I'm having some thoughts and I'd like to have your opinion about it. Given a basic, normal instance of SMP (men-women, men choose), we define $\mu$ to be the set of all stable matches.
Each member of $\mu$ consists of men-rank, which describes the satisfaction index of the men - which in this case the member with the highest rank is the male-optimal match.
So if we run the stable-match known algorithm, the output would be the match with the highest rank, because the algorithm always returns the men-optimal and female-pessimal match - lets denote this match by $A$.
Now, lets replace the roles, and run the algorithm again. We denote the output match by $B$ - which is the female-optimal and male-pessimal match.
- Can we say that this instance has a unique stable match?
- What else can we say about this instance?