The version of the secretary problem to be considered here is the one where the hirer must pick the best or second-best secretary using exactly one choice. This answer provides the optimal strategy, along with a few papers on the problem, but it doesn't quite address the question below.
The asymptotic formula in the case of the traditional secretary problem, given some percentage of applicants to reject (as I understand it), is given by $P(x)=x\int_{x}^{1}\frac{1}{t}dt$. The motivation behind this question is the observation that the asymptotic formula is almost intuitive if you stare at it long enough. Almost, just to be sure, but one can be convinced that it does in fact return the asymptotic probability without getting into the full derivation.
Noting this, I would think there should be an analogous formula for the modified version of the secretary problem, whereby one can derive optimal percentages. Since the search space would be $[0,1]\times[0,1]$, a reasonable guess should be that a new asymptotic formula would be a single double integral.
Further, there are more options for new variables to be created. If, say, $a$ were the first cutoff, and $b$, the second ($1\leq a,b\leq N$ and $a<b$), then some new variables one could define might be in terms of $\frac{a}{N}$,$\frac{b}{N}$, or $\frac{b-a}{N}$. Any two of those should provide enough information to find the probability, so I suppose the choice is arbitrary. Of course, there could be some other variables one would have to create to make this work, but I don't know of any.
Now, to get to the actual question: is there some intuitive double integral one could write down that would provide the asymptotic probability of picking the best or second-best secretary? Is it possible to derive such a formula without starting with the discrete case and then approximating sums with integrals? Note that I pose the latter question as a criterion by which to measure the intuitiveness of any formula one could come up with.
