The Shapley value of phylogenetic trees

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.

Evolutionarily distinctive species often capture more phylogenetic diversity than expected

“…We propose that more work be done investigating the properties of QE, perhaps expanding it from a pairwise to a multi-species framework (cf. Haake et al, 2005;Hartmann and Steel, 2006).…”

Equality of Shapley value and fair proportion index in phylogenetic trees

“…The Shapley value, a parameter long studied in cooperative game theory, has recently been suggested as a prioritization tool for taxa in phylogenetics; see Haake, Kashiwada and Su [3]. However, this parameter has the drawback that its definition is relatively complicated making its computation complicated as well.…”

Investing in evolutionary history: implementing a phylogenetic approach for mammal conservation

“…In addition to the original concept of PD [4], the two most widely followed scoring methods in the recent literature are equal splits (ES; [30,31]) and evolutionary distinctiveness (ED; [28]), also known as fair proportion (and which is very close to another, evolutionarily stable strategy game inspired measure, the Shapley index; [32], see [33] for wider review). ES hierarchically partitions branch lengths by the number of descendent edges, such that for a given branch, a descendent species receives credit equal to 0.5 to a power equal to the number of splits between the branch and the species.…”