Dating two guys that know about it

What is the best strategy in order to find your perfect match? Is the first guy the best? Do we have to date as many guys as possible to find the best?

But you can easily understand that these two strategies are risky and not optimal: in the first case, you can regret it later if you marry the first guy you date; in the second case, you can lose some good matches opportunity and end up just alone.

To determine what is the best strategy for you, you have to know the total number (n) of guys you want to date because maybe you want to marry before a certain deadline or maybe you don’t have the time or don’t feel like dating many persons.

It is worth noting that the notion of the best guy is only the best one among the total number of dates (n) you have chosen to have. So if you choose to have one date, then by definition, you have chosen the best one.

And to simplify the mathematical model, here are several assumptions to be specified:

• All of them can be compared and rated, and you can say who is better without ambiguity.
• If you reject a guy, you can not come back on your decision. Otherwise, it would be too easy.
• They arrived one by one, at random order and you can only date one for a given time.
• Of course, you can only choose once.

Now the question is: how to choose the guy so that the probability that (the chosen guy is the best one) is the highest?

First let’s look at the most trivial way of choosing: random strategy.

If your strategy is to stop randomly at any step, pick the guy, without comparing them, then the reasoning is as follows: for each step, the probability that he is the best is 1/n. And the probability at stopping at the step i and selecting him is also 1/n.

Since you don’t compare or study the guys, then the two events (the guy is the best and the guy is selected) can be considered as independent. So we can have the following results below:

Now let’s see another way of choosing: fixed strategy.

To simplify the vocabulary, I will say guy i, for the guy that you meet at step i. The index i is related to the i-th date, not the guy. It is worth noting that: what is random here is the number of dates you choose to have, not the choice of a guy among all the guys.

If you choose to always stay with the first choice. Then the probability of him being the best one is: 1 / total number = 1/ n

If you choose to always stay with the second choice, whatever the reason. Then the probability would be:

• First, the guy 1 should not be the best one. But hey, this probability is high : (n-1)/n
• Then knowing this, the probability of the second guy being the best is then 1/(n-1)
• In the end, the final probability of always choosing the second guy and him being the best would also be 1/n.

The reasoning is very similar if you choose to always stay with the third choice:

We can now generalize the reasoning if you choose to always stay with your ith choice: 1 / n.

So without any meaningful strategy (with the random strategy or the fixed strategy), the probability of selecting the best guy among 10 guys is 10%.

Intuitively you know that you can do better: you want to be able to compare the current guy to the previous ones.

So let’s detail the comparison strategy now!

This strategy consists of two phases:

• Phase 1: we can call it the observation phase, this is the phase during which you always end up leaving the person, but you will then have a basis of comparison for other people. The best guy during this observation phase, let’s call him the reference guy.
• Phase 2: we can call it the selection phase, we will compare the current guy to all previous guys, and if she or he is better than all the previous ones, then you select this one. This is the same as saying that you will select the first one that is better than the guys you dated during the observation phase.

In order to choose the best one, or in other terms, in order to have the highest probability when selecting our guy, the only parameter to be determined is: how many guys should we date during the observation phase?

If the total number of your dates is 10, then the answer is 3!

As callous as it may sound, with this strategy, you should date with 3 guys and then reject them (because they are just part of the observation phase). Then starting from the 4th guy, if he is better than all the previous ones (which is also the same as comparing with the reference guy), you can stay with him.

And with this strategy, the probability of selecting the best one is almost 40%!

Which is 4 times better than the random strategy.

Why? Let’s do a demonstration.

In order to demonstrate the comparison strategy for 10 people, we will simplify by iterating first with one guy, then with two guys, then with three guys, etc.

Only one guy

If you choose to date only one guy, then we cannot say if he is the best or the worst, actually, he is both!

Two guys

If you choose to date 2 guys, then you only have 2 choices. You can stay with the first guy, or you can stay with the second guy. There is no difference. So you have 50% probability of selecting the best one.

It is worth noting that the random strategy produces the same result.

Three guy

If you choose to date 3 guys, then we have to do some probability calculation. Let’s say that we can rate them by a number of ★, and since there are 3 guys: ★★★is the best, and ★is the worst.

There are 6 possible scenarios. Note that the choice 1 means the first time that you meet a guy, choice 2 means the second time you meet a guy, etc…

Staying with your first choice

If you decide to stay with your first choice, then the probability of choosing the best one is 2 out of 6. (the scenarios 1 and 2 in our scenario table).

So the probability is 33%, which is the same as the probability of random strategy.

Rejecting the first choice and then selecting

If you decide to reject the first one, then your first choice will be your observation phase. And starting with the second choice, you will choose : if he is better than all the previous ones, then you stay with him, and if not, you continue (and end up with the third one).

Now you can see that with this strategy, the probability that you end up with the best one is 3 out of 6.

Or in other words, the probability is 50%.

Rejecting the first and the second choices and choosing the third one

For this strategy, the probability is simple to calculate, it is 2 out of 6, or 33%.

In the case of 3 total guys:

• If you follow the random strategy, then the probability of selecting the best guy is 33%.
• If you follow our best strategy with the comparison strategy, then the probability is 50%!

Now let’s try to generalize the calculation of the probability P for two given parameters

• n, the total number of guys
• r, the number of rejections

And this time, we have to use conditional probability because in this strategy the events are dependent.