Let's consider a system of $n$ men and women. Each woman is paired with one man (there are only pairings between a woman and a man in this system). There are $n!$ possible distinct pairings. I refer to a pairing as the state of the system. Every man and every woman has a preference list for their potential partners. It is not possible that a woman is equally attracted to two men and it is also impossible that a man is equally attracted to two women.
When a woman is higher on the preference list of a man than his current partner and at the same time this man is higher on the preference list than her current partner they will leave their current partners and form a new pair. Their former partners also form a new pair (although they potentially will be less satisfied with their new partner than before the split). In the following I will refer to such a change in the state of the system as an allowed move. I define a state in which no moves are possible as stable.
I have the following questions about this system:
When I start with a state there might be a number of allowed moves (according to the above definition). Will the system always end up in a stable state, no matter in which order I carry out these moves? If no, will it at least end up in a stable state if I carry out the moves randomly?
If the answer to the first question is no, is there at least always a stable state?
If there is always a stable state, does it have to be unique? If no, can I evolve into different stable states depending on the order in which I carry out the allowed moves starting from the same initial state?