Run both variants of Gale-Shapley side by side: when men propose (men-optimal)
vs when women propose (women-optimal). Then compare to see who benefits under each.
Men Propose (Men-Optimal)
Women Propose (Women-Optimal)
Comparison of Results
Conway's Lattice Theorem
The set of all stable matchings forms a distributive lattice (Conway, 1976).
The men-optimal matching sits at the top of this lattice (best for all men simultaneously),
and the women-optimal matching sits at the bottom (best for all women simultaneously).
Every other stable matching lies somewhere between these two extremes.