The James function, also known as the log5 method, assigns a probability to the result of a competition between two teams based on their respective winning percentages. This paper, which builds on earlier work of the authors and Steven J. Miller, exp
lores the analogous situation where a single team or player competes simultaneously against multiple opponents.
Building on techniques developed by Cowen and Gallardo-Guti{e}rrez, we find a concrete formula for the adjoint of a composition operator with rational symbol acting on the Hardy space $H^{2}$. We consider some specific examples, comparing our formula with several results that were previously known.
