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The Shapley Value of Phylogenetic Trees

Author

Listed:
  • Haake, Claus-Jochen

    (Center for Mathematical Economics, Bielefeld University)

  • Kashiwada, Akemi

    (Center for Mathematical Economics, Bielefeld University)

  • Su, Francis Edward

    (Center for Mathematical Economics, Bielefeld University)

Abstract

Every weighted tree corresponds naturally to a cooperative game that we call a tree game; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley value on the edge weights of the tree, and we also compute a null space basis of M. Finally, we characterize the Shapley value on tree games by five axioms, a counterpart to Shapley's original theorem on the larger class of cooperative games. We also include a brief discussion of the core of tree games.

Suggested Citation

  • Haake, Claus-Jochen & Kashiwada, Akemi & Su, Francis Edward, 2016. "The Shapley Value of Phylogenetic Trees," Center for Mathematical Economics Working Papers 363, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:363
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    File URL: https://pub.uni-bielefeld.de/download/1875964/2319741
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    References listed on IDEAS

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    1. Martin L. Weitzman, 1992. "On Diversity," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 107(2), pages 363-405.
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    Cited by:

    1. Hubert Stahn, 2017. "Biodiversity, Shapely Value and Phylogenetic Trees: Some Remarks," AMSE Working Papers 1741, Aix-Marseille School of Economics, France.
    2. Ricardo Martínez & Joaquín Sánchez-Soriano, 2021. "Mathematical indices for the influence of risk factors on the lethality of a disease," ThE Papers 21/02, Department of Economic Theory and Economic History of the University of Granada..
    3. Claus-Jochen Haake, 2008. "Comments on: Transversality of the Shapley value," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 16(1), pages 48-50, July.

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