We're Down $700 Billion. Let's Go Double or Nothing!
How the financial markets fell for a 400-year-old sucker bet.
Read more about Wall Street's ongoing crisis.
On the other hand, if you start with a big bankroll (or generous lenders), it's pretty unlikely you'll encounter a run of luck bad enough to knock you out of the game. It's a little messy to compute exactly how unlikely, but we don't need exact figures to make the main point. (If exact figures are your bag, though, I've worked them out in a short PDF.) To simplify matters, let's say there's a 99 percent chance you wind up $100 ahead. Then the expected value of the martingale is
(0.99) x ($100) + (0.01) x (catastrophic outcome) = 0
But we already know the expected value is 0! Simple algebra suffices to solve the resulting equation—for the bet to have a value of 0, "catastrophic outcome" must be -$9,900.
In other words, the martingale strategy doesn't eliminate risk—it just takes your risk and squeezes it all into one improbable but hideous scenario. The expected value computation is unforgiving. No matter what ultrasophisticated betting strategy you adopt, you can't expect to make money in the long run by flipping a fair coin. There's always a risk of loss—and the smaller the chance of losing, the uglier the potential loss becomes. The result is a kind of "upside-down lottery." If you play the Powerball, you'll probably lose the cost of a ticket, but you might win big. In the martingale, you'll probably win a little, but if all six numbered balls match your ticket, then the bank comes around and takes away everything you've got.
You probably wouldn't sign up for that game. But the news of the last few weeks confirms that we've been playing it for years. And it looks like the balls just lined up. Oh, and there's one more difference between the thickly interwoven financial markets and the lottery: If one person wins the Powerball, just one person gets rich. If one massively leveraged financial firm loses while playing the martingale, it can bring the whole system down with it.
The complex derivatives behind the current financial havoc aren't literally martingales, but what's wrong with the martingale is one of the things that's wrong with the derivatives. There's no question that you can reduce risk drastically by combining different investments in a single portfolio; that's what plain-Jane instruments like index funds do. What sounds an alarm is the claim that you can get low risk and high returns in the same happy package. "Once the limits of diversification have been reached," John Quiggin, an economist at the University of Queensland, told me, "rearranging the set of claims involved isn't going to reduce risk any further, so if all parties appear to be making risk-free profits, the risk must have been shifted to some low-probability, high-consequence event." In other words, if it sounds too good to be true, it's probably heading toward some outcome too bad to be borne. Or, as financial skeptic Nassim Nicholas Taleb wrote last week, "It appears that financial institutions earn money on transactions (say fees on your mother-in-law's checking account) and lose everything taking risks they don't understand."
The martingale's bad reputation is just about as old as the martingale itself; the word, which dates back almost five centuries, is said to come from the hinterland town of Martigues in southern France, whose residents weren't known for their gambling savvy. The quantitative superstars who inhabit the back offices of the financial industry, and the people who regulate them, are no star-struck hicks. So why did they fling themselves so boldly into martingale-style investments?
One way the banks got fooled was by convincing themselves that the coin wasn't really fair. The only way to make money in the long term by betting on coin flips is to have some reliable way of predicting the outcome—for example, if you know that a flipped coin will land on the side it was flipped from about 51 percent of the time. Not long ago, the credit market was convinced that the upward trajectory of house prices had reached some kind of escape velocity and that the usual laws of finance were powerless to bring prices back down. It was supposed to be like betting on a coin that was heads on both sides.
A better way to account for the financial markets' irrational behavior is to concede that it's not as irrational as it looks. There's one kind of game in which a martingale strategy makes sense: a game in which it matters whether you win or lose, but not by how much. If you're a hockey team down by a goal with a minute left, you pull your goalie; that strategy has a negative expected value, but losing by two or three goals is no worse than losing by one. If you're a presidential candidate behind in the polls with time running short, you choose an unknown small-state governor for your running mate, or you suspend and then reanimate your campaign in a 48-hour period. What's the downside? If the magnitude of the loss doesn't matter, trading a big probability of a narrow loss for a smaller probability of a truly spectacular flameout is just smart play.
And this is what makes some people queasy about the federal bailout of the banks. It just might be that the prospect of a bailout—which could make a total collapse no worse for the banks than a garden-variety bear market—could have helped cause the martingale boom. There seems to be little question that the country needs the bailout now. But unless some real pain for the martingalers is built in, we'd better be ready for a return to maverick finance down the road.
Jordan Ellenberg is a professor of mathematics at the University of Wisconsin. His book How Not To Be Wrong is forthcoming. He blogs at Quomodocumque.
Illustration by Robert Neubecker.