Love in the Time of Dice Games

Uncertainty dominates our lives. From what the Catch of the Day will be (we may never truly know) to generally accepted interpretations of the mathematics describing the fundamental nature of matter, the world is rife with possibility. In some sense, it’s amazing that we aren’t paralyzed in our decision-making by the unknowability of the future—though you might have felt as much as you stared down your iPhone’s gray ellipses bubble, desperately hoping for a "yes" to some follow-up Netflix and Chill with your Valentine.

Indeed, love and uncertainty inevitably go hand in hand. And while we are comfortable using mathematics to quantify risk everywhere from the stock market to weather forecasts, it strikes most as an affront to even suggest that economics and statistics could say anything about the institution of love.

I’ll be the first to admit that these fields could never come close to reducing the human experience to a set of equations. In spite of their names, solutions to the stable marriage and secretary/marriage problems won’t and can’t reveal the secrets of affection; however, these and other ideas from financial theory can offer new perspectives, orthogonal to those of Márquez or The Bachelor, on navigating the travails of romance.

Take the idea of expectation. Intuitively, and with an apology to probability theorists, the expectation of an unknown outcome—think the roll of a die—is simply the outcome on average. More precisely, it is equal to the sum of every possible outcome multiplied by the probability that it occurs; hence, the expected number of dots thrown on a die is three and a half, since each side of the die has a one-in-six chance of being thrown.

Expectation plays a central role in finance because it is the foundation for valuing stocks, bonds, and financial derivatives. To a first order, a financial security is worth the expected cash flows it offers you: You should be willing to pay up to about fifty cents for a one-shot machine that prints out a dollar bill half the time and nothing the other half. You’d be giving away money on average if you paid any more.

In our everyday lives, expectation provides a baseline for luck. You might consider yourself lucky if you like big numbers and throw a four, five, or six on a given die roll, and unlucky if you throw below your expectation of three and a half. With this in mind, consider a game in which you have two chances to roll a die with the option of walking away after either throw. I will pay you, in dollars, the number of dots showing on the die you just rolled. What’s your strategy?

Since you know that a fresh die roll will pay out $3.50 on average, you would only want to roll again if you unluckily threw a one, two, or three at first; otherwise, you would forgo the second roll to lock in your above-average winnings. If you were given more chances to roll, you would be become weakly more willing to pass on any given number, and with infinitely many rolls, you would pass on every throw until you hit a six.

So what’s love got to do with it?

Imagine you knew that there were only two people left on Earth who would date you (perhaps a rosy view of reality for me), and suppose you found yourself slightly happier than expected upon dating the first of the two. Then, you probably wouldn’t want to take your chances with the second, since you would be less happy on average than you are now. Even if your current partner isn’t quite your soulmate, the fact that you did better than expected means you’ve, in some sense, won this dating game.

Like with dice, you can be more discerning if you know you’ll have more chances for love later on. We know this intuitively: college students jump from relationship to relationship while middle-aged singles take what they can get. But the (somewhat unscientific) evidence loosely suggests that we might not have as many chances as we think, and that the scenario above might not be completely abstracted from reality: On average, we only have two long-term relationships—two rolls of the die—before settling down for good.

Of course, dice games are far from a correctly specified model for love: The expectation of our happiness with our next romantic partner is near-impossible to pinpoint and ever-changing, and our relationships themselves aren’t just points on some goodness scale. It’s not a simple thing to leave a relationship as soon as you realize you could be better off on average with a new significant other, nor should we be indifferent to the risk of being forever alone at the end of the day.

As this column will note over and over again, all models are wrong, but some can be useful. We should not (and frankly could not) compute expectations over all our potential dating partners; rather, what we can take from rolling dice is an understanding that we ought to think a little less about what our perfect partner or best-case relationship might look like. It doesn’t make sense to wait for our soulmate to show up at our doorstep, just as it doesn’t make sense to wait until we hit a six when we only have a few rolls of the die. Though it might sound less-than-romantic to settle for better-than-expected, perhaps thinking about how lucky we are to be in our current relationships, and being consciously grateful for when we are happier than we otherwise should have been, is yet another path towards figuring out what love really is.

Marshall Zhang ’16 is a statistics and math joint-concentrator in Mather House and does not subscribe to Netflix.