SCIENCE CHINA Information Sciences, Volume 62, Issue 2: 022201(2019) https://doi.org/10.1007/s11432-018-9414-5

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

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  • ReceivedJan 27, 2018
  • AcceptedMar 5, 2018
  • PublishedDec 25, 2018


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.


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


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  • Table 1   Payoff function using symmetric Shapley value
  • Table 2   Global objective $W(a)$
  • Table 3   Payoff function using the weighted Shapley value

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