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So one of my good friends is starting to date again (after being out of the country for two years), and I think that it might be helpful, or at least fun, to keep track of her dates in a ranked fashion so that we can always be on the look-out for the optimum stopping point (i.e. who she should marry) in a semi-rigourous fashion (yes, we're nerding out about this). So I understand what the procedure is for the secretary problem with a known n, but since we're going to be doing this on the fly, how do we know when to accept the new best ranked guy as the one? Thanks!

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If there is an infinite number of "applicants", it will always be better to wait until we find some better guy. Hence, there needs to be some constraint on the number of "applicants": for instance, they are examined at some fixed rate, and the output gets smaller as we wait more time. However, this requires to fix a decay rate on the output (exponential, sigmoid, with thresholds, etc.) and a way to rate objectively the guys. – D. Thomine Jul 8 '12 at 22:51

As asked, you should estimate how many candidates there will be, then divide by e. It is clearly not 1,000,000 and probably not 10, either. I think if you study it, the optimum is rather flat, so being off somewhat is not that big a deal. There are many "real life" things that modify the problem. The two largest that I think of are first, that as you meet candidates, you get an idea of the distribution, so can make a more informed decision and second, there is an opportunity cost of waiting, which should bias you early.

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This is probably one of the most reasonable answers. However, for the purposes of "nerding", a complete (or at least detailed) but unreasonable model might be better. Especially since, as you hint, there are some very nice mathematics behind. The author will have to judge, of course. – D. Thomine Jul 8 '12 at 23:07

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