Tanya Khovanova's Math Blog

The first time I heard about the “stopping problem,” many years ago, it was in this version: The king announces that it is time for his only daughter to marry. Shortly thereafter 100 suitors line up in a random order behind the castle walls. Each suitor is invited to the throne room in front of the eyes of the princess and the king. At this point, the princess has to either reject the suitor and send him away, or accept the suitor and marry him. If she doesn’t accept anyone from the first 99, she must marry the last one. The princess is very greedy and wants to marry the richest suitor. The moment she sees the suitor she can estimate his wealth by his clothes and his gifts. What strategy should she use to maximize the probability of marrying the richest person?

The correct strategy is to reject the first 37 suitors and then marry the one who is better than anyone else before him. Generally, if there are N suitors the number of people to skip is about N/e — slightly more than one third of the whole group.

However, the greedy princess never received a good mathematical education. It is clear that her goal should have been to maximize the expected wealth of the future husband. In this case the strategy would be different; in particular, it changes significantly at the end. Let’s imagine that when the line of leftover suitors thins, she realizes that she’s already rejected the best one. In this case, it would be in her interest to consider the second best and, closer to the end, even the third best.

In real life, marrying the second best creates a burden of regret and bitterness. Let us assume that we all want to marry the best we can. But of course in our cases, the best does not necessarily mean the wealthiest. Also, we do not know how many suitors are lining up for us. Back in Russia I heard that a woman, on average, receives ten proposals. So, we should skip the first three, then marry the best person after that.

I do not know how many proposals an average American woman gets, but nowadays she doesn’t have to wait for an official proposal before deciding whether a guy is right for her. The question becomes: which marriage candidate should a woman try to marry?

If we assume that marriage candidates are distributed evenly in time and girls are seriously hunting for husbands between ages 20 and 35, then the above math advice can be applied in the following way: From age 20 to about 25 or 26, just look around and see what life offers you. After that, marry the one who is better than all the previous ones.

This is a very mathematical piece of advice. The idea makes sense: first you sample your options then you target the best. The problem is that these assumptions do not cover our real-life situations. Let’s look at some realistic adjustments and how they affect the age at which you stop sampling and become more active.

• You might be OK without marrying at all. In this case you can afford to be much pickier and skip the first 60% of the candidates.
• Potential husbands are not spread evenly in time. In this case try to estimate the distribution and act accordingly. For example, if you expect more men around you while you are in college, your cut-off age goes down.
• You change with time. If you think that you will lose your freshness and charm with age, your cut-off age goes down. If you think that you’ll become more experienced and effective at seducing men later in life, your cut-off age goes up.
• Your values change with time. You might be interested in looks now and value a big heart later. Because of this, it is better to delay your choice until you know yourself well and your opinion of life has stabilized.
• The value of a man changes with time. A high school drop out can become a very successful businessman and the brightest student in college can become an alcoholic. You might benefit from waiting to see how he stabilizes, thus moving your cut-off age up.
• You can divorce. This allows you to make a mistake on the first try and moves your cut-off age for this first try down.

Based on the mathematics and discussion above, here is the advice I would give to my teenage daughter — if I had one:

Take your time looking around and sampling your boyfriends. Constantly analyze the pool of your boyfriends as a whole. If there are strange patterns — for example, all of them look exactly like your father or you always pay for their dinners — start psychotherapy and work it out. As soon as your boyfriends start to look different from each other — except for things that are important to you, like education — compare your dream husband to the pool. If you dream about a Nobel prize winner who is not older than 30 and on the list of the 100 sexiest men alive, you should adjust your expectations to your chances. You will choose a guy who is better than anyone before, but it is unrealistic to expect him to be much better. As soon as you get to a cut-off age, which you have estimated using the suggestions above, stop sampling and start deciding. As soon as you find someone who is better than anyone else before, go for it — marry him.

That was advice from my brain, now I will give you advice from my heart:

Use mathematics to guide you, but make the final decision with your heart.