The Shapley Value of Phylogenetic Trees

Authors

Claus-Jochen Haake, Universitat Bielefeld
Akemi Kashiwada '05, Harvey Mudd CollegeFollow
Francis E. Su, Harvey Mudd CollegeFollow

Student Co-author

HMC Undergraduate

Document Type

Article - preprint

Department

Mathematics (HMC)

Publication Date

4-2008

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. 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. We also include a brief discussion of the core of tree games.

Comments

Author's pre-print manuscript available for download.

For the publisher's version, please visit http://dx.doi.org/10.1007/s00285-007-0126-2.

Rights Information

© 2008 Springer-Verlag

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

DOI

10.1007/s00285-007-0126-2

Recommended Citation

Claus-Jochen Haake, Akemi Kashiwada, and Francis Edward Su. The Shapley value of phylogenetic trees. J. Math. Biol., 56(4):479–497, 2008.