The famed secretary problem asks when you should decide to hire, given that you cannot postpone the decision, and that you can objectively rank the candidates. I've seen that the solution for getting the best expected value candidate is to stop after $\sqrt{N}$ candidates, for a given $N$ total candidates.
I'm wondering if there are any closed solutions for the following modification:
There are an infinite amount of secretaries, but the rate they apply is constant, say $f$.
Time has value - so that one is willing to compromise in exchange for less time without a secretary. That is, one is interested in maximizing the expected value of $X_t - g(t)$, where $g(t)$ is some increasing function of time.
I feel this problem is a more realistic model of the real world problem, but I haven’t seen a solution for it. Any ideas?
This kind of problem is known as "optimal stopping," and there's an entire theory of optimal stopping with lots of textbooks and courses available. Some of the variants you've described above are analyzed here, for instance:
http://www.math.ucla.edu/~tom/Stopping/sr6.pdf