Equality of Shapley value and fair proportion index in phylogenetic trees

Abstract

The Shapley value and the fair proportion index of phylogenetic trees have been introduced recently for the purpose of making conservation decisions in genetics. Moreover, also very recently, Hartmann (J Math Biol 67:1163–1170, 2013) has presented data which shows that there is a strong correlation between a slightly modified version of the Shapley value (which we call the modified Shapley value) and the fair proportion index. He gave an explanation of this correlation by showing that the contribution of both indices to an edge of the tree becomes identical as the number of taxa tends to infinity. In this note, we show that the Shapley value and the fair proportion index are in fact the same. Moreover, we also consider the modified Shapley value and show that its covariance with the fair proportion index in random phylogenetic trees under the Yule-Harding model and uniform model is indeed close to one.

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References

  1. Chang H, Fuchs M (2010) Limit theorems for patterns in phylogenetic trees. J Math Biol 60(4):481–512

    MATH  MathSciNet  Article  Google Scholar 

  2. Flajolet P, Sedgewick R (2009) Analytic Combinatorics. Cambridge University Press

  3. Haake C, Kashiwada A, Su FE (2008) The Shapley value of phylogenetic trees. J Math Biol 56:479–497

    MATH  MathSciNet  Article  Google Scholar 

  4. Hartmann K (2013) The equivalence of two phylogenetic biodiversity measures: the Shapley value and fair proportion index. J Math Biol 67:1163–1170

    MATH  MathSciNet  Article  Google Scholar 

  5. Mahmoud HM (1992) Evolution of Random Search Trees. Wiley, New York

    Google Scholar 

  6. Meir A, Moon JW (1978) On the altitude of nodes in random trees. Canad J Math 30(5):997–1015

    MATH  MathSciNet  Article  Google Scholar 

  7. Redding D, Hartmann K, Mimotot A, Bokal D, DeVos M, Mooers AO (2008) The most “original species” often capture more phylogenetic diversity than expected. J Theor Biol 251:606–615

    Article  Google Scholar 

  8. Semple C, Steel M (2003) Phylogenetics, Oxford lecture series in Mathematics and its applications, vol 24. Oxford University Press

Download references

Acknowledgments

The authors are grateful to the anonymous referees for many good suggestions. This research was done while both authors visited the Institut für Diskrete Mathematik und Geometrie, Technical University of Vienna. The authors thank the department for hospitality. The first author was partially supported by NSC grants NSC-102-2918-I-009-012 and MOST-103-2115-M-009-007-MY2; the second author was supported by the grant from German Research Foundation (DFG), JI 207/1-1.

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Correspondence to Emma Yu Jin.

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Fuchs, M., Jin, E.Y. Equality of Shapley value and fair proportion index in phylogenetic trees. J. Math. Biol. 71, 1133–1147 (2015). https://doi.org/10.1007/s00285-014-0853-0

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Keywords

  • Phylogenetic trees
  • Shapley value
  • Fair proportion index
  • Moments
  • Correlation

Mathematics Subject Classification

  • 92B99
  • 05C05
  • 60C05