The Shapley value of phylogenetic trees

Area of Science:

• Cooperative game theory
• Phylogenetic analysis
• Mathematical biology

Background:

• Weighted trees naturally form cooperative games called tree games.
• Tree games assign values to coalitions of leaves based on subtree weights.
• In phylogenetics, tree games model species diversity within coalitions.

Purpose of the Study:

• To explore the Shapley value of tree games and propose a biological interpretation.
• To determine the linear transformation (M) influencing Shapley values based on tree edge weights.
• To characterize Shapley values on tree games using a novel set of four axioms.

Main Methods:

• Defined tree games based on weighted trees and subsets of leaves.
• Investigated the Shapley value within the framework of tree games.
• Determined the linear transformation M and computed its null space basis.
• Developed an axiomatic characterization for Shapley values on tree games.

Main Results:

• The Shapley value of tree games was analyzed with a proposed biological interpretation.
• The linear transformation M and its null space basis were computed, depending on tree split counts.
• A new set of four axioms was established to characterize the Shapley value on tree games.
• A discussion on the core of tree games was included.

Conclusions:

• The study provides a mathematical framework (tree games) to analyze diversity in phylogenetic contexts using cooperative game theory.
• The findings offer new insights into the Shapley value's properties and its axiomatic characterization in this specific game class.
• This work bridges concepts from graph theory, game theory, and evolutionary biology.