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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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    1. Kjell Hausken & Matthias Mohr, 2001. "The value of a player in n-person games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(3), pages 465-483.
    2. Kongo, Takumi, 2018. "Balanced contributions based on indirect claims and the Shapley value," Economics Letters, Elsevier, vol. 167(C), pages 48-50.
    3. van den Brink, Rene, 2007. "Null or nullifying players: The difference between the Shapley value and equal division solutions," Journal of Economic Theory, Elsevier, vol. 136(1), pages 767-775, September.
    4. Yokote, Koji & Funaki, Yukihiko & Kamijo, Yoshio, 2017. "Coincidence of the Shapley value with other solutions satisfying covariance," Mathematical Social Sciences, Elsevier, vol. 89(C), pages 1-9.
    5. AUMANN, Robert J. & DREZE, Jacques H., 1974. "Cooperative games with coalition structures," LIDAM Reprints CORE 217, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Radzik, Tadeusz, 2012. "A new look at the role of players’ weights in the weighted Shapley value," European Journal of Operational Research, Elsevier, vol. 223(2), pages 407-416.
    7. Skibski, Oskar & Michalak, Tomasz P. & Wooldridge, Michael, 2018. "The Stochastic Shapley Value for coalitional games with externalities," Games and Economic Behavior, Elsevier, vol. 108(C), pages 65-80.
    8. Michael Maschler, 1963. "The Power of a Coalition," Management Science, INFORMS, vol. 10(1), pages 8-29, October.
    9. Ilya Segal, 2003. "Collusion, Exclusion, and Inclusion in Random-Order Bargaining," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 70(2), pages 439-460.
    10. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
    11. Casajus, André & Huettner, Frank, 2014. "Null, nullifying, or dummifying players: The difference between the Shapley value, the equal division value, and the equal surplus division value," Economics Letters, Elsevier, vol. 122(2), pages 167-169.
    12. Guillermo Owen, 1972. "Multilinear Extensions of Games," Management Science, INFORMS, vol. 18(5-Part-2), pages 64-79, January.
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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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