Let me tell you about my favorite mathematical proof.
It’s for a puzzle called the marriage problem  (alternatively called the secretary problem or the sultan’s dowry). It’s my favorite for belonging to a rare category of “mathematical proofs which also have to do with sex.”
The problem is very simple:
Suppose you are looking to get hitched. You have many people you could potentially marry. How do you find the best spouse?
Well, as a mathematical puzzle, rather than a human one, we need to define the process for finding and evaluating suitors a bit more rigidly to answer that question.
The rules of this particular puzzle are defined as follows:
- Each potential suitor you meet, you can go on a date (or a few) with them. During this time, you’re able to evaluate their suitability.
- After each dating period, assuming the person is willing to propose to you, you can either accept or reject the suitor.
- If you accept the person, great, you’re now hitched.
- If you reject the person, move onto the next.
- In either case, accept or reject, you’re not allowed to go back and change your mind later. Once you reject someone, you can’t go back to them if you realize they were the one for you. Once you accept someone, you can’t change your mind if you meet someone else.
- You aren’t allowed to date more than one person at a time.
This puzzle has a more formal, mathematical definition, that can work as an analogy for a lot more than just spouses. You could, for example, also think of this as an employer looking for the best employee, but is required to either hire or reject each applicant, one at a time.
Interestingly enough, this puzzle has a solution. As in, there is a mathematically provable optimal algorithm for deciding the best spouse.
The Algorithm for the Best Spouse
The algorithm has two parts: what I’ll call a “rejection” phase and a “choosing” phase.
During the initial rejection phase, you reject every single applicant who proposes to you. It doesn’t matter how good they are, you just reject them. (In the formal definition you do this for the first n/e candidates, or roughly the first 40% of people.)
Then, after the rejection phase, you enter a new, choosing phase. Now you agree to marry the first candidate who is better than every other suitor you dated who also agreed to marry you.
With this algorithm, you can demonstrate that you will, in fact, select the best possible spouse a whopping 37% of the time, regardless of whether there are ten billion applicants or only ten.
No, You Can’t Actually Use This Algorithm to Find a Spouse
Obviously the mathematical solution to this puzzle won’t work strictly in real life. Many of the assumptions of the model are violated: you don’t know how many people you might potentially date, you don’t know whether the suitability of the suitors is time-dependent, you have access to information about people you aren’t currently dating which can inform you of the relative merits of those you are, etc.
It also goes without saying that basing your love life on an algorithm is a pretty poor way to live.
However, there is something I like about this algorithm, and I believe it can offer an analogy, if not a solution, for thinking about many areas of life.
Searching and Building
In short, the algorithm does two things. First, it has a searching capability. It spends a certain amount of time not making a decision at all, but simply gathering information about the overall range of suitability of the different options.
Second, it has a deciding capability. This is where you have gathered enough information and now need to make a choice.
While the algorithm only deals with the decision phase, most areas of life also have a follow-up part. They have not only a part where you must choose the best option, but also a point when you have to build on that choice you’ve made. A good marriage isn’t just selecting the right spouse, after all, but years of investment into building a relationship with that person.
Therefore, in a real context, I’d describe the split as being between searching and building. Searching, when you lack enough experience to know what to choose, and building, when you have enough data and now need to just make a choice and run with it.
What interests me about the ideal algorithm is that it divides itself neatly into these two phases. Search for awhile and then, abruptly, switch to deciding (or in our real world case, building).
Should You Search or Build?
Unfortunately, real life doesn’t offer a precise point to switch from one phase to the other, like in our idealized mathematical problem. But I do think the puzzle does illustrate the need for both searching and building in different areas of life.
Consider many decisions: which city to live in, what career to go into, what friends to associate with or habits to create. In many ways, they suffer from the same problems as the original puzzle I outlined: you have many options and you don’t know which to pick. Yet, at the same time, you know that once you do pick you’ll have to put in a lot of effort to make them work anyways.
Ask yourself whether your problem is a lack of information? If so, entering a searching phase where you don’t choose anything but explore options might be best. Spend some time living in different cities before picking a home. Spend some time in different jobs before picking a career. Spend some time with different groups of people before finding a tribe.
Do you have information, but can’t make a choice? If so, maybe you need to stop shuffling around, pick something and start building on it. Fretting over what is the right business idea? Maybe you already have enough information and just need to make a choice and commit to it.
What interests me about the algorithm is that the ideal solution may have two distinct phases, depending on where you sit. Which is better depends crucially on how much information you already possess, hence the seemingly endless contradictory advice between gathering more information and taking action.
Please note the “reject the first 40% of all applicants” really only applies in this formal puzzle. If you got married early or late, live in your hometown or haven’t settled into one by middle age, that shouldn’t imply you made an incorrect choice. Changing any of the assumptions can lead to very different outcomes for the algorithm.
That being said, look over the areas of your life. Could they benefit from more searching or do they need commitment? Share your thoughts in the comments.