Matrix Analysis for the Shapley Value and Its Inverse Problem

Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 758)


In the framework of cooperative game theory, any linear value of games is a linear operator on game space, implying that algebraic representations and matrix analysis are possibly justifiable techniques for studying linear values. For any linear value, the payoff vector of any game is represented algebraically by the product of a column-coalitional representation matrix and the worth vector. The analysis of the structure of these representation matrices covers the study of the class of linear values. We achieve a matrix approach for characterizing linear values with some essential properties. Also, some properties are described for the Shapley standard matrix, which is the representation matrix of the Shapley value. Furthermore, the inverse problem of the Shapley value is studied in terms of the null space of the Shapley standard matrix.


Matrix analysis The Shapley value The Shapley standard matrix Inverse problem 



This research has been supported by the National Natural Science Foundation of China (Grant Nos. 11402194, 71671140). And Jun Su presented the work on the Ninth International Conference on Matrix Theory and its Applications in Shanghai.


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© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.School of ScienceXi’an University of Science and TechnologyXi’anPeople’s Republic of China
  2. 2.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China

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