You’re single and looking for love. In front of you are three doors. Behind each door is a prospective partner. Your mission is to couple up with your best possible match.
Imagine you chose a door at random. If you couple up with the person behind it, the chance you get your best match is 1 in 3. (You can assume that of the three people who are waiting behind the doors, there is a best match, a not so good match, and a least good match for you.)
Thankfully, I’m going to let you open some doors. But there are rules:
To start, you select a door. Any door. It will open to reveal one of your potential suitors. If you want to choose this person as your match, you can. Job done. You couple up with them. But if you decide to go with the first person, you don’t get to see either of the others. Risky! The person might be just your type, but maybe the others are even more so…
(You can assume that you are able to correctly judge the person’s suitability for you as soon as you see them.)
If you decide to move on, you discard the first person. They’re history. You select one of the two remaining doors. It opens to reveal someone else. Again, you can choose this person as your match. But just like before, if you choose this person, you don’t get to see who is behind the final door. Nor are you allowed to return to the person you discarded behind the first door.
If you decide to move on a second time, you select the final door. You must choose the person behind it as your match, whatever you think of them. Also risky!
To summarise: you only have one chance to decide on each person. If you decide to reject someone, you can’t return to them after. If you choose the first, you don’t get to see any of the others. If you want to see them all, you must choose the last.
What strategy gives you the best chance of finding your best match? Can you improve your odds from 1 in 3?
(Further clarification: You must assume that you know nothing about the average attractiveness of the group that the suitors are selected from. If you see the most attractive person you have ever seen behind the first door, there is still a 50 per cent chance that the person behind the second door is even more attractive. The question is not psychological or sociological: you can’t settle with the first one you think is acceptable. The puzzle is mathematical: it asks you to maximise your chance of getting the best possible match of those three on offer.)
The most concise answer was suggested by reader Michael John:
“Take the second if they are better than the first, else take the third.”
The insight here is that you must never choose the first person. You must use the first person as a reference for what comes next.
In more detail, and with some maths, the strategy is this:
Select a door, and reject the person behind it.
Select a second door, and proceed as follows.
If you think that this person is a better match than the person behind the first door, choose them.
If you think that this person is a worse match than the person behind the first door, reject them. Your choice is the person behind the final door.
Let’s see why this works. Let’s call the suitors A, B and C in order of suitability.
There are three possible cases for the person behind the first door:
It’s A. The strategy tells us to reject A. Since A is the best match, in this case you wont get your best match.
It’s B. The strategy tells us to reject B, and choose another door:
If we get C, we reject them, since C is a worse match than B. We end up with A, our best match.
If we get A, we choose them, since A is a better match than B. We end up with A, our best match.
It’s C. The strategy tells us to reject C, and choose another door:
If we get B, we choose B, since B is a better match than C. We don’t get our best match.
If we get A, we choose A, since A is a better match than C. We end up with A, our best match.
In summary, we win if we see B first, or if we see C and then A. The probability of the former is 1/3 and the probability of the latter 1/6. The combined probability is 1/3 + 1/6 = 1/2, or 50 per cent. We have increased our chances of getting the best match from 1 in 3, to 1 in 2.
I adapted the puzzle from a well-known mathematical problem, the Marriage problem, in which the set-up is similar: a man dates n different women one after the other. If he must choose to marry a woman while he is dating her, what’s his best strategy if he wants to end up with his best possible match. (The context makes many sexist assumptions, of course. The problem is also known as the Secretary problem, in which a boss sees n different candidates for the position of secretary.)
I asked you the Marriage problem with n = 3. But what’s the answer when n is bigger than 3? The calculation is too involved for publication here, but if you are interested there are many web pages on it. Start with the Wiki page, for example.
The answer, however, is simple to state. You date 36.8 per cent of the women, whom you use as a sample group, and then choose to marry the next person you date who you rate higher than any of the ones that came before. (The percentage 36.8 comes from e, the exponential constant, which is 2.718 to three decimal places. 1/e = 0.368.)
As a rule of thumb this strategy is often helpful in many situations, not just dating. Reader daze mentioned below the line this morning that they use it to decide which pub to stop in: you see a bunch of pubs, then choose the next one you see that is better than what came before.
Please discuss any other applications below!
I hope you enjoyed today’s puzzle and I’ll be back in two weeks.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
I also co-write Football School, the children’s book series that explains the world through football. The third volume (left) is just out. In it there are chapters on the maths of the coin-toss, the chemistry of pitch markings, the biophysics of dribbling, the biology of footballers’ feet, the history of the first football club, the design and technology of trophies, the politics of Spain, and much much more.