In the secretary problem, we assume that your ability to distinguish a good secretary from a bad one is fully developed from the beginning. In the examples given, you spent a significant amount of time (maybe even your whole life) figuring out what makes a good partner, apartment, or job. It's a neat problem, but I would be concerned for anyone who uses it as a life guide.
My favorite math problem that offers a realistic solution to a practical problem is Sperner's Lemma (implemented here [1])
Yes, the Algorithm changes if you can go back to an earlier partner. You can keep looking further beyond 37% and defer decision making.
I wouldn't recommend dating multiple people at once.
Say your goal is to find your soulmate in next 2 years, let's define n =24 months. Then you keep looking for the next 9 months without committing to any one. Let's say the name of your perfect partner in the first 37% is Max. Then you start looking beyond the first 37%, the first person better than Max is your soulmate. Considering there is chance of refusal or rejection, you can start earlier (follow the above algorithm at 33% )
Although this result sets up in some way a baseline, n (the number of choices) should be a random variable N in order to make things a bit more realistic.
The `1/e-law of best choice` mentioned on there was interesting, and helped me accept the conclusion regarding finding love. With the previous examples, love and similar concepts didn't seem like they would work well when N could theoretically be infinite. Though, after some help understanding the wiki explanation [1], applying the rule over the finite _period of time_ during which you see options, rather than the total number of options, makes much more sense and seems more agreeable.
My favorite math problem that offers a realistic solution to a practical problem is Sperner's Lemma (implemented here [1])
[1] https://www.nytimes.com/interactive/2014/science/rent-divisi...