The Waiting Game

Dating is the worst.  Smiling, nodding, pretending to foster an intense love of indie movies.  Sharing meals with total strangers you generally wish would've remained more total.  Not everyone is looking to settle down, of course, but for all those seeking out their one-and-onlys, dating is  a necessary evil --- a means to an end.  The key word being end.  As in, when can I stop doing this already?

Since there's no better tool for managing personal relationships than math, let's set down some rules for our (idealized) dating game.1

  1. You will meet a finite number of datable human beings in your lifetime.  You are able to at least estimate this number with some back-of-the-envelope calculations.
  2. You're going to date a certain number of people without seriously considering settling down.  You want to get a feel for what's out there, partner-wise.
  3. You are able to unambiguously rank each date's awesomeness relative to all the others.  If, for instance, you prefer Jack to Pete and Pete to Kenneth, then you are required to be a non-crazy person and prefer Jack to Kenneth.
  4. After your trial period ends, each new date is potentially The One.  If you decide (s)he is The One, the two of you will live happily ever after.  If not, that person is gone from your romantic life forever.
  5. Here is your criterion for deciding whether a person is The One.  Remember everybody you dated in your trial period?  One of them was the best.  For convenience, let's call him Chris.  Once you start thinking about settling down, you will choose the very first person who is better than Chris.  If it turns out that no one is better than Chris (a real possibility), then you will settle for the last person you ever meet --- someone is better than no one.

So you're going to date casually for a little while, and then you're going to get serious.  The real question is, when should you start?  How long should your trial period be in order to maximize the chances of ending up with your soul mate?

Let's say you figure your dating pool consists of eight people.  If your trial period contains zero people, then the probability of ending up with your soul mate is just the probability that your soul mate is the very first person you ever date: 1/8.  If your trial period contains seven people, then the probability of ending up with your soul mate is just the probability that your soul mate is the last person you meet: also 1/8.  In other words, if your trial period is too short, you risk pulling the trigger before you meet The One; and if your trial period is too long, you risk skipping over The One during your casual phase.  There has to be a better answer somewhere in the middle.

Play around with the widget below and see whether you can come up with some circumstances that determine when you will or will not end up with The One (marked #1).  Drag the triangle to change the length of your trial period.  Drag the individual people to change their order, or hit Shuffle to get a random ordering.  The person you end up with is highlighted in green.

 Come up with anything good?  For instance, you probably noticed that if The One is in your trial period, then you can never be successful (which we already talked about).  You may also have noticed that if #2 is in your trial period, then you're always successful!  If you're a rock star, you will notice that you are successful in precisely the case that the best person out of all the people before The One was someone you dated in your trial period.

In other words, and to generalize a bit: if your trial period contains k people, and your soul mate is in the jth position, you are successful if and only if the best of the first j — 1 people is also in the first k — 1 people.  Once you realize that, you can write down some probabilities and sum them...or you can simulate!

Eventually, you get enough data to build up a picture like this graph, which shows your probability of success as a function of your trial period size.

Notice that, regardless of the size of your dating pool, your best bet is to have a little over 1/3 of those people2 in your trial period!  And your probability of success using this strategy also converges to a little over 1/3, which is heartening.  Adding more potential mates into the mix doesn't do anything to diminish your chances of finding true love.  Isn't that beautiful?

Teachers: want to have this conversation in your classroom?  Check out the lesson materials on our site!

1.  You may recognize this in its more general form as the Secretary Problem, which has applications in other contexts as well, including hiring (as the name suggests) and even home-buying.

2.  For those of you who are Calculus-ly inclined: in the limit you should have 1/e of your dating pool in your trial period, and your probability of success with this strategy also converges to 1/e.  Cool, right?

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