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Biodiversity, Shapley value and phylogenetic trees: some remarks

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  • Hubert Stahn

    (AMSE - Aix-Marseille Sciences Economiques - EHESS - École des hautes études en sciences sociales - AMU - Aix Marseille Université - ECM - École Centrale de Marseille - CNRS - Centre National de la Recherche Scientifique)

Abstract

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.lama.univ-amu.fr/10.1007/s00285-007-0126-2) and Fuchs and Jin (J Math Biol 71:1133–1147, 2015. https://doi-org.lama.univ-amu.fr/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.lama.univ-amu.fr/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.lama.univ-amu.fr/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.lama.univ-amu.fr/10.1016/j.jtbi.2017.07.010), who reach similar conclusions with a different simulation method.

Suggested Citation

  • Hubert Stahn, 2020. "Biodiversity, Shapley value and phylogenetic trees: some remarks," Post-Print hal-02417290, HAL.
    • Handle: RePEc:hal:journl:hal-02417290
    DOI: 10.1007/s00285-019-01439-z
    Note: View the original document on HAL open archive server: https://amu.hal.science/hal-02417290
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    References listed on IDEAS

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    1. Kleinberg, Norman L. & Weiss, Jeffrey H., 1985. "A new formula for the Shapley value," Economics Letters, Elsevier, vol. 17(4), pages 311-315.
    2. Haake, Claus-Jochen & Kashiwada, Akemi & Su, Francis Edward, 2016. "The Shapley Value of Phylogenetic Trees," Center for Mathematical Economics Working Papers 363, Center for Mathematical Economics, Bielefeld University.
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    Cited by:

    1. Ricardo Martínez & Joaquín Sánchez-Soriano, 2021. "Mathematical indices for the influence of risk factors on the lethality of a disease," ThE Papers 21/02, Department of Economic Theory and Economic History of the University of Granada..

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    More about this item

    Keywords

    Fair Proportion index; Biodiversity; Phylogenetic trees; Shapley value;
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