The Calculus Diaries - Jennifer Ouellette [35]
Born in Texas, John L. Kelly was a Naval Air Force pilot during World War II who survived a plane crash into the ocean and eventually earned a PhD in physics from the University of Texas-Austin. He found work in the oil industry, using his scientific training to identify likely oil sites. But his employer’s instincts were better than Kelly’s models, so Kelly decided the oil business was best left to those with a nose for hidden deposits and found himself working for Bell Labs, one of the most prestigious research centers in the United States. He cut a colorful figure among his fellow physicists, with his Texas drawl, passion for guns, and penchant for taking calculated risks.
It was a hugely popular television game show called The $64,000 Question that inspired Kelly to devise his famous formula in the 1950s. People would place bets on the most likely contestants to win. But there is a three-hour time difference between New York City—where the show was produced and aired live—and the West Coast. Kelly heard a rumor that one gambler on the West Coast had a partner back east tell him the winners by phone so that he could place bets before the show aired in the West, giving said gambler an inside track. This spurred Kelly to ponder probabilities and gambling. He reasoned that if a gambler with an inside track bets everything he or she has on the basis of those tips, the gambler will lose everything the first time he or she gets a bad tip. But if the same gambler makes just the minimum bet for each tip, that insider information no longer confers much of an advantage. Recognizing the importance of how much someone bets in fashioning a winning strategy, Kelly determined that dividing your edge by the odds tells you what percentage of your bank roll you should bet each time.
The odds determine how much profit you make if you win; the edge describes the amount you expect to win on average if you make the same wager repeatedly under the same probabilities. Remember the lesson of gambler’s ruin: Even if the odds are in your favor, you still don’t want to bet your entire bankroll in one fell swoop; your odds of losing everything on one roll are much higher. Play it safe and bet too little, however, and your return won’t be sufficient to make up for the inevitable losses. Kelly’s formula reveals the optimal betting strategy for maximizing long-term returns. For a bet with even odds, Kelly tells us to bet a fraction of our bankroll that is determined by 2p -1, where p is our probability of winning.
When it comes to playing craps in a Vegas casino, it will be a discouraging answer unless you have the good fortune to be the house. Players usually have an edge of zero at best (a 50/50 chance) and more often it is slightly less. In either case, the Kelly criterion says that the best way to maximize your long-term return in craps is to bet 0 percent of your bankroll—that is, not to play. But that is just a detached, mathematical analysis that doesn’t take into account the fun factor, the sheer pleasure one derives from playing craps.
We can tweak this problem a little to take that subjective quality into account by assigning it a quantitative value: Let’s say the odds are 49/51, giving the house a 2 percent edge, but the fun factor is 3 percent, giving us a net edge over the casino of 1 percent. That corresponds to a winning probability of 0.51, so the Kelly criterion tells us to bet 2 percent of our bankroll. Now, we can place our bets accordingly to optimize our fun—that is, play as long as possible by maximizing our long-term gains. We’ll still most likely lose in the end, but we will be getting the most bang for our buck.
There is a downside to the Kelly criterion, or rather, a kind of trade-off: Following the Kelly criterion exactly leads to a lot of volatility in the outcomes. In the long term, it works; in the short term, it can lead to intense anxiety over the wild fluctuations in one’s fortunes. For those who prefer a bit less drama in their gambling, a popular