I wrote before on a surprising resolution to the famous St. Petersburg Paradox using the fact that the game is non-ergodic. The term ergodic refers to systems that explore their “phase space” well enough so that an average over all of phase space would be equal to an average over time. Simply put, in the St. Petersberg wager the expected value, which is the gain averaged over playing the game many times, is not representative of how a person would actually experience the game, since no one would (or could) play the game enough times. That is, you have a really tiny chance of winning a lot of money, and these effects balance just enough when calculating to give a series that diverges to infinity. Recently, I came across another site that uses the same approach to solve the paradox, but extends the reasoning in an attempt to answer a more general question. The wager described in the St. Petersberg paradox is just one example of a gamble that has a positive expected value (in fact, infinite expected value), and yet we “know” that we should stay away. This contradicts the basic premise of rational self-interest, which would seem to encourage taking any wager in which you expect a positive average return. The behavior heuristic of risk aversion developed in humans, for, among other reasons, the simple fact that if we lose enough times in a row we will be out of the game, and not have the opportunity to try to win it back (“gambler’s ruin“). We might then define a “risky” game as one in which there is a strong probability of being wiped out from one bad round or a brief streak of your luck running cold. The author of the blog post points out that no matter how favorable the odds, if you play a risky game enough times, you will lose your shirt. Even a well thought-out strategy for betting on games in your favor, like the Kelly Criterion, which basically says that you should wager a fraction of your bankroll equal to your “edge,” or how favorable the odds are, divided by the odds, will leave the gambler at risk of losing everything. That is, you still have to balance your desire to win quickly when at an advantage with the reality that sometimes “The race is not always to the swift.” Interestingly, the system was developed by J. L. Kelly, Jr, when he found a way to count cards at blackjack so that he would have a slightly better than 50:50 chance of winning.
I’ve been thinking about the way most wagers are structured by casinos in order to get around this risk aversion. Usually, your bet is a small fraction of your net worth (eg. penny slots, or one dollar lotto ticket) with a small chance to win a big jackpot. Rarely do you see a bet in Vegas like 99% chance to win one dollar, 1% chance to lose 110 dollars, and not just because of the house edge. Even a fair game with so much downside would trigger the risk-aversion klaxons in patrons’ minds. However, if you are so inclined, you can structure your betting so that you have the equivalent wager. Using a martingale system, you can win $1 with very high probability, with a small chance (depending on how much money you start with) of losing it all. All you have to do is play a game for even money starting with a $1 wager. If you win, stop and enjoy your unit increase in wealth. If you lose the first game, play again but increase the stakes so that you will be up $1 overall if you win. While it might seem like easy money, if you keep repeating this system, eventually you will have a losing streak big enough to wipe you out. Nicolas Taleb describes stock traders who did essentially the same wager using highly leveraged positions. He calls it “picking up pennies in front of a train,” in the sense that each time you make a little gain, but eventually a “tail event” will arrive to blow everything up, with a loss more than overwhelming all of the previous gains. Many traders were lulled into a false sense of security, in which small and steady gains carried a very real, but unseen, risk.