Note that you can think of this as a theme (marriages) used to explain an abstract mathematical problem in an intuitive way. The math isn't used for marriages; it's used for other things. These...
Note that you can think of this as a theme (marriages) used to explain an abstract mathematical problem in an intuitive way. The math isn't used for marriages; it's used for other things.
These days the theme is badly out of date (no same sex-marriages) so it should probably be swapped out for something less distracting, like maybe one of the real-world examples.
I wager in principle it's not hard to add in modern complexities. Just add for some subset of populations have preferences for the same sex. Then in some sense it's just an extra loop to handle...
I wager in principle it's not hard to add in modern complexities. Just add for some subset of populations have preferences for the same sex. Then in some sense it's just an extra loop to handle these pairings.
Actually, you can start to get really complicated, adding in ace people, poly, enby etc. Simple models can always be extended.
Well, sort of, not really. Without the strict two groups of people, it basically devolves into the stable roommate problem, which unlike stable marriage, is not guaranteed to have a solution at all!
Well, sort of, not really. Without the strict two groups of people, it basically devolves into the stable roommate problem, which unlike stable marriage, is not guaranteed to have a solution at all!
Sure, I suppose that's true. But then maybe you can analyse the non-equilibrium behaviour, which could be interesting (I actually don't know, I've never looked into this in detail). Hey, random...
Sure, I suppose that's true. But then maybe you can analyse the non-equilibrium behaviour, which could be interesting (I actually don't know, I've never looked into this in detail). Hey, random thought, add a driving term to the model (i.e. Add people to the population over time) and then see if there's an equilibrium solution.
It's not really an empirical problem (i.e we don't know if it can be solved), but rather it's proven that for all n > 2 that there exists preference rankings such that no stable pairing can exist,...
It's not really an empirical problem (i.e we don't know if it can be solved), but rather it's proven that for all n > 2 that there exists preference rankings such that no stable pairing can exist, at all. The non-equilibria behavior is not particularly interesting, in fact it's pretty simple for a 4 person group to come up with preferences such that they loop.
For anyone interested in real-world applications, the largest I'm aware of is matching soon-to-graduate MDs to residency programs in the US. (Because doctors are amazing at naming things, this...
For anyone interested in real-world applications, the largest I'm aware of is matching soon-to-graduate MDs to residency programs in the US. (Because doctors are amazing at naming things, this process is called "the match".) Applicants rank programs, programs rank applicants, and in March the whole shebang is processed and each applicant is assigned to a program in a stable pairing. (Hopefully. Also there are some complications to handle real-world cases like "hey I'm married to another new MD and we're not willing to go long-distance for three to seven years of post-doctoral training".)
There are definitely odd practical consequences to this procedure. Applicants are told which program they match to (and thus where they're going to live for the next three to seven years), for instance, rather than getting to choose between offers as in undergraduate admissions. It's generally agreed to be far preferable to the previous regime, though, which was essentially a free-for-all in which programs wielded a disproportionate amount of power (e.g. programs were known to make exploding offers—"accept in the next 24 hours or this offer is rescinded"—which are horribly coercive to applicants in an extremely competitive process).
An interesting thing to note is that the graduating Doctors are the men (the proposing group) in stable marriage, and hospitals are the women. Stable marriage is a male-optimal algorithm (or...
An interesting thing to note is that the graduating Doctors are the men (the proposing group) in stable marriage, and hospitals are the women. Stable marriage is a male-optimal algorithm (or proposal-optimal since male and female are arbitrary distinctions for the algorithm, I guess to the people who made it having the men propose just made more sense, make of that what you will) somewhat unintuitively, after all the proposing group is the one getting rejected over and over again.
So this produces the best stable matching for doctors and the worst stable matching for hospitals (with worst and best being the end preference ranking they get).
“The kinds of people who wait till their thirties to get married may be the kinds of people who aren’t predisposed toward doing well in their marriages,” he writes. This also means “people who marry later face a pool of potential spouses that has been winnowed down to exclude the individuals most predisposed to succeed at matrimony.”
Other sociologists who cover this waterfront were quick to weigh in with doubts. The University of Maryland’s Phillip Cohen used a different set of data, from the American Community Survey, to say that getting older didn’t mean your marriage had less chance of survival. According to his analysis, the perfect age to get married if you don’t want to get divorced is 45 to 49, which, he notes, is why people shouldn’t make life decisions based on statistical analyses on the Internet.
Note that you can think of this as a theme (marriages) used to explain an abstract mathematical problem in an intuitive way. The math isn't used for marriages; it's used for other things.
These days the theme is badly out of date (no same sex-marriages) so it should probably be swapped out for something less distracting, like maybe one of the real-world examples.
I wager in principle it's not hard to add in modern complexities. Just add for some subset of populations have preferences for the same sex. Then in some sense it's just an extra loop to handle these pairings.
Actually, you can start to get really complicated, adding in ace people, poly, enby etc. Simple models can always be extended.
Well, sort of, not really. Without the strict two groups of people, it basically devolves into the stable roommate problem, which unlike stable marriage, is not guaranteed to have a solution at all!
Sure, I suppose that's true. But then maybe you can analyse the non-equilibrium behaviour, which could be interesting (I actually don't know, I've never looked into this in detail). Hey, random thought, add a driving term to the model (i.e. Add people to the population over time) and then see if there's an equilibrium solution.
It's not really an empirical problem (i.e we don't know if it can be solved), but rather it's proven that for all n > 2 that there exists preference rankings such that no stable pairing can exist, at all. The non-equilibria behavior is not particularly interesting, in fact it's pretty simple for a 4 person group to come up with preferences such that they loop.
For anyone interested in real-world applications, the largest I'm aware of is matching soon-to-graduate MDs to residency programs in the US. (Because doctors are amazing at naming things, this process is called "the match".) Applicants rank programs, programs rank applicants, and in March the whole shebang is processed and each applicant is assigned to a program in a stable pairing. (Hopefully. Also there are some complications to handle real-world cases like "hey I'm married to another new MD and we're not willing to go long-distance for three to seven years of post-doctoral training".)
There are definitely odd practical consequences to this procedure. Applicants are told which program they match to (and thus where they're going to live for the next three to seven years), for instance, rather than getting to choose between offers as in undergraduate admissions. It's generally agreed to be far preferable to the previous regime, though, which was essentially a free-for-all in which programs wielded a disproportionate amount of power (e.g. programs were known to make exploding offers—"accept in the next 24 hours or this offer is rescinded"—which are horribly coercive to applicants in an extremely competitive process).
An interesting thing to note is that the graduating Doctors are the men (the proposing group) in stable marriage, and hospitals are the women. Stable marriage is a male-optimal algorithm (or proposal-optimal since male and female are arbitrary distinctions for the algorithm, I guess to the people who made it having the men propose just made more sense, make of that what you will) somewhat unintuitively, after all the proposing group is the one getting rejected over and over again.
So this produces the best stable matching for doctors and the worst stable matching for hospitals (with worst and best being the end preference ranking they get).
The follow up, relating this to modern dating, is at https://www.arvarik.com/the-stable-marriage-problem-and-modern-dating.
Fred & Wilma? Really? :)
The more interesting problem is when to get married
Related: https://en.m.wikipedia.org/wiki/Secretary_problem
Well now I’m just bummed out.
Maybe you didn't read the entire article?