The Shapley value of phylogenetic trees.
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.
Haake
Claus-Jochen
CJ
Institute of Mathematical Economics, Bielefeld University, P. O. Box 100131, 33501 Bielefeld, Germany. chaake@wiwi.uni-bielefeld.de
Kashiwada
Akemi
A
Su
Francis Edward
FE
D000818:Animals; D005716:Game Theory; D049810:Genetic Speciation; D014644:Genetic Variation; D006801:Humans; D008957:Models, Genetic; D016571:Neural Networks, Computer; D010802:Phylogeny; D011157:Population Dynamics; D013045:Species Specificity