I have a piece in Slate today about the “martingale strategy” and how it relates to the financial crisis:
Here’s how to make money flipping a coin. Bet 100 bucks on heads. If you win, you walk away $100 richer. If you lose, no problem; on the next flip, bet $200 on heads, and if you win this time, take your $100 profit and quit. If you lose, you’re down $300 on the day; so you double down again and bet $400. The coin can’t come up tails forever! Eventually, you’ve got to win your $100 back.
Or not. If you want to see how well this strategy works in practice (answer: not very) try a few runs of the martingale applet at UIUC.
My colleague Timo Seppalainen explained to me a nice way of seeing of the long-term failure of the martingale. Let X_j be the length of the jth run of tails. Then X_j is 0 with probability 1/2, 1 with probability 1/4, 2 with probability 1/8, and so on. The chance that X_j >= n is 1/2^n. In particular, the probability that X_j is at least (log_2 j + 1) is about 1/2j.
But the amount of money you lose on a run of n tails is about 2^n, while the amount of money you’ve won prior to the start of the jth run is about j. In particular, if X_j > (log_2 j + 1) then you’re at least j dollars down after the jth run of tails. Since the sum of 1/2j as j goes to infinity diverges, you almost always have infinitely many occurences of X_j > (log_2 j + 1); as I learned from Timo, this follows from the second Borel-Cantelli Lemma.
So there are infinitely many j such that, after the jth run of tails, you’re at least j dollars down. Even if you start with a million dollars, that means you’re eventually going broke.