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.
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)
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.
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.
I will ask around and get back to you.
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.
After reading that you can probably see the application to real life.
No, off the top of my head I can't think of any situation where all possible choices other than the best are equally bad.
they don't even like the same music
There are plenty of people the same age who don't like the same music, too.
cute, and fair points; This feels like an attack on the whimsical anec-datas and choice of language that I threw in there to keep people amused and reading further; not the actual things that I wanted to share as content... Is that what you meant to do?
The first point seems like a genuine and substantial objection to the secretary problem's framing of the issue.
You've sort of addressed it by suggesting that maybe we should try a bunch of partners and then choose the next one that's better than 90% so far, rather than the best that's better than all so far, but it seems to me that once you've noticed that P is the wrong problem you should be trying to figure out the right problem and solve that, rather than solving P and then applying ad hoc tweaks to the answer.
I believe people have analysed variants of the secretary problem where, e.g., each candidate has a quality score (drawn independently from some known distribution, I guess) and you want to choose a candidate to maximize expected quality.
[EDITED to add: I see that anon85 has made similar points.]
Is that what you meant to do?
In my second paragraph, yes. In my first paragraph my point was like "garbage in, garbage out".
What if there are prior probabilities over "secretary ability" eg reading their resume or getting references? Has anyone worked out that variant?
Not that I know of. It would be possible to add in a factor. I imagine that you've just increased the "size of the pool". And can proceed accordingly.
What is the nature of the prior knowledge? I suppose predictions are possible.
I know people who read 100 resumes and interview the ones they think would be fit for the role. Anywhere from 0-15 candidates. It depends on how specific the role is and how urgent the job is.
The confusion I'm trying to resolve is it feels like saving the best for last, if you have priors (assuming no attrition in the pool, which is obviously false). The intuition here is you're using some % of the pool for calibration purposes, and should not be using your best prospects for calibration.
This assumes you have no means of estimating how good your current secretary/partner is, other than directly comparing to past options. While it's nice to know what the optimal strategy is in that situation, don't forget that it's not an assumption which holds in practice.
Definitely relevant; most of our experience comes to us through just that - experiencing things. but also it is possible to read books; or borrow knowledge from other people (and we do that) as to good or bad relationships. ultimately so many people still do so much of their learning by trial and error. I hope many people here have found pathways to shortcut the very long process of trial and error.
There are still some errors that are very difficult for one person to trust other people about making. If we could learn what leads so many marriages to divorce (noting that divorce might not be a bad thing), (noting the intent of a marriage is usually to stay together for a long and maybe infinite time), the failure to stay together of so many humans all the time still is an indication that humans are really bad at learning what causes marriages to end in divorce over and over and over again (hopefully a related area to this problem). I want to agree with you; but I see the assumptions here valid in certain ways; after all - the world is a complicated beast :)
I hope you actually liked the writing and also got something out of it. A hard model is no representative of the real world; I tried to start somewhere (in the secretary problem) and diverge. To me there is value in considering the similarities, although limited. I am not sure that you feel the same... (and we may disagree about how much value there is to be had - I felt: enough to mention it)
There seems to be an assumtion that once a relationship has had its active go and there are further relationships that previous partner can't be chosen as the one married. It seems the possibility of getting back to your ex is a option that should be covered. That is after X trials you can retry any sampled so far althought they might not be open to the idea anymore. Is there any additional reason to treat rejection as instantly permanent?
This is entirely true. On a basic level; the assumption that humans will nearly always be in relationships (and often exclusive ones); will lead one to conclude that the opportunity will potentially be gone. I wouldn't be relying on that option coming up; but it is definitely a thing that can happen.
On two levels:
I tried to simplify and skip points like this but this is a notable point. I did mention poly briefly, but knowing how badly it can mess with the basic assumptions means it would need its own post to cover it.
If the reason why you are breaking up is that you want to try to find better candidates the breakup is not that good evidence that the match is bad. While finding the match bad will lead to a breakup, having a breakup doesn't neccesarily mean that the match is bad.
I agree with what you are saying - absolutely!
The situation you are describing is more like 2 (in my response above) in that the reason you broke up (to look for other good matches) is no longer there.
Was there a particular wording of the original post that you would suggest I change or add to clarify this point? (all my posts are open to improvement)
As someone with experience will know - we probably test and rule out bad partners in a single conversation, where we don't even get so far as a date. Or don't last more than a week. (I. E the experience set is growing through various means).
I consider this number to be much more useful than number of long-terms partners due to sample size. You should be able to evaluate most partners in a lot less than 6 months. I think if it takes more than 3 months to evaluate a relationship's potential for marriage, you're doing something wrong. For my part, I've never been in a relationship with someone I thought I might marry. I've only twice known someone I thought I might marry, and neither of those resulted in a relationship.
You have raised some interesting alternative problems to applying this solution to real life. I suspect that most people don't think very hard about it; and they like to enjoy relationships/life/existence for the purpose of existing.
now that I think about it a bit harder; as long as you have a determining factor for how frequently you can process trials (i.e. how long a relationship is, or more specifically how many you can have in each year of your trial-time-period), you can still work out the 1/e time-point, and possibly use that for a heuristic of exploration/exploitation time-point...
You have raised some interesting alternative problems to applying this solution to real life. I suspect that most people don't think very hard about it; and they like to enjoy relationships/life/existence for the purpose of existing.
Apart from enjoyment a relationship also provides for personal growth.
Yea; I had to skip all the other reasons to have relationships in order to specifically attack the reduced problem of the assumption of:
you plan to have children and you will probably be doing so like billions of humans have done so far in a monogamous relationship while married
There are certainly companionship values for having relationships that are worthwhile doing; but that's not what this post was about.
I don't think the plan to have children is simply about finding a suitable mate. It's also about developing the skill set of being a good parent and having an enduring relationship that doesn't break apart after a few years.
https://www.ted.com/talks/amy_webb_how_i_hacked_online_dating
this is a relevant ted talk. She does a similar process to what I worked through.
Just noticed that the secretary problem is not applicable if you have multiple irons in the fire... Not exactly an unheard of strategy...
(the entire set of assumptions does not break down for poly relationships or relationship-anarchy, but it gets more complicated).
I tried to mention this above; but didn't want to cover it entirely as its no longer part of the largest subset of people to which the rules can apply, and requires specific considerations for this set. I would rather leave it as an un-tackled piece than a badly-tackled piece.
It seems that several of the "edge but big enough" case have been brought up by people in relation to what has been missing from my set of rules-to-try-to fit to. I considered these edge-but-big-enough cases not big enough to mention; but I suspect individual variation on these considerations causes people to define big-enough differently and decide them to be more worth mentioning. This is an interesting pattern to note and in future I will list edge cases that I have acknowledged but not tackled in the main post.
One problem I don't see covered is that your preferences, and your ability to evaluate a partner by those preferences, likely changes significantly over the sampling period.
I highly recommend the movie Mr. Nobody as a meditation on knowledge and choice.
A boy grows up living a Many Worlds life, aware of the different paths. He's interviewed as an old man. I try to keep Old Nemo in mind when I'm hemming and hawing over some choice.
Nemo age 9: You have to make the right choice. As long as you don't choose, everything remains possible.
Young journalist: Everything you say is contradictory. You can't have been in one place and another at the same time. Of all those lives, which one is the right one?
Nemo Nobody aged 118: Each of these lives is the right one! Every path is the right path. Everything could have been anything else and it would have just as much meaning.
Nemo Nobody aged 118: Before he was unable to make a choice because he didn't know what would happen. Now that he knows what will happen, he is unable to make a choice.
Yes. An entirely different challenge is:
your ability to evaluate a partner by those preferences,
and I hope to give everyone some consideration in that area in a future post.
Assume that once you find the right person - getting married (the process of organising a wedding from the day you have the engagement rings on fingers); and falling pregnant (successfully starting a viable pregnancy) takes at least a year. Maybe two depending on how long you want to be "we just got married and we aren't having kids just yet".
While getting married might be useful when you have children together there no reason in this day and age to make not having yet been maried to stop you from getting children.
This is true, but I would hope for a measure of stability in order to provide a stable environment for children, something easier done with the income of two humans, not one.
A significant matter than is:
easier done with...
I don't think I could say what it is
mostly
a matter of.
You appear to have some idea of what it is of; and what it is "mostly" of. I am interested in what sits on that list and what you find "most" or less "most" significant on that list.
I am not sure what are you asking -- what conditions/factors do I consider to be useful/necessary for successful child upbringing?
apologies.
what conditions/factors do I consider to be useful/necessary for successful child upbringing?
yes. this.
Well, that's an awfully general question. Out of factors that you have some influence over (as opposed to e.g. your own IQ or temperament):
First, good genes. Yes, you can't change yours but you have a choice of a set out of which the second half will be picked. Use it wisely :-)
Second, money in its usual function of being able to get you what you need including such things as e.g. safety and, to a limited extent, time. Of course there are the usual problems in that money itself is commonly a trade-off against your own time and stress, so this is highly dependent on the circumstances. But children are expensive :-/
Third, a support network of relatives and friends (hopefully including your spouse).
Fourth, later, a good street/school environment.
Of course, people managed to bring up kids lacking all of the above, so it's not a set of necessary things, only useful ones.
The following problem is best when not described by me:
https://en.wikipedia.org/wiki/Secretary_problem
Although there are many variations, the basic problem can be stated as follows:
There is a single secretarial position to fill.
There are n applicants for the position, and the value of n is known.
The applicants, if seen altogether, can be ranked from best to worst unambiguously.
The applicants are interviewed sequentially in random order, with each order being equally likely.
Immediately after an interview, the interviewed applicant is either accepted or rejected, and the decision is irrevocable.
The decision to accept or reject an applicant can be based only on the relative ranks of the applicants interviewed so far.
The objective of the general solution is to have the highest probability of selecting the best applicant of the whole group. This is the same as maximizing the expected payoff, with payoff defined to be one for the best applicant and zero otherwise.
Application
After reading that you can probably see the application to real life. There are a series of bad and good assumptions following, some are fair, some are not going to be representative of you. I am going to try to name them all as I go so that you can adapt them with better ones for yourself. Assuming that you plan to have children and you will probably be doing so like billions of humans have done so far in a monogamous relationship while married (the entire set of assumptions does not break down for poly relationships or relationship-anarchy, but it gets more complicated). These assumptions help us populate the Secretary problem with numbers in relation to dating for the purpose of children.
If you assume that a biological female's clock ends at 40. (in that its hard and not healthy for the baby if you try to have a kid past that age), that is effectively the end of the pure and simple biological purpose of relationships. (environment, IVF and adoption aside for a moment). (yes there are a few more years on that)
For the purpose of this exercise – as a guy – you can add a few years for the potential age gap you would tolerate. (i.e. my parents are 7 years apart, but that seems like a big understanding and maturity gap – they don't even like the same music), I personally expect I could tolerate an age gap of 4-5 years.
If you make the assumption that you start your dating life around the ages of 16-18. that gives you about [40-18=22] 22-24 (+5 for me as a male), years of expected dating potential time.
If you estimate the number of kids you want to have, and count either:
3 years for each kid OR
2 years for each kid (+1 kid – AKA 2 years)
(Twins will throw this number off, but estimate that they take longer to recover from, or more time raising them to manageable age before you have time to have another kid)
My worked example is myself – as a child of 3, with two siblings of my own I am going to plan to have 3 children. Or 8-9 years of child-having time. If we subtract that from the number above we end up with 11-16 (16-21 for me being a male) years of dating time.
Also if you happen to know someone with a number of siblings (or children) and a family dynamic that you like; then you should consider that number of children for yourself. Remember that as a grown-up you are probably travelling through the world with your siblings beside you. Which can be beneficial (or detrimental) as well, I would be using the known working model of yourself or the people around you to try to predict whether you will benefit or be at a disadvantage by having siblings. As they say; You can't pick your family - for better and worse. You can pick your friends, if you want them to be as close as a default family - that connection goes both ways - it is possible to cultivate friends that are closer than some families. However you choose to live your life is up to you.
Assume that once you find the right person - getting married (the process of organising a wedding from the day you have the engagement rings on fingers); and falling pregnant (successfully starting a viable pregnancy) takes at least a year. Maybe two depending on how long you want to be "we just got married and we aren't having kids just yet". It looks like 9-15 (15-20 for male adjusted) years of dating.
With my 9-15 years; I estimate a good relationship of working out whether I want to marry someone, is between 6 months and 2 years, (considering as a guy I will probably be proposing and putting an engagement ring on someone's finger - I get higher say about how long this might take than my significant other does.), (This is about the time it takes to evaluate whether you should put the ring on someone's finger). For a total of 4 serious relationships on the low and long end and 30 serious relationships on the upper end. (7-40 male adjusted relationships)
Of course that's not how real life works. Some relationships will be longer and some will be shorter. I am fairly confident that all my relationships will fall around those numbers.
I have a lucky circumstance; I have already had a few serious relationships (substitute your own numbers in here). With my existing relationships I can estimate how long I usually spend in a relationship. (2year + 6 year + 2month + 2month /4 = 2.1 years). Which is to say that I probably have a maximum and total of around 7-15 relationships before I gotta stop expecting to have kids, or start compromising on having 3 kids.
A solution to the secretary equation
A known solution that gives you the best possible candidate the most of the time is to try out 1/e candidates (or roughly 36%), then choose the next candidate that is better than the existing candidates. For my numbers that means to go through 3-7 relationships and then choose the next relationship that is better than all the ones before.
I don't quite like that. It depends on how big your set is; as to what the chance of you having the best candidate in the first 1/e trials and then sticking it out till the last candidate, and settling on them. (this strategy has a ((1/n)*(1/e)) chance of just giving you the last person in the set - which is another opportunity cost risk - what if they are rubbish? Compromise on the age gap, the number of kids or the partners quality...) If the set is 7, the chance that the best candidate is in the first 1/e is 5.26% (if the set is 15 - the chance is much lower at 2.45%).
Opportunity cost
Each further relationship you have might be costing you another 2 years to get further out of touch with the next generation (kids these days!) I tend to think about how old I will be when my kids are 15-20 am I growing rapidly out of touch with the next younger generation? Two years is a very big opportunity spend - another 2 years could see you successfully running a startup and achieving lifelong stability at the cost of the opportunity to have another kid. I don't say this to crush you with fear of inaction; but it should factor in along with other details of your situation.
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. Incidentally this probably happens in real life quite often. In a stroke of, "you'll do"...
Where it breaks down
Real life is more complicated than that. I would like to think that subsequent relationships that I get into will already not suffer the stupid mistakes of the last ones; As well as the potential opportunity cost of exploration. The more time you spend looking for different partners – you might lose your early soul mate, or might waste time looking for a better one when you can follow a "good enough" policy. No one likes to know they are "good enough", but we do race the clock in our lifetimes. Life is what happens when you are busy making plans.
As someone with experience will know - we probably test and rule out bad partners in a single conversation, where we don't even get so far as a date. Or don't last more than a week. (I. E the experience set is growing through various means).
People have a tendency to overrate the quality of a relationship while they are in it, versus the ones that already failed.
Did I do something wrong?
“I got married early - did I do something wrong (or irrational)?”
No. equations are not real life. It might have been nice to have the equation, but you obviously didn't need it. Also this equation assumes a monogamous relationship. In real life people have overlapping relationships, you can date a few people and you can be poly. These are all factors that can change the simple assumptions of the equation.
Where does the equation stop working?
Real life is hard. It doesn't fall neatly into line, it’s complicated, it’s ugly, it’s rough and smooth and clunky. But people still get by. Don’t be afraid to break the rule.
Disclaimer: If this equation is the only thing you are using to evaluate a relationship - it’s not going to go very well for you. I consider this and many other techniques as part of my toolbox for evaluating decisions.
Should I break up with my partner?
What? no! Following an equation is not a good reason to live your life.
Does your partner make you miserable? Then yes you should break up.
Do you feel like they are not ready to have kids yet and you want to settle down? Tough call. Even if they were agents also doing the equation; An equation is not real life. Go by your brain; go by your gut. Don’t go by just one equation.
Expect another post soon about reasonable considerations that should be made when evaluating relationships.
The given problem makes the assumption that you are able to evaluate partners in the sense that the secretary problem expects. Humans are not all strategic and can’t really do that. This is why the world is not going to perfectly follow this equation. Life is complicated; there are several metrics that make a good partner and they don’t always trade off between one another.
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Meta: writing time - 3 hours over a week; 5+ conversations with people about the idea, bothering a handful of programmers and mathematicians for commentary on my thoughts, and generally a whole bunch of fun talking about it. This post was started on the slack channel when someone asked a related question.
My table of contents for other posts in my series.
Let me know if this post was helpful or if it worked for you or why not.