Even though the dating service ChannelFireball Heart was announced on April 1, I took it as a suggestion for a new content direction. For today’s first installment, I considered doing a graph-theoretical analysis of romantic pairings in the Gatewatch, but I figured that looking for Hamiltonian paths in the love triangle between Nissa, Gideon, and Chandra wouldn’t be particularly interesting to the average reader.
Instead, I’ll describe the mathematics of finding the perfect partner. With a little imagination, we can cast this problem into a sequential search model with the following assumptions:
- There is a given number, let’s say 100, of possible partners you can date over your lifetime.
- Partners can be ranked from good to bad. Effectively, this means that each person can be represented as a single number that indicates how good of a match they are for you. But each person’s number is completely random, and you don’t know the underlying population distribution.
- One by one, you can screen successive partners by dating or entering a relationship with them. The model doesn’t care whether you take minutes or months to learn a person’s value, but after doing so you have to make a choice: marry them or continue looking. At the point you’re making this choice, you don’t yet know how passionately you might feel about unmet people.
- Once you pass over a prospective partner, they are gone forever. You can never get them back because they have gone off and married someone else.
- Your goal is to maximize the probability of choosing the best partner.
Finding the best strategy for this setting is a classic problem in optimal stopping theory. Mathematicians often refer to it as the secretary problem, perhaps because the people who named it were more interested in choosing secretaries than romantic partners. I wonder what it would have been called if a Magic player had thought of it first.
No matter what you call it, the problem has an elegant solution: Reject the first 37 candidates unconditionally to learn about the range of values—you could view this as your dating period before you’d consider serious relationships—and then stop at the first person who is better than every potential partner seen so far (or continuing to the last one if you never find anyone better).
In general, if you are presented with N rather than 100 successive options, you should reject the first N/e, where e=2.718… is the base of the natural logarithm. (Mathematical proofs of this and other results can be found here. To properly round N/e to an integer, you can use dynamic programming.)
As an example, if you expect that you’ll only meet three interesting potential partners in your life, then this model would suggest rejecting the first one, proposing to the second one if you believe they’re a better match than the first, and proposing to the third person otherwise.
For sufficiently large N, the resulting probability of stopping at the best candidate converges to 1/e (about 37%). Sadly, this is the best you can do. Even with the best possible strategy, you’ll miss your best possible partner over half of the time.
While this is a fascinating model and result, the real world is of course not as dramatic. For one, the model disregards the value of time or the fact that you can get information about your ideal partner from other sources than just your own dates. And if you would already be happy with a partner in, say, the top 10% rather than just the best one, then the optimal strategy would change completely.
But still, this model provides some insight into when you should settle down, at least in a stylized setting, and the solution is essentially one that Magic players have known for a long time: Don’t get married to your first pick! Never let anyone tell you that drafting couldn’t teach you anything about finding your perfect partner.