Journal of Mathematical Biology

, Volume 56, Issue 4, pp 479–497 | Cite as

The Shapley value of phylogenetic trees

  • Claus-Jochen Haake
  • Akemi Kashiwada
  • Francis Edward Su


Every weighted tree corresponds naturally to a cooperative game that we call a tree game; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley value on the edge weights of the tree, and we also compute a null space basis of M. Both depend on the split counts of the tree. Finally, we characterize the Shapley value on tree games by four axioms, a counterpart to Shapley’s original theorem on the larger class of cooperative games. We also include a brief discussion of the core of tree games.


Shapley value Core Phylogenetic trees Biodiversity 

Mathematics Subject Classification (2000)

Primary 92D15 Secondary 91A12 05C05 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Claus-Jochen Haake
    • 1
  • Akemi Kashiwada
    • 2
  • Francis Edward Su
    • 2
  1. 1.Institute of Mathematical EconomicsBielefeld UniversityBielefeldGermany
  2. 2.Department of MathematicsHarvey Mudd CollegeClaremontUSA

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