The biggest problem with defining θ, there is a high likelihood of an attack stat soft cap common to grindy MMOs. Unlike other MMOs, Empires does not tell us the damage range. So a soft cap might be complicating our data set.
(Has SG 100% nerfed color stacking? - #63 by Garanwyn)
([Research] Does Color Stacking have a soft cap? or Attack stat research)