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Matrix expression of Shapley values and its application to distributed resource allocation

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Abstract

The symmetric and weighted Shapley values for cooperative n-person games are studied. Using the semi-tensor product of matrices, it is first shown that a characteristic function can be expressed as a pseudo-Boolean function. Then, two simple matrix formulas are obtained for calculating the symmetric and weighted Shapley values. Finally, using these new formulas, a design technique for the agents’ payoff functions in distributed resource allocation problems is proposed. It is possible to design payoff functions with the weighted Shapley value by the nonsymmetric weights defined on the players, thus ensuring that the optimal allocation is a pure Nash equilibrium. Practical examples are presented to illustrate the theoretical results.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 61773371).

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Authors and Affiliations

  1. School of Management Science and Engineering, Shandong Normal University, Jinan, 250014, China

    Yuanhua Wang & Xiyu Liu

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Daizhan Cheng

Authors
  1. Yuanhua Wang
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  2. Daizhan Cheng
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  3. Xiyu Liu
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Correspondence to Yuanhua Wang.

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Wang, Y., Cheng, D. & Liu, X. Matrix expression of Shapley values and its application to distributed resource allocation. Sci. China Inf. Sci. 62, 22201 (2019). https://doi.org/10.1007/s11432-018-9414-5

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  • DOI: https://doi.org/10.1007/s11432-018-9414-5

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