The secretary problem is way overused, and very rarely has any application in practice. This is because it maximizes the probability of finding the best match, and NOT the expectation over the utility of the match your get. This is almost never what you want in practice; in practice, you don’t care much between a match with utility 1000 and a match with utility 999, you just want to avoid a match with utility −1000.
Does the secretary problem say anything about getting the second-best match? That seems to be the crux and this is the counter-argument I’m most swayed by.
is an interesting alternative. But still not great, basically choose the best candidate after sqrt(n) and a non-trivial sqrt(n) chance of ending up at the last candidate.
A solution to the risk of having the best candidate in your test phase; or to the risk of lost opportunity—is to lower the bar; instead of choosing the next candidate that is better than all the other candidates; choose the next candidate that is better than 90% of the candidates so far.
This was my solution to that problem. I find the secretary problem to have a very high risk for low numbers. by the time you get to higher numbers; (>73) the chance of ending up at the last candidate is down at 0.5% or 1/1000.
By using my solution your greatly lower that risk, because not only does the top candidate need to be in the first 1/e trials but the top n% of candidates need to be in the 1/e trials (in my example 10%) to cause you to be stuck with the last candidate.
Of course the reasons for doing that extra weighting are more salient when concern for lost opportunity of exploration time is also factored in. Unfortunately “how much do I care about searching through the partner space for 10 years more” is a different answer for different people (read: different utilities of care/pain/effort factor)
By changing the strategy from “first candidate better than the ones seen in the first n/e” to anything else, you lose all the rigorous mathematical backing that made the secretary problem cool in the first place. Is your solution optimal? Near-optimal? Who knows; it depends on your utility function and the distribution of candidates, and probably involves ugly integrals with no closed-form solution.
The whole point of the secretary problem is that a very precise way of stating the problem has a cool mathematical answer (the n/e strategy). But this precise statement of the problem is almost always useless in practice, so there’s very little insight gained.
Update: yes; secretary problem has a cool and clean mathematical “next bestest candidate after 1/e trials” solution. Real life is a lot more complicated. if you work off that solution it has a 1/e chance of selecting the last candidate. which personally is atrocious odds to be playing around with. Considering the above mentioned opportunity cost ticking-time race, I need better odds than that. even if it sacrifices my chances of finding the best candidate.
if you are at the 1/e*n point and you have passed the best candidate you will end up at the last candidate. If you have any suspicion that you have passed the best (or very good candidates) maybe its time to change your rule to select the best candidate excluding one known candidate. or (again) the candidate that is better than 90% of existing trials.
By changing the strategy from “first candidate better than the ones seen in the first n/e” to anything else, you lose all the rigorous mathematical backing that made the secretary problem cool in the first place.
I didn’t mathematically model my variation. I strongly considered doing so...
The reason being that the difference in my head was as clear as an addition sum. It really seems like a minor change.
I am going to explain the difference again and I’d like you to try to explain what I might have overlooked in modifying the original solution.
existing solution: choose the next candidate after 1/e trials that is better than all existing trials.
my variation: choose the next candidate after 1/e trials that is better than 90% of existing trials.
Why?: if you have a low number of candidates: worked solution − 10 candidates. you should (according to the secretary problem) interview 4 candidates, then select the next one that is better than the ones before.
I believe the sum from ((1/e)*n)->n of 1/n will yield the chance of passing the best candidate in the 1/e trials.
This was the largest factor I was trying to avoid by changing the criteria to >90% of all trials.
my variation: choose the next candidate after 1/e trials that is better than 90% of existing trials. Why?: if you have a low number of candidates: worked solution − 10 candidates. you should (according to the secretary problem) interview 4 candidates, then select the next one that is better than the ones before.
Why n/e, and not some other number? Why 90%, and not some other amount? Come to think of it, shouldn’t the value of the candidates matter, and not just the rank? For example, if I know my candidates’ utility is sampled from [-1000,1000] and the first candidate I see has value 1000, would you recommend that I discard her? Or if I don’t know the range, do I at least have a prior distribution for it?
yes; the problem of distribution is going to mess with things. If for example you knew that candidates utilities spiked at age 25, you should game the system and aim for 25 year olds.
If you had prior knowledge of the candidates being all −1000 utility except for one which is +1000, then you shouldn’t rely on this system at all.
The premise of the problem is that n candidates can be ranked 1 to n. (which is not necessarily true for real life). The nature of the standard solution to the secretary problem is to give you the best candidate 1/e of the time; the last candidate 1/e of the time and other candidates the rest of the time.
Without challenging half of the known world of my Internet friends to derive their own mathematical theory as to solving the secretary problem; starting with a neat solution and dragging an applicable one out of it is my best option.
If you think that I am putting too much pseudo into the mathematics of the application; I’d encourage you to say so. If you think this is too far from applicable then I’d also encourage you to say so. (Please tell me I am wrong, I would rather be wrong than nice, and wrong than vague)
There are certainly flaws in applying this known problem/solution to real life. As many other people have pointed out other prominent edge cases, (returning to partners after a few years; having children without being married). I came up with this concept when talking to a person who needed clarity on a similar issue of deciding whether he should settle down; and found it applicable enough to help him; and also while honing it down I found it applicable enough to myself to help me.
Are you suggesting it shouldn’t be applicable at all? Or also that it doesn’t work for you? I will concede that this idea will not work for many many many people.
The secretary problem is way overused, and very rarely has any application in practice. This is because it maximizes the probability of finding the best match, and NOT the expectation over the utility of the match your get. This is almost never what you want in practice; in practice, you don’t care much between a match with utility 1000 and a match with utility 999, you just want to avoid a match with utility −1000.
Does the secretary problem say anything about getting the second-best match? That seems to be the crux and this is the counter-argument I’m most swayed by.
https://en.wikipedia.org/wiki/Secretary_problem#Cardinal_payoff_variant
is an interesting alternative. But still not great, basically choose the best candidate after sqrt(n) and a non-trivial sqrt(n) chance of ending up at the last candidate.
This was my solution to that problem. I find the secretary problem to have a very high risk for low numbers. by the time you get to higher numbers; (>73) the chance of ending up at the last candidate is down at 0.5% or 1/1000.
By using my solution your greatly lower that risk, because not only does the top candidate need to be in the first 1/e trials but the top n% of candidates need to be in the 1/e trials (in my example 10%) to cause you to be stuck with the last candidate.
Of course the reasons for doing that extra weighting are more salient when concern for lost opportunity of exploration time is also factored in. Unfortunately “how much do I care about searching through the partner space for 10 years more” is a different answer for different people (read: different utilities of care/pain/effort factor)
Does this make sense? (/answer your comment?)
By changing the strategy from “first candidate better than the ones seen in the first n/e” to anything else, you lose all the rigorous mathematical backing that made the secretary problem cool in the first place. Is your solution optimal? Near-optimal? Who knows; it depends on your utility function and the distribution of candidates, and probably involves ugly integrals with no closed-form solution.
The whole point of the secretary problem is that a very precise way of stating the problem has a cool mathematical answer (the n/e strategy). But this precise statement of the problem is almost always useless in practice, so there’s very little insight gained.
Update: yes; secretary problem has a cool and clean mathematical “next bestest candidate after 1/e trials” solution. Real life is a lot more complicated. if you work off that solution it has a 1/e chance of selecting the last candidate. which personally is atrocious odds to be playing around with. Considering the above mentioned opportunity cost ticking-time race, I need better odds than that. even if it sacrifices my chances of finding the best candidate.
if you are at the 1/e*n point and you have passed the best candidate you will end up at the last candidate. If you have any suspicion that you have passed the best (or very good candidates) maybe its time to change your rule to select the best candidate excluding one known candidate. or (again) the candidate that is better than 90% of existing trials.
I didn’t mathematically model my variation. I strongly considered doing so...
The reason being that the difference in my head was as clear as an addition sum. It really seems like a minor change.
I am going to explain the difference again and I’d like you to try to explain what I might have overlooked in modifying the original solution.
existing solution: choose the next candidate after 1/e trials that is better than all existing trials.
my variation: choose the next candidate after 1/e trials that is better than 90% of existing trials. Why?: if you have a low number of candidates: worked solution − 10 candidates. you should (according to the secretary problem) interview 4 candidates, then select the next one that is better than the ones before.
I believe the sum from ((1/e)*n)->n of 1/n will yield the chance of passing the best candidate in the 1/e trials.
This was the largest factor I was trying to avoid by changing the criteria to >90% of all trials.
I will ask around and get back to you.
Why n/e, and not some other number? Why 90%, and not some other amount? Come to think of it, shouldn’t the value of the candidates matter, and not just the rank? For example, if I know my candidates’ utility is sampled from [-1000,1000] and the first candidate I see has value 1000, would you recommend that I discard her? Or if I don’t know the range, do I at least have a prior distribution for it?
yes; the problem of distribution is going to mess with things. If for example you knew that candidates utilities spiked at age 25, you should game the system and aim for 25 year olds.
If you had prior knowledge of the candidates being all −1000 utility except for one which is +1000, then you shouldn’t rely on this system at all.
The premise of the problem is that n candidates can be ranked 1 to n. (which is not necessarily true for real life). The nature of the standard solution to the secretary problem is to give you the best candidate 1/e of the time; the last candidate 1/e of the time and other candidates the rest of the time.
Without challenging half of the known world of my Internet friends to derive their own mathematical theory as to solving the secretary problem; starting with a neat solution and dragging an applicable one out of it is my best option.
If you think that I am putting too much pseudo into the mathematics of the application; I’d encourage you to say so. If you think this is too far from applicable then I’d also encourage you to say so. (Please tell me I am wrong, I would rather be wrong than nice, and wrong than vague)
There are certainly flaws in applying this known problem/solution to real life. As many other people have pointed out other prominent edge cases, (returning to partners after a few years; having children without being married). I came up with this concept when talking to a person who needed clarity on a similar issue of deciding whether he should settle down; and found it applicable enough to help him; and also while honing it down I found it applicable enough to myself to help me.
Are you suggesting it shouldn’t be applicable at all? Or also that it doesn’t work for you? I will concede that this idea will not work for many many many people.