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The Family of Ideal Values for Cooperative Games

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Abstract

In view of the nature of pursuing profit, a selfish coefficient function is employed to describe the degrees of selfishness of players in different coalitions, which is the desired rate of return to the worth of coalitions. This function brings in the concept of individual expected reward to every player. Built on different selfish coefficient functions, the family of ideal values can be obtained by minimizing deviations from the individual expected rewards. Then, we show the relationships between the family of ideal values and two other classical families of values: the procedural values and the least square values. For any selfish coefficient function, the corresponding ideal value is characterized by efficiency, linearity, an equal-expectation player property and a nullifying player punishment property, and also interpreted by a dynamic process. As two dual cases in the family of ideal values, the center of gravity of imputation set value and the equal allocation of nonseparable costs value are raised from new axiomatic angles.

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Notes

  1. The Shapley value allocates the dividends of every coalition equally over the players in the coalition, and since the sum of the dividends over all coalitions equals the worth of the grand coalition, the Shapley value is efficient.

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Acknowledgements

This research has been supported by the National Natural Science Foundation of China (Grant Nos. 71571143 and 71671140), the China Scholarship Council (Grant No. 201706290181).

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Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, People’s Republic of China

    Wenna Wang, Hao Sun & Genjiu Xu

  2. Department of Econometrics and Operations Research, VU University, De Boelelaan 1105, 1081 HV, Amsterdam, The Netherlands

    Wenna Wang & René van den Brink

Authors
  1. Wenna Wang
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  2. Hao Sun
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  3. René van den Brink
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  4. Genjiu Xu
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Correspondence to Genjiu Xu.

Additional information

Communicated by Irinel Chiril Dragan.

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Wang, W., Sun, H., van den Brink, R. et al. The Family of Ideal Values for Cooperative Games. J Optim Theory Appl 180, 1065–1086 (2019). https://doi.org/10.1007/s10957-018-1259-8

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