When To Stop — According To Maths
I have a friend from my university days who is engaged to her childhood sweetheart. They’re truly wonderful together — the fairytale we all dreamed of as kids. But, from a philosophical point of view, I regularly wondered: how can they know that they’re the right person for each other, when they have never been with anyone else?
As my twenties drift by, I have asked myself a similar question. When well I know when I should stop looking and start leaping? Indeed, when will anyone?
The Optimal Stopping Problem
This dilemma; when to stop looking and when to start leaping; faces the vast majority of us during the course of our lives, whether it be searching for a new home or hiring an employee. To continue with the theme of matchmaking, how will we know when we have met enough people to know who our best match is? It’s plainly true that gathering as much data as possible is the most helpful way to establish who our best match is.
But this doesn’t take into account the very nature of the human condition: being subject to time. By the time we’ve gathered the data, the chance to romantically whisk away our first choice has almost certainly passed.
So how do we balance this desire to be matched with the best possible person with the desire to find that person before it’s too late?
When should we stop?
The 37% Rule
Let’s say you’re 20 years old and want to marry when you’re 30, and that in that time it’s realistic for you to date 1 person every year. You could date 10 people in that time. Now, if you were to marry the first person you dated, your odds of marrying the best person would be 10% — certainly not outstanding. To get a sense of what’s out there, you would need to date a few more people.
In fact, according to the 37% rule, you shouldn’t marry the first 3 people you date. However, after 37% of the time has passed between your twentieth and thirtieth birthdays, you should aim to marry the next person who is as least as wonderful as the best person you dated beforehand.
Stopping optimally means selecting the next best choice after seeing and eliminating 37% of your options. The applications for this knowledge don’t stop at finding love. My choice of flat, in which I’m sat at this particular moment, was informed by optimal stopping. I’m also in the habit of choosing parking spaces based on optimal stopping — driving around for what I hope is around 37% of the time that I estimate I can be bothered to persevere for. One could also use optimal stopping for your job search, hiring new staff, certain types of trading, test-driving cars; and undoubtedly many more things besides.
There is a really complicated formula for working out the optimal point at which to stop looking and start leaping — but I found this graph explanatory enough!
A useful nuance of the optimal stopping rule is that you can choose whether to apply it based on the number of attempts (interview 100 people, “leap” after 37) or based on the time taken (interview for 100 days, “leap” after 37). Your success rate will remain the same.
The Best Possible Outcome
Curiously, optimal stopping will only get us the best possible outcome 37% of the time. If you’ve decided you’re only going to interview 2 people before “leaping”, this doesn’t sound very impressive. But what if you were willing to interview 15? 37% would be strong odds indeed!
What if you could — hypothetically — interview a million people? As bizarre and counterintuitive as this may seem, your odds are still 37%. And that’s surely incredible.
No matter how many options you consider or how long you look for, your chances of selecting the best possible outcome will always be 37%.
Whilst optimal stopping isn’t foolproof, next time you’re searching for something, you will know how to get maths on your side to optimise your chances of getting the very best partner, house, new job or parking space!
I first discovered optimal stopping in Brian Christian and Tom Griffiths’ excellent book Algorithms To Live By, which I could not recommend more highly.
You might enjoy Professor Michael Trick’s entertaining blog post on his experience of Finding Love Optimally!
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Software Engineer | Growth Hacker | Digital Marketer