The (square) Root of Love

All right, fellas, huddle up.  We’re going to talk about the best way to find true love.  I mean, you can’t just go running around all willy-nilly hoping to bump into somebody great.  The world is a big place.  You need a strategy, man.  A dating plan of attack.

First, some ground rules, some general observations about romantic life, and a few restrictions in the interest of mathematical well-behavedness:

  1. You are only going to meet a finite number of datable women over the course of your lifetime.  It will be a depressingly low number.
  2. You are going to be an upstanding citizen and date only one woman at a time.
  3. You will date a woman for some finite period of time, at which point you’ll make a decision either to pull the trigger and propose, or cut her loose.  Or, more likely, she’ll dump you first.
  4. Once you propose, no takesies-backsies.  And once you cut a woman loose, you can’t ever reconsider her for marriage; she will hate you forever.
  5. You are able to perfectly rank the women you have dated according to a strict, unambiguous order of preference.  Tie goes to the blonde.
  6. You will encounter these women in random order.  That is, you are completely ignorant of where the next potential wife will stand in the overall rankings.
  7. You will date a certain number of women without really considering any of them for a proposal.  In other words, you’ll take some time getting a feel for who’s out there.  Setting the bar.

In the world of mathematics, this is what’s known as an optimal stopping problem.  You’re going to date, and date, and date…, and stop.  Hopefully on the woman of your dreams (hence the optimal part).  In fact, this is one of those problems that’s so famous it goes by several (mildly sexist) names: the secretary problem, the sultan’s dowry problem, the fussy suitor problem.  Because it’s Valentine’s Day, we’ll call it the marriage problem.

Strategy One: The Soul Mate

Out of all the datable candidates you’re ever going to meet, one of them is going to be objectively best.  For you economists out there, she will uniquely maximize your utility function (hubba hubba).  You want her, and only her.  No one else will do.  How do you maximize the probability you’ll end up together?  When should you stop dating?

Notice that, according to our rules above, you will begin your search with a period of casual, commitment-phobic dating (which doesn’t seem particularly unrealistic, actually).  After this initial period, you will propose to the first woman who’s better than all the ones before.  Assuming, of course, she’s still out there.

Suppose that you will meet n datable women in your life, and that you’ll see w of those women casually as part of your initial feeling-out phase.  Since you’ve already decided to marry the first woman that’s better than any of the first w, the question really boils down to, What is the best possible value of w How many women should you casually date before starting to think about marriage?

Now if w = 0, you’re going to propose to the very first woman who comes along, in which case the probability of marrying your soul mate is just the probability that she is the first woman you randomly meet, which is 1/n.  That’s not much of a strategy.  By the same reasoning, if you wait too long and set w = n – 1, you’re still only successful with probability 1/n (your soul mate just happens to be the last woman you randomly meet).  Clearly the optimal value of w occurs somewhere in between those two.  But where?  Make w too small, and you risk marrying too early, make it too large and you risk passing over your soul mate in the casual phase, which dooms you to a lonely death.

It turns out that this is one of those magical places where the number e shows up out of the clear, blue sky.  You should (optimally) date n/e of your potential mates before looking for a wife, which maxes out your probability of successfully soul-mating at a whopping 1/e, or about 37%.  Not too shabby.

There’s a problem with this number, though.  For one thing, e shows up because the harmonic series makes an appearance in the probability calculations (which is almost romantic), and e and the harmonic series are bound together in an intimate way.  Actually, e shows up (as the base of the natural logarithm) alongside Euler’s number in an approximation to the harmonic series.  But the harmonic series diverges.  And it diverges verrrrrry…verrrrrry…sloooooowly.  In fact, even with n as large as 1000, Euler’s number is only accurate to two decimal places.  Because of this, we need slightly different threshold numbers for small and large values of n, and there’s no clear line of demarcation.  Besides, I’m not entirely convinced there are 1000 datable people in North America.

In addition to the mathematical objections, there is an economic one.  In the Soul Mate model, the payoff for ending up with anyone besides your soul mate is zero, which is pretty harsh.  Even if you don’t end up with the absolute best possible wife, certainly you’ll derive some nontrivial level of utility from ending up with a good wife, or a great one.  With that in mind, we turn our attention to…

Strategy Two: Settling Down

In this model, not only does every woman come with a lineal ranking, she also comes equipped with…let’s call it a “happiness value,” i.e., the amount of happiness you would derive from marrying her.  (Let’s assume everybody has a positive happiness value, otherwise you wouldn’t really consider her to be “datable,” would you?)  Your soul mate is still out there, and she still has the highest happiness value, but now we acknowledge that other women are perfectly capable of making you happy as well, albeit to a lesser extent.

Notice that our focus has shifted a bit.  Instead of trying to maximize the chances of meeting your one-and-only, now you’re trying to maximize your expected happiness, which may or may not involve any one woman in particular.

As before, you’re going to date w women before the marriage search begins in earnest.  And, as before, you’ll propose to the first woman who’s better than any of the first w.  If, however you don’t run into any (you already passed over your soul mate), then you’ll settle for the nth woman; after all, in this model you’re better off with anybody (with a nonzero happiness value) than alone.  How high should you set w now?

The answer this time is √n, rounded to the nearest integer.  Good news if you’re impatient: in general, √n is going to be smaller than n/e (actually, as long as you’re going to meet more than seven datable women in your life).  Also good news if you’re risk-averse: you trade some of the likelihood of finding your soul mate for the certainty that you won’t end up alone and totally unhappy, which is closer to actual human behavior.  Also good news if you’re mathematically fastidious: this solution isn’t asymptotic; it holds for all n, even small values.

In Which I Can Hear Your Objections Already

Obviously these are very simplistic models with some glaring shortcomings.  For one thing, we’ve been assuming that it’s possible to know the true value of n, which is ridiculous.  It turns out that, for an unknown number of potential mates, the marriage problem becomes fairly difficult and completely resistant to tidy, blogworthy solutions.  There are more realistic, time-dependent models that presume, for instance, you have finite time to marry one of the women who arrives at your doorstep, according to a Poisson distribution, of course…but that sounds too geeky to be heuristically useful, even for me.

For another thing, the rules I laid out in the beginning are pretty rigid and unrealistic.  People routinely multi-date, get back together with exes, break engagements/marriages, etc.  And, while I’ll bet most men go through an initial dating period without too much thought toward marriage, they probably don’t wait to meet someone who’s better than all of the women they’ve dated before.  Better than the last couple might be enough.

So I won’t make any guarantees that this post will score you true love, but it makes for an interesting exercise.  Try to estimate how many single, demographically and geographically appropriate women you’re likely to meet during your lifetime, punch it into your calculator, hit the square root button, and pony up for some flowers.  Oh, and iron your shirt.  You look like a slob.

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