*Robert Krulwich/NPR*

Poor Johannes Kepler. One of the greatest astronomers ever, the man who figured out the laws of planetary motion, a genius, scholar and mathematician — in 1611, he needed a wife. The previous Mrs. Kepler had died of Hungarian spotted fever, so, with kids to raise and a household to manage, he decided to line up some candidates — but it wasn't going very well.

Being an orderly man, he decided to interview 11 women. As Alex Bellos describes it in his new book** ***The Grapes of Math*, Kepler kept notes as he wooed. It's a catalog of small disappointments. The first candidate, he wrote, had "stinking breath."

*Robert Krulwich/NPR*

The second "had been brought up in luxury that was above her station" — she had expensive tastes. Not promising.

*Robert Krulwich/NPR*

The third was engaged to a man — definitely a problem. Plus, that man had sired a child with a prostitute. So ... complicated.

*Robert Krulwich/NPR*

The fourth woman was nice to look at — of "tall stature and athletic build" ...

*Robert Krulwich/NPR*

... but Kepler wanted to check out the next one (the fifth), who, he'd been told, was "modest, thrifty, diligent and [said] to love her stepchildren," so he hesitated. He hesitated so long, that both No. 4 and No. 5 got impatient and took themselves out of the running (bummer), leaving him with No. 6, who scared him. She was a grand lady, and he "feared the expense of a sumptuous wedding ... "

*Robert Krulwich/NPR*

The seventh was very fetching. He liked her. But he hadn't yet completed his list, so he kept her waiting, and she wasn't the waiting type. She rejected him.

*Robert Krulwich/NPR*

The eighth he didn't much care for, though he thought her mother "was a mostly worthy person ... "

*Robert Krulwich/NPR*

The ninth was sickly, the 10th had a shape not suitable "even for a man of simple tastes," and the last one, the 11th, was too young. What to do? Having run through all his candidates, totally wooed-out, he decided that maybe he'd done this all wrong.

"Was it Divine Providence or my own moral guilt," he wrote, "which, for two years or longer, tore me in so many different directions and made me consider the possibility of such different unions?"

**Game On**

What Kepler needed, Alex Bellos writes, was an optimal strategy — a way, not to guarantee success, but to maximize the likelihood of satisfaction. And, as it turns out, mathematicians think they have such a formula.

It works any time you have a list of potential wives, husbands, prom dates, job applicants, garage mechanics. The rules are simple: You start with a situation where you have a fixed number of options (if, say, you live in a small town and there aren't unlimited men to date, garages to go to), so you make a list — that's your final list — and you interview each candidate one by one. Again, what I'm about to describe doesn't always produce a happy result, but it does so more often than would occur randomly. For mathematicians, that's enough.

They even have a name for it. In the 1960s it was called (a la Kepler) "The Marriage Problem." Later, it was dubbed The Secretary Problem.

*Robert Krulwich/NPR*

**How To Do It**

Alex writes: "Imagine that you are interviewing 20 people to be your secretary [or your spouse or your garage mechanic] with the rule that you must decide at the end of each interview whether or not to give that applicant the job." If you offer the job to somebody, game's up. You can't go on and meet the others. "If you haven't chosen anyone by the time you see the last candidate, you must offer the job to her," Alex writes (not assuming that all secretaries are female — he's just adapting the attitudes of the early '60s).

So remember: At the end of each interview, you either make an offer or you move on.

If you don't make an offer, no going back. Once you make an offer, the game stops.

According to Martin Gardner, who in 1960 described the formula (partly worked out earlier by others), the best way to proceed is to interview (or date) the first 36.8 percent of the candidates. Don't hire (or marry) any of them, but as soon as you meet a candidate who's better than the best of that first group — that's the one you choose! Yes, the Very Best Candidate might show up in that first 36.8 percent — in which case you'll be stuck with second best, but still, if you like favorable odds, this is the best way to go.

Why 36.8 percent? The answer involves a number mathematicians call "e" – which, reduced to a fraction 1/e = 0.368 or 36.8 percent. For the specific details, check here, or Alex's book, but apparently this formula has proved itself over and over in all kinds of controlled situations. While it doesn't guarantee happiness or satisfaction, it does give you a 36.8 percent chance — which, in a field of 11 possible wives — is a pretty good success rate.

**Try It, Johannes ...**

What would have happened if Johannes Kepler had used this formula? Well, he would have interviewed but made no offers to the first 36.8 percent of his sample, which in a group of 11 ladies means he'd skip past the first four candidates. But the moment he'd met somebody (starting with lady No. 5) that he liked better than anyone in the first group, he'd have said, "Will you marry me?"

In real life, after a period of reflection, Johannes Kepler re-wooed and then married the fifth woman.

The way Alex figures it, if Kepler had known about this formula (which today is an example of what mathematicians call optimal stopping), he could have skipped the last batch of ladies — the sickly one, the unshapely one, the too-young one, the lung-disease one — and, all in all, "Kepler would have saved himself six bad dates."

Instead, he just followed his heart (which, of course, is another tolerable option, even for great mathematicians). His marriage to No. 5, by the way, turned out to be a very happy one.

## Comments [12]

I wish I was surprised to find sexist content here. I challenge you to challenge your own white male privilege. Listen and learn. Start with allowing Kao Kalia Yang ti speak in her own words. Here they are: http://hyphenmagazine.com/blog/archive/2012/10/science-racism-radiolabs-treatment-hmong-experience

If this story is accurate, then Kepler still might not have married the fifth woman by following the "optimal" strategy, since the problem was that he was having trouble deciding whether #5 was better than #4, and it was his delay that complicated matters. A simple mathematical rule like this assumes that you can make quick and accurate judgments, so it doesn't really model real life where you only have limited information or need time and experience before you can know your own mind.

I agree with the critique posted by Nathan and Hector that seeing the best one in the first 37% would leave you not with second best but with the last one, which on average would be someone around the 50% mark. To me that seems a fatal flaw with the strategy, so I would question its designation as "optimal". We intuitively know how do better than that by "settling" for someone "good enough" when we know we're approaching the end of the lineup. For example, here's a quick strategy to mitigate against the risk of seeing the best in the first 37% (and therefore according to the strategy ending up with the last one): if someone who's "good enough" (in top 10% of candidates overall) pops up in the last, say 15% of applicants, choose that person rather than waiting for the last one. But a better mathematician than I can probably come up with something even better than that, maybe a continuous function that more elegantly and efficiently adjusts the "desperation factor" as you approach the end of the line.

Another limitation of the "optimal" strategy is that there are lots of other ways to end up with someone who's not that close to the top. For example, if the top five were all destined to show up after #6 and #7, and if #7 arrived within the first 37% of candidates and #6 comes after that, you'd end up with #6. Of course, you'd never know it since you'd never meet #1-5, so I suppose you'd always THINK you ended up with #1!

So if I were making judgments about whether to accept this mathematical rule as "optimal" or keep looking for a better one....I think I would keep looking. But maybe only because I happened to see this rule within the first 37% of all the rules I'll see!

If this story is accurate, then Kepler still might not have married the fifth woman by following the "optimal" strategy, since the problem was that he was having trouble deciding whether #5 was better than #4, and it was his delay that complicated matters. A simple mathematical rule like this assumes that you can make quick and accurate judgments, so it doesn't really model real life where you only have limited information or need time and experience before you can know your own mind.

I agree with the critique posted by Nathan and Hector that seeing the best one in the first 37% would leave you not with second best but with the last one, which on average would be someone around the 50% mark. To me that seems a fatal flaw with the strategy, so I would question its designation as "optimal". We intuitively know how do better than that by "settling" for someone "good enough" when we know we're approaching the end of the lineup. For example, here's a quick strategy to mitigate against the risk of seeing the best in the first 37% (and therefore according to the strategy ending up with the last one): if someone who's "good enough" (in top 10% of candidates overall) pops up in the last, say 15% of applicants, choose that person rather than waiting for the last one. But a better mathematician than I can probably come up with something even better than that, maybe a continuously function that more elegantly and efficiently adjusts the "desperation factor" as you approach the end of the line.

Another limitation of the "optimal" strategy is that there are lots of other ways to end up with someone who's not that close to the top. For example, if the top five were all destined to show up after #6 and #7, and if #7 arrived within the first 37% of candidates and #6 comes after that, you'd end up with #6. Of course, you'd never know it since you'd never meet #1-5, so I suppose you'd always THINK you ended up with #2!

So if I were making judgments about whether to accept this mathematical rule as "optimal" or keep looking for a better one....I think I would keep looking. But maybe only because I happened to see this rule within the first 37% of all the rules I'll see!

It's an interesting anecdote historically-speaking, and the article makes it clear that, theoretically, it can apply to any selection process. But, really, don't you think the title is a bit, um, exclsusive towards certain members of the population? This isn't the first time I've seen Radiolab, which I do believe to be well-intentioned, get stuck in some rather unflattering dude-centric pitfalls. There are a lot of different kinds of Radiolab listeners. It might do you well to remember that.

This is inaccurate, right?

"Don't hire (or marry) any of them, but as soon as you meet a candidate who's better than the best of that first group — that's the one you choose!"

What if no one in the second group is 'better than the best of that first group'? I must be missing something, but I don't understand how this guarantees anything; you may never meet the criteria beyond the first group.

This is inaccurate, right?

"Don't hire (or marry) any of them, but as soon as you meet a candidate who's better than the best of that first group — that's the one you choose!"

What if no one in the second group is 'better than the best of that first group'? I must be missing something, but I don't understand how this guarantees anything; you may never meet the criteria beyond the first group.

The way I heard this problem was: Put 100 slips of paper in a hat, each with a number. Any numbers, 1,3, 0.0000000007, -4X10^50, whatever. Your job is to pick the largest number as it's read. What are the odds you need to make this game work in your favor? IIRC, it's 3:1. Using the same logic, watch the first 37 numbers pass. Then as soon as a number larger than the largest in that group is called, choose it. If the largest was in the first 37, then you loose by never calling a number. But with 3:1, the odds are in your favor.

What Nathan said...

It only works at all if he/she isn't in the first rejected block. You don't end up with second best if she is, you end with whoever's last, which is as likely to be among the worst, as among the best...

"...the Very Best Candidate might show up in that first 36.8 percent — in which case you'll be stuck with second best"

Actually in this case I think you wind up, not with the second best, but with the last person you interview. Since you've already seen the best, everyone else goes in the reject pile until you default to hiring the person at the end of the list.

Who knew that a mathematical formula could be applied to dating? It's an interesting concept to learn about honestly, and the article provided a catching story that intrigued my interest in such an endeavor. Maybe the mathematician's formula would be helpful in the dating environment, but when it comes down to choosing a future partner and achieving happiness with that one person, my personal opinion would to follow your gut instinct, basically your heart. However, for the heck of it I'd try the 36.8 formula for interesting results.

Commenting on Chris' remark : ... only if you know they're using this method.

If you interview too late in the "queue", you start to be compared to all the others before you so you might not look as good. In that case, it might be better to interview near the beginning.

I guess, if you had a choice, it'd be better to get interviewed as close, but after, the 36.8% point. *shrugs*

So remember, when going to a job interview or on a date ... don't try to be one of the first 1/3 - 1/2 who apply.

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