# Combinatorial properties of phylogenetic diversity indices

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## Abstract

Phylogenetic diversity indices provide a formal way to apportion ‘evolutionary heritage’ across species. Two natural diversity indices are Fair Proportion (FP) and Equal Splits (ES). FP is also called ‘evolutionary distinctiveness’ and, for rooted trees, is identical to the Shapley Value (SV), which arises from cooperative game theory. In this paper, we investigate the extent to which FP and ES can differ, characterise tree shapes on which the indices are identical, and study the equivalence of FP and SV and its implications in more detail. We also define and investigate analogues of these indices on unrooted trees (where SV was originally defined), including an index that is closely related to the Pauplin representation of phylogenetic diversity.

## Keywords

Phylogenetic tree Diversity index Shapley value Biodiversity measures## Mathematics Subject Classification

05-C05 05-C35 91-A12 92-D15## Notes

### Acknowledgements

We thank Arne Mooers for a number of helpful suggestions, and the two anonymous reviewers for detailed comments on an earlier version of this manuscript. We also thank François Bienvenu for pointing out an alternative proof of Lemma 1, and Mareike Fischer for helpful comments concerning Sect. 4. The first author also thanks the German Academic Scholarship Foundation for a doctoral scholarship.

## References

- Dubey P (1975) On the uniqueness of the Shapley value. Int J Game Theory 4:131–139MathSciNetCrossRefGoogle Scholar
- Faith DP (1992) Conservation evaluation and phylogenetic diversity. Biol Conserv 61:1–10CrossRefGoogle Scholar
- Fuchs M, Jin EY (2015) Equality of Shapley value and fair proportion index in phylogenetic trees. J Math Biol 71:1133–1147MathSciNetCrossRefGoogle Scholar
- Haake CJ, Kashiwada A, Su FE (2008) The Shapley value of phylogenetic trees. J Math Biol 56:479–497MathSciNetCrossRefGoogle Scholar
- Isaac N, Turvey ST, Collen B, Waterman C, Baillie J (2007) Mammals on the edge: conservation priorities based on threat and phylogeny. PLoS One 2:e296CrossRefGoogle Scholar
- Pauplin Y (2000) Direct calculation of a tree length using a distance matrix. J Mol Evolut 51:41–47CrossRefGoogle Scholar
- Redding DW (2003) Incorporating genetic distinctness and reserve occupancy into a conservation priorisation approach. Master’s thesis. University Of East Anglia, Norwich.Google Scholar
- Redding DW, Hartmann K, Mimoto A, Bokal D, DeVos M, Mooers AØ (2008) Evolutionarily distinctive species often capture more phylogenetic diversity than expected. J Theor Biol 251:606–615MathSciNetCrossRefGoogle Scholar
- Redding DW, Mazel F, Mooers AO (2014) Measuring evolutionary isolation for conservation. PLoS One 9:1–15CrossRefGoogle Scholar
- Redding DW, Mooers AØ (2006) Incorporating evolutionary measures into conservation prioritization. Conserv Biol 20:1670–1678CrossRefGoogle Scholar
- Semple C, Steel M (2004) Cyclic permutations and evolutionary trees. Adv Appl Math 32:669–680MathSciNetCrossRefGoogle Scholar
- Shapley LS (1953) A value for \(n\)-person games. Contributions to the theory of games (AM-28), vol II. Princeton University Press, Princeton, pp 307–317Google Scholar
- Stahn H (2017) Biodiversity, Shapley value and phylogenetic trees: Some remarks. WP2017- Nr 41. AMSE. URL: https://halshs.archives-ouvertes.fr/halshs-01630069/document
- Steel M (2016) Phylogeny: discrete and random processes in evolution. Society for Industrial and Applied Mathematic, PhiladelphiaCrossRefGoogle Scholar
- Steele JM (2004) The Cauchy–Schwarz master class. Cambridge University Press, New YorkCrossRefGoogle Scholar
- Vellend M, Cornwell WK, Magnuson-Ford K, Mooers AO (2011) Measuring phylogenetic biodiversity. Biological diversity frontiers in measurement and assessment, vol 14. Oxford University Press, Oxford, pp 194–207Google Scholar
- Winter E (2002) The Shapley value. In: Aumann R, Hart S (eds) Handbook of game theory with economic applications, 53, vol 3, 1st edn. Elsevier, Amsterdam, pp 2025–2054Google Scholar