Finding the right partner from 3,812,261,000 females (or 7,692,335,072 humans, if you're bisexual) is difficult. You never really know how one partner would compare to all the other people you might meet in the future. Settle down early, and you might forgo the chance of a more perfect match later on. Wait too long to commit, and all the good ones might be gone. You don't want to marry the first person you meet, but you also don't want to wait too long because you'll run the risk of missing your ideal partner and being forced to make do with whoever is available at the end. It's a tricky one.
This is what's called "the optimal stopping problem". It is also known as "the secretary problem", "the marriage problem", "the sultan's dowry problem", "the fussy suitor problem", "the googol game", and "the best choice problem". The problem has been studied extensively in the fields of applied probability, statistics, and decision theory.
The problem is as follows:
"Imagine an administrator who wants to hire the best secretary out of
nrankable applicants for a position. The applicants are interviewed one by one in random order. A decision about each particular applicant is to be made immediately after the interview. Once rejected, an applicant cannot be recalled. During the interview, the administrator gains information sufficient to rank the applicant among all applicants interviewed so far, but is unaware of the quality of yet unseen applicants." - The Secretary Problem
At the core of the secretary problem lies the same problem as when dating, apartment hunting (or selling) or many other real life scenarios; what is the optimal stopping strategy to maximize the probability of selecting the best applicant? Well, in reality, the problem is not about choosing secretaries or finding the ideal partner, but about decision making under uncertainty.
The solution to this problem turns out to be quite elegant. Let's say you can rate each partner/secretary from 1-10 according to how good they are:
Had we known the full information beforehand, the problem would be trivial; choose either Alissa or Lucy. Unfortunately, we cannot look-ahead and there's no going back. When you're evaluating one partner, you are unable to look forward into the future and consider other opportunities. Similarly, if you date a great girl for a while, but leave her in a misguided attempt to find a better one and you fail, there's a good chance she'll be unavailable in the future.
So, how do you find the best one?
Well, you have to gamble. Like in casino games, there's a strong element of chance but the Secretary Problem helps us improve the probability of getting the best partner.
The magic figure turns out to be 37% (1/e=0.368). If you want to delve into the details of how this is achieved, I suggest you to read the paper by Thomas S. Ferguson named "Who Solved the Secretary Problem". The solution to the problem states that to increase the probability of finding the best partner, you should date and reject the first 37% of your total group of admirers. Then you follow this simple rule: You pick the next best person who is better than anyone you're ever dated before.
So if we take the example above, we have 10 lovers. If we chose 1 at random, we have approximately a 10% chance of finding "the right one". But if we use the method above, the probability of picking the best of the bunch increases significantly, to 37% - much better than random!
In our case, we end up with Lucy (9). Yes she's not an Alissa (10), but we didn't do badly.
Unfortunately, this method is not a 100% successful, as mathematician Hannah Fry discusses in this entertaining 2014 TED talk:
Variations of the Problem
In the Secretary Problem, the goal was to get the very best partner possible. Realistically, getting someone that is slightly below the best option will leave you only slightly less happy. You could still be quite happy with the second (or third-best) option, and you'd also have a lower chance of ending up alone. Matt Parker argues this in his book "Things to Make and Do in the Fourth Dimension: A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More".
In reality, many of the variations of the Secretary Problem are quite more accurate in solving specific problems.
In laboratory experiments, people often stop searching too soon when solving optimal stopping problems.
At the end of the day, the secretary problem is a mathematical abstraction and there is more to finding the "right" person than dating a certain number of people.
Although applying the Secretary Problem for finding true love should be taken with a pinch of salt, Optimal Stopping problems are real and can be found in areas of statistics, economics, and mathematical finance and you should take them seriously if you ever want to:
- Sell a House
- Hire someone in a difficult position
- Look for Parking
- Trade Options
- Just know when to stop in general
Real life is much more messy than we've assumed. Sadly, not everybody is there for you to accept or reject, when you meet them, they might actually reject you! In real life people do sometimes go back to someone they have previously rejected, which our model doesn't allow. It's hard to compare people on the basis of a date, let alone estimate the total number of people available for you to date. And we haven't addressed the biggest problem of them all: that someone who appears great on a date doesn't necessarily make a good partner. Like all mathematical models our approach simplifies reality, but it does, perhaps, give you a general guideline; if you are mathematically inclined.
A great article "Knowing When to Stop: How to gamble if you must—the mathematics of optimal stopping" in American Scientist by Theodore P. Hill goes into more details about this topic and is quite an interesting read.