THE ALGEBRA OF TRUE LOVE

By RANDOLPH COHEN

[Illustration]

By now, lawyers are quite familiar with attempts to apply mathematical formulas to certain settlement decisions -- the ones that concern when, and for how much, to settle civil cases. But what about the type of settlement decision that really obsesses twenty- and thirty-something lawyers -- the decision when to "settle" on a particular partner, and get married? Could mathematics apply there too?

You might think not. You might think that deciding when to settle on a particular partner, and pop the question, is the type of highly subjective judgment call that is tailor-made for hours of angsting on Ally McBeal -- not the type of cold, rational calculation that is tailor-made for a quick mathematical fix.

You might think that, sure. But you'd be wrong.

on a marriage partner. For the marriage-minded, I'll present that solution first. Then, for the math-minded, I'll detail the reasons this solution is objectively correct.

When to Marry: The Mathematical Solution

I'll start by assuming, for simplicity, that you start dating at age 18 and want to marry by 37 (perhaps for biological clock reasons, if you're a woman, or, if you're a guy, because the club kids will be mocking your bald spot by then). That's a 19-year period. Applying the formula set forth in the math section below, you should date for about 7 years -- from the ages of 18 to 25 -- without marrying.

Don't even consider it! Enjoy your single life. Sow your wild oats. Feel no guilt during this period. Others may call you a cad, a Casanova, someone with Peter Pan Syndrome -- or a flirt, a party girl, a heartbreaker, a femme fatale. Or, if you get really extreme, a sexual predator. But don't give in to the peer pressure, the name-calling, the suicide attempts, the tears. Stay strong. And remain secure in the knowledge that your behavior, callous and hedonistic as it may appear, is defensible according to immortal mathematical theorems.

If you follow this advice, you should still be single when you hit age 25. But you should also think to yourself, at that point: Who is the best prospect I've met so far? Call that person the Best Ex. Keep a picture of the Best Ex next to your bed. He or she is going to set your standard.

On the day of your 25th birthday, you should suddenly be in the market for a serious relationship leading to marriage. Yes, this should be an overnight transformation. And, yes, it is just as rational as it is abrupt -- and potentially shocking to your girl/boyfriend, who may have written you off as marriage material based on your past behavior. It's a new kind of Seven Year Itch -- not an irrational itch to cheat, but an entirely rational itch to marry.

Listen up, Mr. or Ms. Twenty-Five: The party's over. For commitment-phobic though you might be, it's time to swallow your fear and get ready to shop for rings when 25 rolls around. Because now fear of marriage just isn't rational anymore -- assuming you do want to marry by 37, that is. For the more commitment-phobic you are, after age 25, the more you hurt your prospects of marrying the best catch you can get and ending up with someone who, by your own lights, is second-best instead.

Of course, even after the magical 25th birthday, you shouldn't marry just anyone. Instead, marry the first person you meet who blows the Best Ex out of the water. Get down on one knee and propose. And whatever you do, don't look back. You made the optimal choice -- by definition (and the math section below explains why that is).

Look at the case of Oscar winner Gwyneth Paltrow. A few years ago, in her early 20s, she was engaged to the dashing Brad Pitt (probably her Best Ex). Realizing she was not yet 25 (and obviously displaying a strong command of mathematics), she called it off. Now 26, however, she should be searching for permanent love. Will her off-and-on flame Ben Affleck fill the bill? Or should it be relationship Armageddon? It will be easy for Gwynnie to decide: she need only compare Affleck to Pitt, her best pre-25 catch. If she finds Ben to be Brad's superior, time to head for the chapel. If not, it's back to the bars (or, in her case, the cool Hollywood parties).

I'll leave it to you to figure out whether Winona Ryder should have married Matt Damon (did they break up because she applied the formula and Matt fell short of Johnny Depp -- perhaps her Best Ex?). Now for the math that justifies this solution to the problem of when -- and whom -- to marry, and explains why the solution is objectively correct.

Part I - The Optimal Stopping Problem

Kudos for this approach are due to the great British mathematician Arthur Cayley -- who first considered this type of settling problem in the late 1800s. Although Cayley never knew Ally McBeal, nor could he watch her show, he still felt her pain. Indeed, Cayley felt the pain of any person who, like Ally, is confronted with a series of options over time (for Ally, the options would be Bachelors #1, #2, #3 and so on until marriage or cancellation). Anyone, that is, who must forgo one option forever in order to move on to the next; anyone who cuts and runs -- holding close the hope that the next relationship will be better but staving off the fear that the last was the best there is.

In short, although he lived in the 1800's, Cayley anticipated the dilemma of the serial monogamist. Better than that, he took steps toward solving it -- thereby furthering the chances that the love thang that turns into a marriage thang will be the very best it can be. And who says mathematicians aren't romantics?

To mathematicians, a dilemma of this sort is known as an Optimal Stopping Problem. It's easiest to illustrate, at least initially, by imagining you're offered a series of bags of cash that vary in amount -- rather than, say, a series of persons of varying degrees of fantastic-ness.

Once you open a bag o' cash (labeled, in the best cartoon fashion, with a large dollar sign), you can choose to either (1) accept it, or (2) reject it and go for a new bag. An important rule: You can never go back to bags that you've already opened and rejected. Another important rule: At some point, the bags o' cash are going to run out, and if you go through all of them and reject them all, you get the last one by default.

You want to maximize your chance of taking the most valuable bag. The question is, when should you stick, and accept a particular bag of money -- rather than rejecting that bag, and moving on to the next one?

Getting a rough, intuitive answer to this question is easy if you have some information about the kinds of amounts you are likely to be offered (what mathematicians call the "distribution function"). For example, if you knew you were going to see 30 bags, and you knew the amount in each was a random number of dollars that is equally likely to be anything from 1 to 1000, then you'd pretty much wait until you saw something up over 900 and jump on it. If your luck was bad and after 26 bags you'd seen no $900+ bags, you'd grab an $800 or maybe $750 if you saw one, just to play it safe. Nothing to it.

But let's add a twist -- one which gets us closer to the marriage scenarios I discussed above.

Part II - A Twist On The Optimal Stopping Problem

-- is that you have no idea what amounts of money to expect. (This wrinkle was added by Merrill Flood in 1950.) Of course, this new feature -- that you don't know in advance the quality of opportunities you can expect to have -- is also characteristic of the search for that one perfect love.

Now you need a new strategy: (1) Watch the money bags for a while, to get a feel for what constitutes a nice chunk of change, and (2) then pick one that looks good, relative to this newly developed definition of what, indeed, a "nice chunk of change" is. But how long should you watch and wait, developing your sense of a "nice chunk of change" before choosing?

In 1954, the statistician Herb Robbins showed, amazingly, that there was a precise, mathematical answer to this very question. Robbins' answer is this: (1) If there are N bags, then you begin by rejecting the first N/e bags, where e is the calculus constant 2.71828 . . . . Since 1/e is about .37, this means you watch 37 percent of the bags go by before even considering choosing one. But (2) you keep a record of the highest cash valued-bag as the first 37% of bags go by -- call it the Best Cash. Then (3) you stick with the next bag you open with cash of a value greater than the Best Cash you've seen so far.

Perhaps you are underwhelmed by this explanation. Perhaps you are having flashbacks to not understanding the numerical solution to the meaning of life provided in The Hitchhiker's Guide to the Galaxy. Perhaps you were a liberal arts major because calculus made you as sick as cotton candy on a rollercoaster. Take heart. Here's an easy application:

Suppose there are 100 bags. Then N=100, and N/e is about 37. That means you should reject the first 37 bags, but be careful to write down the value of the bag, among the first 37, that contained the most cash. Say the value was $10,000. Then, starting with bag 38, keep opening the rest of the bags, and stick with the first bag of cash with a value in excess of $10,000. If none of the bags has that much cash in it, just take the last one -- you did the rational thing, but you still got screwed.

So now you get it, even if you read Keats and Yeats in the back of Calc One -- rather than, say, acing Calc One but (like many math majors) believing Keats and Yeats were pronounced the same way, yet still not knowing what way that was. But you're still saying, What does this mean for me, Al Franken? Or, more likely, for me, Ally McBeal?

That brings me to the marriage application we started with. If you need to marry between 18 and 37, then N is 19 (taking each year in which you might meet a potential mate, rather than each bag of money you're offered, as the relevant unit). 19/e is approximately 19/2.7 -- which is approximately 7. So, applying Robbins' formula, you wait N/e years -- or 7 years -- that is, until you're 25 -- to even consider marrying. And the 7-year period between 18 and 25 is also the period during which you settle on your Best Ex -- the man or woman to beat when you choose the person you'll marry after the age of 25.

Once you do marry, I assure you that you'll realize that, like Pangloss, you're living in the best of all possible worlds. You'll thank Cayley and Robbins for solving all your problems. And for doing so mathematically.

Feel free to live happily ever after.

Randolph Cohen, Ph.D., is an Assistant Professor of Finance at Harvard Business School. He is happily -- and, he believes, optimally -- married.


ARTS MAJOR FRIENDS, AND EX-GIRLFRIENDS, OF THE AUTHOR

  1. Why e?
  2. Well, why not e? Calculus fans will recognize e as the number which all other numbers aspire to be; the point to which all series seem to converge. The nonmathematical think of pi as the great mystical number of mathematics. But mathematicians often think of pi as a mere child's toy, while e is a shining beacon of truth. You could say e is to pi as heroin is to cocaine.
  1. Come on, seriously, why e?
  2. Good question. Beyond all the mysticism, you might ask why should we -- following the cult of e -- watch 37% of our opportunities pass by, rather than, say, half? Well, it turns out that the 1/e value is just a very good approximation of a very messy formula. But the intuition for it is this: if you watch half of the moneybags (or potential spouses) go by, then there's a 50% chance that you let the best one get away before you even started looking. Add to that the possibility that you made a selection before the best choice appeared, and you're well below 50-50 in your odds of choosing the most desirable bag, or mate -- tragic! By only watching the first third or so, you decrease your chance of realizing that you missed out on the best possibility out there by waiting too long to choose. (Sound familiar to all the Peter Pan syndrome guys out there? Your girlfriend is right: You really may have waited too long.)
  1. Don't you assume that you are seeking the best possible mate, rather than simply trying at all costs to avoid being stuck with someone really awful?
  2. Yes, but that's pretty realistic. Otherwise everyone would marry tomorrow just to make really-truly-for-certain-sure that they did not inadvertently marry Roseanne.
  1. You also assume, don't you, that marriageable prospects are evenly spread over one's dating life, that dating is merely a search for a spouse and not at least partly an end in itself, and that one cannot return to past exes (though that may well be the best policy)?
  2. Yes -- you make a good point. Be assured that mathematicians have attacked, solved, and been tenured for more general problems which relax these assumptions -- see UCLA mathematician Thomas Ferguson's elegant book on the subject for more on these, as well as related problems such as when to grab that parking space two blocks from the theater and when to keep driving in hope of getting an even better spot.
  1. No one has offered anyone to me like a bag of money lately. Instead, I have to go on disastrous blind dates. I don't like the assumption that potential spouses are "offered" to you in the same way bags of money are, in the first hypothetical.
  2. True. In point of fact, it's quite unusual for anyone to offer bags of money with no strings attached either. In real life, there's a complicated strategic game being played where "offers" of spousehood are rarely made; instead, hints are dropped which must be interpreted, etc. Yet I think we all pretty much know who among the people we've dated we likely could have married had we wanted to (c'mon, think who's on your list - it's fun!).


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