The Shapley Value of Phylogenetic Trees

Haake C-J, Kashiwada A, Su FE (2005) Working Papers. Institute of Mathematical Economics; 363.
Bielefeld: Universität Bielefeld.

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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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Haake C-J, Kashiwada A, Su FE. The Shapley Value of Phylogenetic Trees. Working Papers. Institute of Mathematical Economics. Vol 363. Bielefeld: Universität Bielefeld; 2005.
Haake, C. - J., Kashiwada, A., & Su, F. E. (2005). The Shapley Value of Phylogenetic Trees (Working Papers. Institute of Mathematical Economics, 363). Bielefeld: Universität Bielefeld.
Haake, C. - J., Kashiwada, A., and Su, F. E. (2005). The Shapley Value of Phylogenetic Trees. Working Papers. Institute of Mathematical Economics, 363, Bielefeld: Universität Bielefeld.
Haake, C.-J., Kashiwada, A., & Su, F.E., 2005. The Shapley Value of Phylogenetic Trees, Working Papers. Institute of Mathematical Economics, no.363, Bielefeld: Universität Bielefeld.
C.-J. Haake, A. Kashiwada, and F.E. Su, The Shapley Value of Phylogenetic Trees, Working Papers. Institute of Mathematical Economics, vol. 363, Bielefeld: Universität Bielefeld, 2005.
Haake, C.-J., Kashiwada, A., Su, F.E.: The Shapley Value of Phylogenetic Trees. Working Papers. Institute of Mathematical Economics, 363. Universität Bielefeld, Bielefeld (2005).
Haake, Claus-Jochen, Kashiwada, Akemi, and Su, Francis Edward. The Shapley Value of Phylogenetic Trees. Bielefeld: Universität Bielefeld, 2005. Working Papers. Institute of Mathematical Economics. 363.
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