Biodiversity, Shapley 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. (J Math Biol 56:479-497, 2007. https://doi.org/10.1007/s00285-007-0126-2) and Fuchs and Jin (J Math Biol 71:1133-1147, 2015. https://doi.org/10.1007/s00285-014-0853-0), the latter having been found identical to the Fair Proportion index (Redding and Mooers in Conserv Biol 20:1670-1678, 2006. https://doi.org/10.1111/j.1523-1739.2006.00555.x). In line with Shapley (in: Kuhn, Tucker (eds) Contributions to to the theory of games, volume II, annals of mathematics studies 28, Princeton University Press, Princeton, 1953), 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 Hartmann's (J Math Biol 67:1163-1170, 2013. https://doi.org/10.1007/s00285-012-0585-y) convergence result. An example illustrates our main argument. Finally, we compare the species ranking induced by these two values. Considering the Kendall and the Spearman rank correlation coefficient, simulations show that these rankings are strongly correlated. These results are consistent with Wicke and Fischer (J Theor Biol 430:207-214, 2017. https://doi.org/10.1016/j.jtbi.2017.07.010), who reach similar conclusions with a different simulation method.