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The Egalitarian Efficient Extension of the Aumann–Drèze Value

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Abstract

In this paper, we propose a new efficient value for transferable utility cooperative games with a coalition structure. It first assigns to every player his Aumann–Drèze value and then allocates the remainder of the worth of the grand coalition among players equally. As it is identical with the Aumann–Drèze value for coalitional games with a singleton coalition structure, we call it the egalitarian efficient extension of the Aumann–Drèze value. We provide three axiomatizations of it and compare it with other well-known efficient coalitional values, especially the Owen value and the two-step Shapley value.

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Notes

  1. For simplicities of notation, we omit the braces for singletons, when no confusion occurs.

  2. Following the name of Xu et al. [17], we may also call it as the additive efficient normalization of the AD-value.

  3. Following the name of Chun and Park [19], we can also call PSBU as population fairness between unions. Here, we follow the name of Calvo and Gutiérrez [14].

  4. The Levy–Mclean value is a weighted version of the Ow-value. It differs from the Ow-value by using the weighted Shapely value [30, 31] to distribute the worth of the grand coalition among unions, with the cardinalities of unions act as weights.

  5. A simple game is a TU game, whose range of characteristic function is \(\{0,1\}\). Since the family of simple games is not closed under addition, the additivity axiom is meaningless in this family.

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Acknowledgements

Great appreciation is given to Hans Peters and three anonymous referees for helpful comments. This work was supported by the Key Program of National Natural Science Foundation of China (71231003), the National Natural Science Foundation of China (71671140), and the Scientific Research Allowance of Guangzhou University (69-18ZX10337).

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Authors and Affiliations

  1. School of Management, Guangzhou University, Guangzhou, 510006, China

    Xun-Feng Hu

  2. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, China

    Gen-Jiu Xu

  3. School of Economics and Management, Fuzhou University, Fuzhou, 350108, China

    Deng-Feng Li

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  1. Xun-Feng Hu
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  2. Gen-Jiu Xu
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Correspondence to Deng-Feng Li.

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Communicated by Dusan Stipanovic.

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Hu, XF., Xu, GJ. & Li, DF. The Egalitarian Efficient Extension of the Aumann–Drèze Value. J Optim Theory Appl 181, 1033–1052 (2019). https://doi.org/10.1007/s10957-018-1440-0

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