Biodiversity, Shapely Value and Phylogenetic Trees: Some Remarks

This paper explores the main differences between the Shapley Values of a set of taxa introduced by Haake et al. [4] and Fuchs and Jin [3], the latter having been found identical to the Fair Proportion Index (Redding and Mooers [10]). In line with Shapley [13], we identify the cooperative game basis for each of these two classes of phylogenetic games and use them (i) to construct simple formulas for these two Shapley values and (ii) to compare these different approaches. Using the set of weights of a phylogenetic tree as a parameter space, we then discuss the conditions under which these two values coincide and, if they are not the same, revisit Hartman's [5] convergence result. Finally, we compare the species ranking induced by these two values. Considering the Kendal and the Spearman rank correlation coefficient, simulations show that these rankings are strongly correlated.