 Ranked university application game

There are $mq$ distinctly ranked students and $m$ labelled universities with $q$ seats each. Each student applies to one university; each university selects the greatest ranks, and admitted students receive their admitted cohort's average rank. At a pure Nash equilibrium, every student is admitted: any rejected student could move to a university with a vacancy. The university with the largest average must contain the top $q$ ranks, since any omitted higher rank could displace its lowest rank and improve. Repeat on the remaining students to obtain consecutive rank blocks. Assigning these blocks to the labelled universities gives exactly $m!$ pure equilibria.