge the comments of two anonymous referees that have been very useful to improve this paper.

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Authors and Affiliations

  1. Grupo MODES, CITIC and Departamento de Matemáticas, Universidade da Coruña, Campus de Elviña, 15071, A Coruña, Spain

    J. C. Gonçalves-Dosantos & I. García-Jurado

  2. Grupo MODES, Departamento de Matemáticas, Universidade da Coruña, Campus de Elviña, 15071, A Coruña, Spain

    J. Costa

  3. Grupo MODESTYA, Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, Campus de Lugo, 27002, Lugo, Spain

    J. M. Alonso-Meijide

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  1. J. C. Gonçalves-Dosantos
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  2. I. García-Jurado
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  3. J. Costa
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  4. J. M. Alonso-Meijide
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Correspondence to J. M. Alonso-Meijide.

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Gonçalves-Dosantos, J.C., García-Jurado, I., Costa, J. et al. Necessary players and values. Ann Oper Res 318, 935–961 (2022). https://doi.org/10.1007/s10479-021-03950-3

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