Reason for the non-proposing gender ending with worst valid partners in Gale-Shapley algorithm?

Question

I have gone through the first chapter from Algorithm Design by Kleinberg and Tardos. The Gale-Shapley algorithm and its various properties have been explored there. I would be citing a few of them to set the context for my query.

Note: I am not providing the body of the algorithm or the proofs for the properties, as they are provided in the book already. This is for the sake of brevity.

1. The goal is to find a stable matching (marriage) between n men and n women. They assume the men would be proposing in order of their preference list.
2. They prove that the men end up with their best valid partners. The fact is intuitive as well because the men do the proposing.
3. It is also shown that the women end up with their worst valid partners. The proof for this is convincing, but the intuitive element is missing.

For this third property I do not see an intuitive reason why the women cannot end up with even their last-but-one preferred valid partner. In the algorithm a woman always jumps to better partners. It is understandable why they might not end up with their best valid partners; after all they are not the ones proposing. But what compels them to end up with their worst valid partners? I would prefer to see some hidden, intuitive reason pulling the women down in the algorithm steps.

I would present an example so that the query becomes more valid. Say, we have 4 men m1, m2, m3, m4 and 4 women w1, w2, w3, w4. Their preference lists are -

Men:

• m1: [w1, w2, w3, w4]
• m2: [w3, w1, w4, w2]
• m3: [w3, w2, w1, w4]
• m4: [w4, w2, w1, w3]

Women:

• w1: [m2, m1, m4, m3]
• w2: [m1, m4, m2, m3]
• w3: [m4, m1, m2, m3]
• w4: [m1, m3, m4, m2]

In this case there are three stable matchings -

• (m1,w1), (m2,w3), (m3,w2), (m4,w4)
• (m1,w1), (m2,w3), (m3,w4), (m4,w2)
• (m1,w2), (m2,w1), (m3,w4), (m4,w3)

The first would be the output from G-S algorithm in which men propose in order of their preference list.

w2 has three valid partners - m1, m4 and m3 - in order of her preference list. What is it about the algorith that forces her to end up with m3, and not even m4?

Why does best valid partners for men imply worst valid partners for women? The book's proof starts with the counter assumption and shows that it cannot be. A more direct and convincing proof would be to follow the algorithm steps and show that the women ultimately end with their worst valid partners.