The Shapley Value of Phylogenetic Trees

Details The Shapley Value of Phylogenetic Trees The Shapley Value of Phylogenetic Trees

2005-06-22

arxiv.org/abs/q-bio/0506034v2

J. Mathematical Biology 56 (2008), 479--497

Biology

Quantitative Biology

Quantitative Methods

References added, and a section (calculating the Shapley value of a tree game from its subtrees) was removed for length reason

Scientific paper

10.1007/s00285-007-0126-2

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. Both depend on the "split counts" of the tree. Finally, we characterize the Shapley value on tree games by four axioms, a counterpart to Shapley's original theorem on the larger class of cooperative games.

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