Martingale Betting: A Metaphor for Too Big to Fail
Martingale betting is a class of betting strategies that originated in 18th century France which can turn any game, whether blackjack, roulette or anything else, ideally something played in short, discrete rounds with a roughly 50-50 win/lose ratio, into one which gave the gambler an almost certain chance of success. The way the algorithm works is like this. Imagine that you are playing a simple slightly biased coin toss game where you can bet any amount of money and if you land heads (49% chance) you win and if you land tails (51% chance) you lose. First, you bet one dollar. If you win, rejoice and go back to step one. If you lose, next turn bet two dollars. If you win, once again go back to step one, but if you lose increase your bet to four dollars. If you are unlucky enough to keep losing, you double your bet each time. Eventually, you are guaranteed to win at least once, at which point no matter how many times you lost you come out having gained one dollar.
However, a strategy that turns a game with a slightly negative expected value each round into one with a guaranteed positive reward each time is mathematically impossible, so there must be some hidden flaw. And the flaw is this: your money is not unlimited. If you are playing with 1 million dollars, once every 705661 rounds you will encounter 20 tails in a row and go bankrupt. Thus, the expected value of a round is in fact -$0.41, negative as expected. However, the strategy is neverthelss a good one to show off to your friends: the 7056 out of 7057 times that you can demonstrate a hundred-round streak of guaranteed profits you will not only gain a small amount of money but also gain the social reward of appearing to your friends to be a mathematical genius.
But there is one circumstance in which the Martingale betting system is hugely successful: if you can count on someone to bail you out if you are in trouble. If you can convince someone that you are systematically important and that allowing you to go bankrupt would lead financial armageddon, your expected value function changes – if, for example, you can expect to only foot half the bill of a crisis in the form of hastily conceived and relatively mild after-the-fact penalties, your expected value each round goes up from -$0.41 to +$0.29, making the losing game a profitable one through artificially externalized costs.
What’s most deceptive of all about the Martingale strategy is just how stable it appears to be. Your profits are guaranteed to be exactly $1 each and every round no matter what happens behind the scenes. Standard deviation: zero. To the outside world, you’re an unstoppable profit-making tycoon. And herein lies the problem. I mentioned earlier that it is, in an informal sense, rational to go play blackjack in a casino because if you win you’re also gaining a reputation among your friends as a mathematical genius. But for hedge fund managers and traders in the financial industry, gaining a reputation as a mathematical genius is not fun and games at all; rather, it’s how they get promoted. If a manager appears to be earning constant above-market returns per year, every year, banks and hedge funds everywhere will look to hire him as an advisor and investors everywhere will eagerly buy up copies of his book is he decides to release one. And if he fails he will simply fade into the background and, if worse comes to worst, may have to find a job elsewhere. Ninety percent chance of a 7% return, ten percent chance of bankruptcy becomes a ninety percent chance of eternal glory and a ten percent chance of simple mediocrity.
It’s human nature to want to soften the harshest blows of what seems like random chance. We see people speeding to get to work faster, but the one in a million times that it doesn’t work out and they end up in an accident we say that they don’t deserve to die for something so trivial. The only way we can maintain such a position and not create perverse incentives, however, is by softening the profits as well, in most healthy societies throughout history through a culture that imposes public shame on such acts. The financial system as it stands, however, is a master of exploiting this tendency. Shame, to them, has no meaning; it cannot be bought or sold, longed or shorted, and has no place on the quarterly balance sheet. Every time we give them the message that we got their backs in case something truly unprecedented happens, what we are giving them at the same time is an unlimited license to print money. Also, by not letting them fail, we are in effect playing a Martingale game of our own. We may avoid having to suffer a moderate amount of pain now, but every crisis where the banks get off scot free, they emerge stronger, and the same incentive structures and the same kind of thinking that led to the crisis in the first place is allowed to remain at the helm and breed new disciples. As with any Martingale game, however, there will eventually be a final round, and I dread to think of what might happen if we lose it.