Depending on the which gender does the initiation; the result is either male optimal (males get the best he can get, females gets the worst among those she would tolerate) or female optimal. I wonder which version this one is.
Imagine A ranks X,Y,Z in that order B ranks Y,X,Z in that order C ranks X,Y,Z
X, Y, Z rank A,B,C in that order.
A will propose to X and match. B will propose to Y and match. C will propose to X and get rejected. C will propose to Y and get rejected. C will propose to Z and match.
Y will never get a proposal from A. Z will never get a proposal from A or B.
Edit: I think I fixed it.
Maybe 4 participants is too few to clearly see what happens.
The minimal example with 4 participants is if each person P has unique preferences and P's 1st choice has P as 2nd choice. So in this case just flip X:
A:XY B:YX X:BA Y:AB
If A,B propose you get AX,BY. If X,Y propose you get BX,AY. Both are stable because the proposers are getting their first choice.
This obviously doesn't factor in most of the complexity of romantic relationships: not knowing your own preferences, evolving preferences, incomplete/imperfect information in general, etc. Plus it assumes a global 1:1 matching across two categories, and people can of course be LGBT or choose to be single.
Which may be largely true for many people but it's definitely not a fixed thing!
When I was dating, I initiated with a lot of women, but the woman I wound up marrying messaged me first.
- proposers descend from their 1st choice, while recipients ascend according to their offers; the outcome can't be recipient-biased, because recipients ascend only when proposers are forced to descend
- a matching is stable if each pair contains at least one party that cannot find a strictly better match
- in particular there are multiple distinct matchings, and if proposer-biased, recipients have no recourse to break the proposer-favored pairs
An applicant (the receiver) will end up with a "worst" match in the sense that if you look at all "stable" pairings (there can be multiple configurations), the applicant is going to have the least favorable one among the different pairings.
There's an important concept of a "blocking pair", and the Gale-Shapley algorithm is trying to eliminate all blocking pairs, and you could say that pairings are stable when no blocking pairs exist. A blocking pair is a job applicant and employer who aren't matched together, but both prefer each other over their current match.
Suppose that after the algorithm is done with its work, Alice doesn't end up with a job offer from her top choice, Google. This must mean that Google never offered her a job because Alice was just too far down on their list, and Google made a deal with someone else they like better than Alice.
I don’t understand why this would always be true. In your example, Google may indeed like Alice, so they both get their top choice.
When Alice and Google are not paired up, then they will always form a “blocking pair” in any matching where they aren’t together so that means any matching that doesn’t have Alice and Google paired is not stable.
I think the confusion is that Alice only has one possible stable employer and thus her worst one happens to be her best one.
If Alice’s top choice wasn’t Google, but some other company, then there might be multiple employers she could be matched with to form a stable pairing.
This algorithm is definitely weird too though because it’s possible for some candidates to be paired with a company they don’t really like, but it’s considered a stable situation because the other companies don’t want them more than their current employees.
I guess in real life, you could try improving yourself and then the matching becomes unstable again if preferences change.
Thus extrapolating to a dating app, it’s possible for someone in the matched pair to not really like the other person. Sounds like a rousingly success.
And doesn't something of the same kind happen in real-life, in some societies or countries or cities male-optimal and in others female-optimal ?