In the marriage problem we have an equal number of men and women who we want to pair up (it is presumed they are all heterosexual). Each one has an ordered list of their prefered mates. The goal is to find the set of marriages such that no two people would rather get divorced and marry each other. That is, not two prefer each other over their assigned mates.
This problem can be solved with a simple algorithm where all the men propose to the women. Then, each woman with more than one proposal rejects all but her most preferred which she temporarily accepts. The bachelors then repeat the process by asking their most favorite woman who has not rejected them. The process continues until all women have a partner. I have a NetLogo implementation of the marriage problem that illustrates this process.
It is interesting to notice that while the women are the ones that get to turn down the men, the men do overwhelmingly better than the women. I mentioned this to my wife and she said that is why she proposed to me. Of course, in the end it is always the women who propose to us.
I am interested in finding distributed algorithms that arrive at stable solutions, that is, those where no two people would rather be married to each other than to their partner. Here is a link to a set of slides that goes into more detail on the marriage problem, in case you want to learn a bit more.