# algorithms – Glae-Shapley Stable matching where one man’s preference list changes

Given $$n>1$$ women and men. Let $$M$$ be the stable matching given by the Gale Shapley algo with men proposing. Is there a stable matching instance such that:

changing one man’s preference list results in matching $$M’$$ – a stable matching given by the Gale Shapley algo with men proposing under this slightly altered preference list – where every man strictly prefers his partner in $$M$$ over his partner in $$M’$$ (according to their original preference list used to get matching $$M$$).

The answer should be yes but I’m not sure how or why. The first thing that came to mind is to try to use induction. I came up with a scenario for $$n=2$$ but I don’t know how to extend it to greater $$n$$.