As director of the president's National Economic Council, Larry Summers is currently facing the world's biggest math problem. It was encouraging, therefore, to read in Monday's New York Times that, when he applied for a job in 2006 with investment firm D.E. Shaw, "Mr. Summers was asked to solve math puzzles. He passed, and the job was his."
It's hard to imagine Summers being subjected to the same brainteasers that entry-level quants have to answer. And a White House spokesperson confirmed that it wasn't the same series of questions. But he did have to answer analytical reasoning problems asked by a member of the company's executive committee. What kinds of questions does D.E. Shaw ask?
The New York-based firm is known for its rigorous, numbers-heavy interview process. Most applicants have sterling academic backgrounds. The goal, therefore, is to see if the person can apply the concepts he learned in school to the real world. "The question is, 'Can they get past their white papers?' " says Richard Rusczyk, a former D.E. Shaw trader who conducted dozens of interviews over four years at the firm.
The type of questions most interviewers ask—and those D.E. Shaw is known for—are those with no right answers. Here's an example:
Ten people are bidding on a stock at 90, while 100 people are offering to sell it at 91. What price is the next trade?
Interviewees often say that since there are more sellers than buyers, the sellers get to determine the price. That logic usually yields an answer between 90 and 91. That's exactly wrong. "They're not thinking about what's going on in the real world," says Rubczyk. In reality, when there are more sellers than buyers, the price falls. So the next sale would probably be in the mid- to low 80s.
"Some candidates would say you can't answer that question, because there's no formula," says Rusczyk. "If that makes their heads explode, that's a problem."
The next level of difficulty is the type of question with no answer at all. One such question, which Rusczyk has asked, is the famous St. Petersburg Paradox:
There's a dollar on the table. I'm going to flip a coin. If it comes up heads, I'll double the money. If it comes up heads again, I'll double it again. Whenever it comes up tails, we stop.
But there's a catch: You have to pay a fee to play. How much are you willing to pay?
The answer: infinity. You should theoretically be willing to pay any amount, since the probability on any given flip is that you win 50 cents. (On the first flip, $1 x 1/2 = $0.50. On the second flip, $2 x 1/4 * = $0.50. On the third, $4 x 1/8 = $0.50. And so on.) So the potential winnings extend infinitely.
Of course, you can't offer the guy infinity dollars. So the interviewee is forced to either settle on a real world number—as much as the player can afford—or delve into marginal utility theory. Either way, the interviewer gets a sense of how the person's mind works. (This answer is understandably baffling to most people. See philosopher Ian Hacking wrestle with it here.)
The most difficult question of all is the kind that the interviewee must first get wrong before he can get it right. Rusczyk described a question in which the interviewer first explains the concept of a call option. (That's when you have a right but not an obligation to buy a stock.) He then asks a series of six or seven questions about the call option's price based on different market scenarios. The point is to create situations where academic math tells you to do one thing but the market tells you to do another. The ideal candidate follows the market. Eventually, you get to a stage where everyone gets the question wrong. "Then you ask them a leading question, after which they realize their last answer was wrong," says Rusczyk. "They'd then say, 'Where did I go wrong?' "
During his tenure at D.E. Shaw, only three candidates Rusczyk interviewed made it to the last question. "One is a partner [at D.E. Shaw], one took a professorship at Harvard, and one is in business," he says.
Rusczyk argues that these questions, while hypothetical, are very relevant to our current economic challenges. "Within financial markets, one of the big failures was assuming all these mortgages were more or less uncorrelated based on historical data," he says. In other words, models didn't take into account the possibility that housing prices would not keep trending up indefinitely. "That's kind of what the St. Petersburg Paradox is about. Theoretically, [the game] is worth an infinite amount of money. But in the real world, it's not worth infinity."
If the point of D.E. Shaw interviews is to make sure the person can repurpose academic models for the real world, their methodology might serve the Obama administration well. In the meantime, here's another one for Summers:
x = the economy
x + y = the economy not all screwed up
Correction, April 7, 2009: Due to a copy-editing error, this article misstated the math behind the St. Petersburg Paradox. (Return to the corrected sentence.)