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The Shapley value of coalitions to other coalitions

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  • Kjell Hausken

    (University of Stavanger)

Abstract

The Shapley value for an n-person game is decomposed into a 2n × 2n value matrix giving the value of every coalition to every other coalition. The cell ϕIJ(v, N) in the symmetric matrix is positive, zero, or negative, dependent on whether row coalition I is beneficial, neutral, or unbeneficial to column coalition J. This enables viewing the values of coalitions from multiple perspectives. The n × 1 Shapley vector, replicated in the bottom row and right column of the 2n × 2n matrix, follows from summing the elements in all columns or all rows in the n × n player value matrix replicated in the upper left part of the 2n × 2n matrix. A proposition is developed, illustrated with an example, revealing desirable matrix properties, and applicable for weighted Shapley values. For example, the Shapley value of a coalition to another coalition equals the sum of the Shapley values of each player in the first coalition to each player in the second coalition.

Suggested Citation

  • Kjell Hausken, 2020. "The Shapley value of coalitions to other coalitions," Palgrave Communications, Palgrave Macmillan, vol. 7(1), pages 1-10, December.
    • Handle: RePEc:pal:palcom:v:7:y:2020:i:1:d:10.1057_s41599-020-00586-9
    DOI: 10.1057/s41599-020-00586-9
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    References listed on IDEAS

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    2. Tobias Hiller, 2023. "Measuring the Difficulties in Forming a Coalition Government," Games, MDPI, vol. 14(2), pages 1-15, March.

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