Correlation between Shapley values of rooted phylogenetic trees under the beta-splitting model

In recent years, several different versions of the Shapley value have been introduced in phylogenetics for the purpose of ranking biodiversity data in order to decide whether to preserve the data or not. Two of these Shapley values are the rooted and unrooted Shapley value which have been compared with the fair proportion index since this index is easier to compute. In particular, it was proved for the former that it is identical with the fair proportion index and numerical data was presented by several authors that the latter is strongly correlated with the fair proportion index. In this paper, we will prove a theoretical result which supports this observation. More precisely, we will prove that in random phylogenetic trees under the β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-splitting model, the correlation coefficient between the unrooted Shapley value and the fair proportion index indeed tends to one for all β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} with β>-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta >-\,1$$\end{document}. We also present data which suggests that the convergence worsens as β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} is approaching -1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\,1$$\end{document}.