Mind Your Decisions

How Can I Find True Love Using Game Theory?

Every Tuesday is a Game Theory article at Mind Your Decisions.

Based on the top search results, you could:

Learn from religious advice (love is what God says it is)

Find your true love’s zodiac sign through a five question quiz (I will love a Taurus)

Consider phone counseling (only to realize you are the real problem)

Man, what a rip-off. How did these answers come up as the most relevant results? They give basically no practical advice.

For all you romantics, there is real hope

It comes from a problem in statistics.

For the sake of this discussion, I define true love as the best person who is willing to date you. Even if that’s not exactly true, I’m wiling to live with that definition. Because if you think your true love is someone that won’t date you, well, I’m not sure any advice can help you.

So for all you reasonable romantics, I offer this hope: if you follow this advice, you’ll maximize your chance of finding true love. The advice will give most people about a 37 percent chance of finding true love.

Yes, it’s not perfect—but I’d say that’s pretty good for a really difficult problem.

Modeling the True Love Game

Dating and relationships are complicated social interactions so they need to be simplified before coming up with any meaningful analysis.

In this statistics game, you search for your true love by dating and having relationships with various people. Your only goal is to find the best person willing to date you—any thing less is a failure.

Here are some ground rules (basically the same rules as on MTV’s show Next):

1. You only date one person at a time.
2. A relationship either ends with you “rejecting” or “selecting” the other person.
3. If you “reject” someone, the person is gone forever. Sorry, old flames cannot be rekindled.
4. You plan on dating some fixed number of people (N) during your lifetime.
5. As you date people, you can only tell relative rank and not true rank. This means you can tell the second person was better than the first person, but you cannot judge whether the second person is your true love. After all, there are people you have not dated yet.

Now, I know some of these rules are not realistic, but I think the game captures many of the dynamics of the dating world. So let’s take this as a starting point.

How does the game play out?

You can start thinking about the solution by wondering what your strategies are. Ultimately, you have to weigh two opposing factors.

–If you pick someone too early, you are making a decision without checking out your options. Sure, you might get lucky, but it’s a big risk.

–If you wait too long, you leave yourself with only a few candidates to pick from. Again, this is a risky strategy.

The game boils down to selecting an optimal stopping time between playing the field and holding out too long. What does the math say?

The basic advice: Reject a certain number of people, no matter how good they are, and then pick the next person better than all the previous ones.

The idea is to lock yourself in to search and then grab a good catch when it comes along. The natural questino is how many people should you reject? It turns out to be proportional to how many people you want to date, so let’s investigate this issue.

To make this concrete, let’s look at an example for someone that wants to date three people.

Example with Three Potential Relationships

A naïve approach is to select the first relationship. What are the odds the first person is the best?

It is equally likely for the first person to be the best, the second best, or the worst. This means by pure luck you have a 1/3 chance of finding true love if you always pick the first person. You also have a 1/3 chance if you always pick the last person, or always pick the second.

Can you do better than pure luck?

Yes, you can.

Consider the following strategy: get to know–but always reject–the first person. Then, select the next person judged to be better than the first person.

How often does this strategy find the best overall person? It turns out it wins 50 percent of the time!

For the specifics, there are 6 possible dating orders, and the strategy wins in three cases.

(The notation 3 1 2 means you dated the worst person first, then the best, and then the second best. I marked the person that the strategy would pick in bold and indicated a win if the strategy picked the best candidate overall.)

1 2 3 Lose

1 3 2 Lose

2 1 3 Win

2 3 1 Win

3 1 2 Win

3 2 1 Lose

You increase your odds by learning information from the first person. Notice that in two of the cases that you win you do not actually date all three people.

As you can see, it is important to date people to learn information, but you do not want to get stuck with fewer options.

So do your odds increase if you date more people? Like 5, or 10, or 100? Does the strategy change?

The answer is both interesting and surprising.

The Best Strategy for the General Case

From the example, you can infer the best strategy is to reject some number of people (k) and then select the next person judged better than the first k people.

When you go through the math, the odds do not change as you date more people. Although you might think meeting more people helps you, there is also a lot of noise since it is actually harder to determine which one is the best overall. So here is the conclusion:

The advice: Reject the first 37 percent of the people you want to date and then pick the next person better anyone before. Surprisingly, you’ll end up with your true love 37 percent of the time.

The advice is unchanged whether you plan to date 5, 10, 50, 100, or even 1,000 people. Here is a table displaying specific numbers:

Number of people you want to date (N)	Number of people you should reject (k)
4	1
5	2
10	3
25	9
50	18
100	37

Now I was simplifying matters just a bit because “rejecting 37 percent” is an approximation. There is some math that goes into the exact answer.

To be precise, the exact answer is to find first value of k such that

The full proof is fascinating, though somewhat technical. I encourage my avid math readers to check it out:

http://courses.ncssm.edu/math/Talks/PDFS/Spouse.pdf

A Lesson Learned?

Don’t settle too early.

Suppose that Americans have between five to ten relationships before marriage. This means most people are going to reject the first two or three people, regardless of the person’s quality.

Sounds odd, but it’s just too important to test the market and find that special person. Besides, this strategy improves a person’s odds from a pure random chance (10-20 percent) to almost 37 percent.

Okay great. There’s just one last thing to consider.

Doesn’t the Other Person Play the Game Too?

Game theory would be a lot easier if you could ignore how other people affect the game. So we’re not done yet.

The HUGE caveat is the other person is also trying to game you.

Imagine this: you date a few people, then finally find a great match, and then try to get more serious. Only, that’s not in the other person’s plan.

The other person happens to be less experienced than you, and you happen to be that person’s first serious relationship. As great as you might be, that person is not ready to settle.

The theory suggests you should not feel hurt if someone rejects you like this. You are likely an early victim.

To take advantage of the theory, you should consider whether the other person is ready to get serious. I guess this is why there are certain age clusters when people get married.

But there are things that counter the problem of timing. In real life, you have other strategies to increase your odds not possible in the game:

You some times can rekindle an old flame.

You cannot date simultaneously, but you can often get to know many people at the same time.

You might be able to figure out a true love without having to date many more people. It happens.

And if you are getting more serious, seek out reasonable people who only want to have one or two relationships in their lifetime. You would be in luck, because the math says that they might not reject anyone.

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