Chances are, you just typed "Kelly formula" or "Kelly criterion" into a search engine. The Kelly criterion is a money-management formula of passionate interest (and controversy) to card players, sports bettors, investors, hedge fund managers, and economists. One topic you won't find much about on the Web is Kelly himself. In the hope of remedying that, I've started this page. I am the author of a book, Fortune's Formula (Hill and Wang, 2005), in which John Kelly is a key figure. Kelly was a Bell Labs physicist and computer scientist of diverse accomplishments and interests. I imagine he would have thought it odd that his name would be linked above all to his "gambling formula."
So who was Kelly? How did he get a money-management formula named after him?
John Larry Kelly, Jr. (1923-65), was born in Corsicana, Texas. He came of age during World War II and spent four years as a flyer for the Naval Air Force. A capable pilot, he survived a plane crash into the ocean. Kelly did undergraduate and graduate work at the University of Texas, Austin. His 1953 Ph.D. topic was an "Investigation of second order elastic properties of various materials." This led to work in the oil industry. As Kelly told the story, his employer, a successful wildcatter, would smell the soil and drill by instinct, ignoring Kelly's carefully prepared scientific recommendations. The oilman's hunches were so unerring that Kelly decided he was in the wrong line of work. He accepted a job offer from Bell Labs.
Bell Labs, in Murray Hill, New Jersey, was one of the world's most prestigious scientific research centers. Kelly was barely 30 when he arrived. His Texas drawl set him apart (oddly, it seemed to grow deeper the more years he lived in New Jersey). So did his interest in guns. Kelly belonged to a gun club and counted a Magnum pistol among his prize possessions. Kelly was married to the former Myldred Parham. Myldred was herself a pilot and had been the executive officer of a MASH unit in India during the war. As a couple, the Kellys were ruthless tournament bridge players. They raised three children -- Patricia, Karen, and David -- in a suburban house in Berkeley Heights, New Jersey.
Kelly was "a lot of fun, the life of the party," I was told. Another associate described him as a "wild man." One tale claims that Kelly once earned a reprimand by prankishly flying a plane under the George Washington bridge. In another story, Kelly was at a conference on Cape Cod where a new rocket-powered ejection seat for pilots was being shown. Kelly decided it would be interesting to see if the seat really worked. He and several others put the seat in the back of a convertible and drove around Cape Cod looking (unsuccessfully) for a suitable place to launch it.
Kelly was a chain-smoker. Even in the family's home movies, Kelly is puffing away as he watches the children in the pool. Daughter Karen Kelly recalls that "Not only did we have guns and rifles in our house, but my father also had equipment to make bullets. He used to entertain people with shooting bullets with plastic or gummy inserts into a stone wall in the house. My mom said it was annoying to get them out."
One of Kelly's best friends at Bell Labs was a fellow Texan, Ben Logan. Each morning, Kelly and Logan would make coffee, then go into Logan's office. Kelly would immediately put his feet up on the chalk rim of the blackboard and light up a cigarette. Faced with a difficult problem, Kelly would think a moment, take another drag, and say something showing the most amazing insight. Many rated Kelly the smartest person at Bell Labs next to Claude Shannon himself.
Shannon was in a class by himself. He had single-handedly created the abstract theory of communication called information theory. Shannon presciently realized that computers could express numbers, words, pictures, audio, and video as strings of digital 1s and 0s. Information theory underlies the Internet and today's wired, and wireless, world.
At Bell Labs, Kelly was working on data compression schemes for the still-young medium of television. This brought him into Shannon's new field. Kelly made an ingenious connection between information theory, gambling -- and television.
On June 7, 1955, American television debuted a new quiz show called The $64,000 Question. The show was a sensation. It captured as much as 85 percent of the viewing audience and led to dozens of copycat shows. Kelly heard about a peculiar scam in the news. Some viewers of The $64,000 Question were placing bets on which contestants would win. The show was produced in New York and aired live on the East Coast. It was delayed three hours on the West Coast. According to the news story, one West Coast gambler learned the winners by phone and placed his bets before the West Coast airing.
Thinking about this convinced Kelly that a gambler with "inside information" could use some of Shannon's equations to achieve the highest possible return on his capital. Shannon was intrigued by this application and urged Kelly to publish his finding. Kelly's article appeared (under the opaque title "A New Interpretation of Information Rate") in a 1956 issue of the Bell System Technical Journal.
Kelly wryly presented his idea as a system for betting on fixed horse races. A "gambler with a private wire" gets advance word of the races' outcomes. The natural impulse is to bet everything you've got on the horse that's supposed to win. But when the gambler adopts this policy, he is sure to lose everything on the first bum tip. Alternatively, the gambler could play it safe and bet a minimal amount on each tip. This squanders the considerable advantage the inside tips supply.
In Kelly's analysis, the smart gambler should be interested in "compound return" on capital. He showed that the same math Shannon used in his theory of noisy communications channels applies to the gambler. The gambler's optimal policy is to maximize the expected logarithm of wealth. Though an aggressive policy, this offers important downside protection. Since log(0) is negative infinity, the ideal Kelly gambler never accepts even a small risk of losing everything.
You don't even have to know what a logarithm is to use the so-called Kelly formula. You should wager this fraction of your bankroll on a favorable bet:
The edge is how much you expect to win, on the average, assuming you could make this wager over and over with the same probabilities. It is a fraction because the profit is always in proportion to how much you wager. At a racetrack, the edge is diminished by the track take. When your edge is zero or negative, the Kelly criterion says not to bet.
Odds means the public or tote-board odds. It measures the profit if you win. The odds will be something like 8:1, meaning that a winning wager receives 8 times the amount wagered plus return of the wager itself.
In the Kelly formula, odds is not necessarily a good measure of probability. Odds are set by market forces, by everyone else's beliefs about the chance of winning. These beliefs may be wrong. In fact, they have to be wrong for the Kelly gambler to have an edge. The odds do not factor in the Kelly gambler's inside tips.
Example: The tote board odds for Seabiscuit are 5:1. Odds are a fraction -- 5:1 means 5/1 or 5. The 5 is all you need.
The tips convince you that Seabiscuit actually has a 1 in 3 chance of winning. Then by betting $100 on Seabiscuit you stand a 1/3 chance of ending up with $600. On the average, that is worth $200, a net profit of $100. The edge is the $100 profit divided by the $100 wager, or simply 1.
The Kelly formula, edge/odds, is 1/5. This means that you should bet one-fifth of your bankroll on Seabiscuit.
This version of the formula does not take into account the effect of one's own bet on the odds. It has the virtue of being easy to remember and applicable to other forms of gambling like blackjack. By always making the Kelly bet, you increase your bankroll faster than with any system. That's the good news. The bad news is that it's a rough ride. Downward plunges of wealth are frequent and steep. This can be rectified through diversification (as in team play in blackjack, or at a hedge fund, where the manager makes many simultaneous "bets" with low correlation). For the lone player betting on a single hand or horse, the Kelly formula demands guts and patience -- hence the controversy. Many have found the "half Kelly" strategy to be a good compromise. You bet half of edge/odds. This achieves ¾ the compound return of Kelly betting with much less volatility.
Kelly had originally titled his article "Information Theory and Gambling." That bothered some AT&T executives, as did his mention of a "private wire." Throughout the twentieth century, AT&T had leased wires to organized crime figures who ran "wire services" reporting racetrack results to bookies. Even in the 1950s, bookies were still big customers. The executives feared the press might conclude from Kelly's article that Bell Labs was doing work to benefit illegal gamblers. They pressured Kelly to change the title of his paper to "A New Interpretation of Information Rate."
In fact, the executives didn't have much to worry about. Virtually no one took much note of the article when it first appeared. The practical application of the Kelly criterion began in the early 1960s, after MIT student Ed Thorp told Shannon about his card-counting system for blackjack. Shannon referred him to Kelly's article. Thorp used it to compute optimal bets in blackjack and later in the securities markets. It was Thorp's success as hedge fund manager that made Wall Street start to take notice of the Kelly criterion.
Kelly died tragically young, of a brain hemorrhage at the age of 41. He was by then the head of Bell Labs' information coding and programming department and the author of several patents. Kelly has one further claim to fame. In 1961 Kelly and colleague Carol Lochbaum demonstrated a new voice synthesis system by making a recording of their machine singing the song "Daisy Bell," better known as "Bicycle Built for Two."
It was the latter that inspired the death scene of the computer HAL in Stanley Kubrick's film 2001: A Space Odyssey. Science-fiction writer Arthur C. Clarke had visited Bell Labs in the mid 1960s and heard Kelly's recording. In Clarke's screenplay, HAL is unplugged and reverts to a childish state, singing the same song that Kelly's computer did.
I would welcome any recollections from those who knew Kelly. You can contact me by clicking here.
Text adapted from Fortune's Formula, © 2005 William Poundstone. Kelly photo courtesy of Ben Logan