The following problem is best when not described by me:
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.
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%).
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.
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.