The Fair Proportion is a Shapley Value on phylogenetic networks too

TL;DR

This paper extends the equivalence of the Fair Proportion measure and the Shapley Value from phylogenetic trees to more complex rooted phylogenetic networks, broadening the theoretical understanding of biodiversity valuation.

Contribution

It proves that the Fair Proportion equals the Shapley Value on rooted phylogenetic networks and generalizes existing formulas for unrooted diversity games.

Findings

Fair Proportion equals the Shapley Value on rooted networks

Generalization of Shapley Value expression to rooted networks

Extension of previous tree-based results to networks

Abstract

The Fair Proportion of a species in a phylogenetic tree is a very simple measure that has been used to assess its value relative to the overall phylogenetic diversity represented by the tree. It has recently been proved by Fuchs and Jin to be equal to the Shapley Value of the coallitional game that sends each subset of species to its rooted Phylogenetic Diversity in the tree. We prove in this paper that this result extends to the natural translations of the Fair Proportion and the rooted Phylogenetic Diversity to rooted phylogenetic networks. We also generalize to rooted phylogenetic networks the expression for the Shapley Value of the unrooted Phylogenetic Diversity game on a phylogenetic tree established by Haake, Kashiwada and Su.