$(1)[5+5+5=15$ points$]$

In this problem, we will consider what happens in stable matching when we are guaranteed additional

structure on the preferences.

$($a$)$ Suppose that there is a $$most popular$$ man, whom all women rank first. Describe who this man

must be matched with in every stable matching, and explain why.

$($b$)$ Use your insight from the previous part to analyze the case when all women have identical rankings$\mathrm{.}1$ Prove that in this case, there is a unique stable matching, and describe it$.$

$($c$)$ For the case of the previous problem, describe a simpler algorithm than Gale$-$Shapley which

never has to break any engagements, and computes the unique stable matching. Prove that your

algorithm outputs a stable matching. $($Your proof may heavily draw on the previous subproblems.$)$