Biodiversity, Shapley value and phylogenetic trees: some remarks

Area of Science:

• Phylogenetics
• Game Theory
• Mathematical Biology

Background:

• Shapley values are used to assess contributions in cooperative games.
• Two distinct Shapley value methods for phylogenetic games exist: Haake et al. and Fuchs and Jin (identical to Fair Proportion index).
• Understanding differences and convergence conditions between these methods is crucial for accurate phylogenetic analysis.

Purpose of the Study:

• To explore the main differences between two classes of Shapley values for phylogenetic games.
• To derive simple formulas for these Shapley values based on their cooperative game basis.
• To compare the species rankings generated by these two methods.

Main Methods:

• Identification of the cooperative game basis for each Shapley value class.
• Construction of explicit formulas for the Shapley values.
• Analysis of conditions for the convergence of the two Shapley values, revisiting Hartmann's results.
• Comparison of species rankings using Kendall and Spearman rank correlation coefficients.
• Simulations to assess the correlation between species rankings.

Main Results:

• The paper derives simple formulas for both Shapley value classes.
• Conditions under which the two Shapley values coincide are discussed.
• Simulations demonstrate a strong correlation between the species rankings produced by the two methods.
• The findings align with previous simulation-based conclusions.

Conclusions:

• Despite mathematical differences, the two Shapley value approaches yield highly correlated species rankings.
• The study provides a theoretical framework for understanding the relationship and convergence of these phylogenetic game values.
• The derived formulas and comparative analysis offer valuable tools for researchers in phylogenetics and evolutionary biology.