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The Shapley value of phylogenetic trees

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  • Claus-Jochen Haake

    ()
    (Institute of Mathematical Economics, Bielefeld University)

  • Akemi Kashiwada

    ()
    (Department of Mathematics, Harvey Mudd College, Claremont)

  • Francis Edward Su

    ()
    (Department of Mathematics, Harvey Mudd College, Claremont)

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    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.

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    File URL: http://www.imw.uni-bielefeld.de/papers/files/imw-wp-363.pdf
    File Function: First version, 2005
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    Bibliographic Info

    Paper provided by Bielefeld University, Center for Mathematical Economics in its series Working Papers with number 363.

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    Length: 21 pages
    Date of creation: Mar 2005
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    Handle: RePEc:bie:wpaper:363

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