# On the Shapley Value of Unrooted Phylogenetic Trees

- 42 Downloads

## Abstract

The Shapley value, a solution concept from cooperative game theory, has recently been considered for both unrooted and rooted phylogenetic trees. Here, we focus on the Shapley value of unrooted trees and first revisit the so-called split counts of a phylogenetic tree and the Shapley transformation matrix that allows for the calculation of the Shapley value from the edge lengths of a tree. We show that non-isomorphic trees may have permutation-equivalent Shapley transformation matrices and permutation-equivalent null spaces. This implies that estimating the split counts associated with a tree or the Shapley values of its leaves does not suffice to reconstruct the correct tree topology. We then turn to the use of the Shapley value as a prioritization criterion in biodiversity conservation and compare it to a greedy solution concept. Here, we show that for certain phylogenetic trees, the Shapley value may fail as a prioritization criterion, meaning that the diversity spanned by the top *k* species (ranked by their Shapley values) cannot approximate the total diversity of all *n* species.

## Keywords

Phylogenetic tree Shapley value Shapley transformation Noah’s ark problem## Notes

### Acknowledgements

The first author thanks the Ernst-Moritz-Arndt-University Greifswald for the Landesgraduiertenförderung studentship, under which this work was conducted, and the Barcelona Graduate school of Mathematics (BGSMath) for financial support for attending the Algebraic and Combinatorial Phylogenetics program in Barcelona in June 2017, during which some of the results presented in this manuscript were obtained.

## References

- Fischer M, Liebscher V (2015) On the balance of unrooted trees. arXiv:1510.07882
- Fuchs M, Jin EY (2015) Equality of Shapley value and fair proportion index in phylogenetic trees. J Math Biol 71(5):1133–1147. https://doi.org/10.1007/s00285-014-0853-0MathSciNetCrossRefMATHGoogle Scholar
- Haake CJ, Kashiwada A, Su FE (2008) The Shapley value of phylogenetic trees. J Math Biol 56(4):479–497. https://doi.org/10.1007/s00285-007-0126-2MathSciNetCrossRefMATHGoogle Scholar
- Hartmann K (2013) The equivalence of two phylogenetic biodiversity measures: the Shapley value and Fair Proportion index. J Math Biol 67(5):1163–1170. https://doi.org/10.1007/s00285-012-0585-yMathSciNetCrossRefMATHGoogle Scholar
- Semple C, Steel M (2003) Phylogenetics. Oxford lecture series in mathematics and its applications. Oxford University Press, OxfordGoogle Scholar
- Steel M (2005) Phylogenetic diversity and the greedy algorithm. Syst Biol 54(4):527–529CrossRefGoogle Scholar
- Steel M (2016) Phylogeny: discrete and random processes in evolution (CBMS-NSF regional conference series). SIAM-Society for Industrial and Applied Mathematics, Philadelphia. ISBN: 978-1-611974-47-8Google Scholar
- Weitzman ML (1998) The Noah’s ark problem. Econometrica 66(6):1279. https://doi.org/10.2307/2999617MathSciNetCrossRefMATHGoogle Scholar
- Wicke K, Fischer M (2017) Comparing the rankings obtained from two biodiversity indices: the Fair Proportion Index and the Shapley value. J Theor Biol 430:207–214. http://www.sciencedirect.com/science/article/pii/S0022519317303430
- Wolfram Research Inc (2017) Mathematica 11.1. http://www.wolfram.com