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The Shapley Value of Phylogenetic Trees

Author

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  • 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
    File Function: First Version, 2005
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    References listed on IDEAS

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    1. Martin L. Weitzman, 1992. "On Diversity," The Quarterly Journal of Economics, Oxford University Press, vol. 107(2), pages 363-405.
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    Keywords

    Phylogenetic trees; Diversity; Shapley value;

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