# Changing preference in Gale-Shapley algorithm?

Suppose, in the context of the classic marriage problem, two equal size groups of $n$ men and $n$ women are being matched, with the GS algorithm.

If a man were to switch the order of a pair of women, low on his preference list, say $x$ and $y$, would this yield any sort of advantage with regards to being matched with a woman higher than $x$ and $y$?

This seems like a fairly simple problem; I don't see how switching a pair low on the preference list would increase the chance of being matched with something higher on the list. How would one prove the existence or non-existence of such an advantage?

I've come to the conclusion that this would not yield an advantage, but would still appreciate some guidance with regards to proving that.

The property your wish to prove is known as strategy proofness: Is it possible for an agent to report a preference $P'$ such that it gets matched to a strictly better result w.r.t. its true preference $P$ than reporting truthfully?