Fortune's Formula (9 page)

I
N
1894
THE CITY FATHERS
of Corsicana, Texas, were drilling a new well. They struck oil instead of water. Corsicana became one of the original petroleum boomtowns. For a time the town was wealthy enough to boast an opera house where Caruso sang. Then the Depression came and changed everything. Oil prices plummeted to as low as ten cents a barrel. The region’s economy fell into chaos. The town’s most enduring industry was and is a mail-order fruitcake.

John Larry Kelly, Jr., was born in Corsicana on December 26, 1923. His mother, Lillian, worked for the state teachers’ retirement program. Of Kelly’s namesake father, I could discover little except that he was a CPA. Kelly rarely if ever spoke of his father to friends. Possibly he never knew him. The 1930 census reports that six-year-old John lived with his mother, Lillian, his maternal grandmother, and an aunt in a $30-a-month apartment.

Kelly came of age during World War II and spent four years as a flier for the Naval Air Force. He then did undergraduate and graduate work at the University of Texas at Austin, segueing into an unglamorous end of physics. The subject of his master’s thesis, “Variation of Elastic Wave Velocity with Water Content in Sedimentary Rocks,” hints at an application to the oil industry. Kelly’s 1953 Ph.D. topic was an “Investigation of Second Order Elastic Properties of Various Materials.” The work was important enough to get Kelly a job offer from Bell Labs.

In no small part due to Shannon, Bell Labs was one of the world’s most prestigious research centers. American Telephone and Telegraph’s benign monopoly gave it the luxury of supporting basic research on a grand scale. It was said that Bell Labs was like a university except that its researchers didn’t have to teach, and there was always enough money for experiments.

Kelly was barely thirty when he arrived at Bell Labs’ Murray Hill site. He was strikingly handsome, although he struck some as slightly unhealthy-looking. Bags under his eyes made him look older, mysterious, and dissipated. Kelly was a chain-smoker and liberal drinker—“a lot of fun, the life of the party.” He was gregarious, loud, and funny, quick to loosen his tie and kick his shoes off.

His Texas drawl set him apart at Bell Labs. So did his interest in guns. Kelly collected guns and belonged to a gun club. Among his prize possessions was a Magnum pistol. Another passion was pro and college football. Kelly built resistor circuits on breadboards to model and predict the results of football matches. A team’s win-loss record would be represented with a resistor of a particular ohm rating.

Kelly was married to Mildred Parham. As a couple, they were ruthless tournament bridge players. The Kellys raised three children—Patricia, Karen, and David—in a suburban house at 17 Holly Glen Lane South, Berkeley Heights, New Jersey.

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. With a wave of his hand, he would flick the ashes in the general direction of the trash can on the other side of the room. The ashes, insensible to Kelly’s cue, fell straight down. When one cigarette burned down, it was time to light the next. Kelly ceremoniously stamped each butt out on Logan’s floor.

Faced with a difficult problem, Kelly would sit back, put his feet up somewhere, take another drag, and say something showing the most amazing insight. Manfred Schroeder and Billy Kluver rated Kelly the smartest person at Bell Labs next to Shannon himself.

Kelly and Shannon did not become well acquainted until just before Shannon left Bell Labs. I came across one anecdote involving them both. Robert Fano remembered the two men visiting MIT circa 1956. One evening after dinner they walked past the school’s Kresge Auditorium. Designed by Eero Saarinen, it is a low, dome-shaped building whose roof is thinner in proportion to its area than an eggshell. Students found it an irresistible climbing challenge. Upon hearing this, Shannon and Kelly kicked off their shoes and began scaling the dome. Campus police showed up to stop them. Fano was barely able to talk them out of arresting the “distinguished visitors from Bell Telephone Laboratories.”

Kelly’s career covered a variety of fields. He started out studying ways to compress television data. This brought him into Shannon’s new discipline of information theory, which Kelly probably absorbed through his own reading.

Kelly was drawn into a line of research that had proven to be a black hole of time, money, and talent. It was voice synthesis—teaching machines to talk. Bell Labs’ people had been interested in that idea since the 1930s. It was like alchemy. The people in the field perpetually felt themselves to be on the verge of a great and profitable breakthrough that required just a few more years and a few more dollars. The breakthrough was never to come, at least not in Kelly’s short life.

The original goal was not talking computers but conserving bandwidth. In the 1930s, Bell Labs’ Homer Dudley determined that phone conversations could be compressed by transmitting phonetic scripts rather than voices. In Dudley’s scheme, the system would break speakers’ words into a series of phonetic sounds and transmit a code for those sounds. At the other end of the line, the phone would reconstitute the words phonetically, with some approximation of the original voice and intonation. This system was called a “vocoder” (for voice coder). Dudley exhibited such a device in a grand art deco pavilion at the 1939 World’s Fair. Dudley’s vocoder could send twenty conversations on a line that previously carried one. The downside was that the reconstituted voices were barely intelligible.

Bell Labs was slow to abandon the vocoder concept. As late as 1961, Betty Shannon’s former boss, John Pierce, half seriously proposed to extend the vocoder concept to television or videophones. “Imagine that we had at the receiver a sort of rubbery model of the human face,” Pierce wrote. The basic idea was that every American home would have an electronic puppet head. When a call came in, the puppet head would morph to the appearance of a distant speaker, and you’d converse with it, as the puppet head mimicked every word and facial expression of the calling party.

Kelly worked on a more sophisticated idea, rule-based speech synthesis. Given the phonetic pronunciation of a dictionary, a human can pronounce almost any word. Kelly was attempting to program a computer to perform the same feat. He would feed a computer phonetic spellings on punch cards. The computer would use that, and a set of rules, to enunciate the words. Kelly and others discovered, however, that spoken language is a slippery, interconnected thing. The way a letter or syllable sounds depends on context. Kelly tried to devise rules to account for this, and an efficient way of encoding not only word sounds but also intonation.

At the same world’s fair where AT&T debuted the vocoder, NBC’s General Sarnoff made the famously misguided prediction that “television drama of high caliber and produced by first-rate artists will materially raise the level of dramatic taste of the nation.” Moe Annenberg’s son, Walter, bet his fortune on the new medium by founding
TV Guide
. For every Paddy Chayevsky, however, there were a thousand hucksters dreaming up new and improved ways for TV to prostitute itself. The latest outrage of the postwar era was “giveaway shows.” A show’s host would phone a random American. The lucky citizen would have to answer the phone with a prescribed catchphrase that had been given out on the broadcast—or else answer a question whose answer had been supplied on the show—in order to win a prize.

The shows were a way of bribing people to stay glued to the TV screen or radio dial. In 1949 the Federal Communications Commission, in one of its periodic turns as guardian of public taste, banned giveaway shows. It did this on the dubious theory that they constituted illegal gambling. The FCC vowed not to renew the license of any station broadcasting giveaway shows. Such programs disappeared from the air.

The three major broadcast networks took the case to the Supreme Court. In 1954 the Court sided with the networks. Giveaway shows were legal.

This ruling opened the floodgates. On June 7, 1955, CBS Television responded by airing a new quiz show,
The $64,000 Question
. It was loosely based on one of the old radio giveaway shows,
Take It or Leave It
. The show’s producers took the Supreme Court decision as license to award vastly bigger prizes than had ever been offered on a game show. The top prize on the old radio show had been $64.

A contestant who answered the first question correctly on the TV show won $1. Prizes doubled with each succeeding question—jumping from $512 to $1,000 to keep the amounts round—and continuing to double, all the way up to a top prize of $64,000. The twist was that the contestants had to risk losing everything they had won in order to have a crack at the next question. It was double or nothing.

The most successful contestants sat in the “Revlon Isolation Booth” to keep them from hearing shouted help from the studio audience. The producers turned off the air-conditioning in the booth so that close-ups would show beads of sweat on the contestants’ foreheads. The quiz show was as big a sensation as the Kefauver hearings had been. It captured as much as 85 percent of the viewing audience and led to dozens of copycat shows.

The show’s contestants became celebrities. There was Redmond O’Hanlon, the Staten Island cop who was an expert on Shakespeare…Joyce Brothers, the psychologist who knew about prizefighters…Gino Prato, the Bronx cobbler who knew opera…Some viewers placed bets on which contestants would win.
The $64,000 Question
was produced in New York and aired live on the East Coast. It was delayed three hours on the West Coast. One West Coast gambler learned the winners by phone. He placed his bets before the West Coast airing, already knowing the winners.

According to the mimeographed notes for a lecture that Shannon gave at MIT in 1956, it was “news reports” of this con that inspired John Kelly to devise his mathematical gambling system. I have looked through back issues of newspapers and magazines trying to find stories about betting on
The $64,000 Question
or the unnamed West Coast bettor, without luck. The only thing I came up with was that similar scams have been reported for the recent reality shows
Survivor
,
The Bachelor
, and
The Apprentice
. All were taped in remote locales or on closed sets, and contestants and crew pledged to keep the winner secret until the airdate. An Internet casino, Antigua-based BetWWTS.com, was taking bets on the shows’ winners. In each case, the casino suspended betting after a number of large bets were placed on one contestant, suggesting that someone had inside information.

In any case, Kelly was able to connect the
$64,000 Question
con to a theoretical question about information theory. Shannon’s theory, born of cryptography, pertains exclusively to
coded
messages. Some wondered whether the theory could apply in situations where no coding was involved. Kelly found one. Though he worked in a different department and did not then know Shannon well, he decided he should tell him.

Shannon urged Kelly to publish his idea. Unlike Shannon, Kelly was prompt at doing so.

K
ELLY DESCRIBED HIS IDEA
this way: A “gambler with a private wire” gets advance word of the outcome of baseball games or horse races. These tips may not be 100 percent reliable. They are accurate enough to give the bettor an edge. The bettor is able to place bets at “fair” odds that have not been adjusted for the secret tips. Kelly asked how the bettor should use this information.

This is not the no-brainer you might think. Take an extreme case. A greedy bettor might be tempted to bet his entire bankroll on a horse on the basis of the inside tips. The more he wagers, the more he can win.

The trouble with this policy is that the tips are not necessarily sure things. Sooner or later, a favored horse will lose. The gambler who
always
stakes his entire bankroll will lose everything the first time that a tip is wrong.

The opposite policy is bad, too. A timid bettor might make the minimum bet on each tip. That way he can’t lose too much on a bum tip. But minimum wagers mean minimum winnings. The timid bettor squanders the advantage his inside information provides.

What should the bettor do? How can he make the most of his tips without going broke?

Those lucky souls who strike it rich at the track do so by
parlaying
. They win, then put some or all of their winnings on another winning horse, and then on another, and so on, increasing their wealth exponentially at each step. Kelly concluded that a gambler should be interested in “compound return,” much as an investor in stocks or bonds is. The gambler should measure success not in dollars but in percentage gain per race. The best strategy is one that offers the highest compound return consistent with no risk of going broke.

Kelly then showed that the same math Shannon used in his theory of noisy communications channels applies to this greedy-though-prudent bettor. Just as it is possible to send messages at a channel’s bandwidth with virtually no chance of error, it is possible for a bettor to compound wealth at a certain maximum rate, with virtually no risk of ruin. The have-your-cake-and-eat-it-too feature of Shannon’s theory also applies to gambling.

Kelly analyzed pari-mutuel betting. At U.S. and many Asian racetracks, the bettors themselves set the odds. The track adds up every “win” wager on a given race, deducts a track take for expenses and taxes, and distributes the remaining money to the people who bet on the winning horse.

The payoffs therefore depend on how much money was wagered on the winning horse. This is easiest to explain in the case of a track with no take. Suppose one-sixth of the money is bet on Smarty Jones, and Smarty Jones wins. Everyone who bet on Smarty Jones to win will then get back six times their wager. This is conventionally expressed as odds: “Smarty Jones is paying 5 to 1.” That means that someone who bets $10 wins a profit of $50
plus
the return of the $10 wager (for a total of $60).

Kelly described a simple way for a gambler with inside tips to bet. It is practical only at a track with no take (there aren’t any!) or in a case where the inside tips are highly reliable. The strategy is to bet your entire bankroll each race, apportioning it among the horses according to your informed estimate of each horse’s chance of winning.

With this system, you bet on every horse running. One horse
has
to win. You are certain to win one bet each race. You can never end up completely broke.

Strangely enough, this is also the
fastest
way to increase your bankroll. Most people find this hard to believe. You don’t get rich in roulette by betting on every number.

That’s because the payoffs in roulette favor the house. The situation is different at our imaginary track with no take—and with inside tips. Look at the tote board. The posted odds reflect the aggregate beliefs of all the poor slobs with no inside information. Should you bet your bankroll according to posted odds, you would invariably win back your bankroll every race (again, assuming no track take). If the odds on Seabiscuit are 2 to 1—meaning that the public believes he has a 1-in-3 chance of winning—you would put 1/3 of your bankroll on Seabiscuit. And if Seabiscuit won, you would get back three times your wager, or 100 percent of your original bankroll. The same goes for any other horse, favorite or long shot.

Kelly’s gambler ignores the posted odds. The private wire gives him a more accurate picture of the
real
chances of the horses winning. He apportions his money according to his superior estimates of the probabilities.

Take the most clear-cut case. The private wire says that Man o’ War is a sure thing. It is known from experience that the wire’s information is
always
right. You can be certain that Man o’ War has a 100 percent chance of winning and the other horses have zero chance. Then that is how you should apportion your money. Bet 100 percent on Man o’ War and zero on the other horses. When Man o’ War wins, you will collect a profit according to the tote-board odds. This is obviously the best way of profiting from a 100 percent sure inside tip.

Kelly’s (and Shannon’s) system more often deals with uncertainty. In the real world, nothing is a sure thing. It might be that the wire service is sometimes wrong or intentionally deceptive—or there’s noise on the line and you can’t be sure you heard the tip right. It might be that the wire service gives only probabilities, like a rain forecast, or it supplies inside information whose significance you must interpret for yourself (“Phar Lap didn’t eat his breakfast”).

Shannon’s theorem of the noisy channel describes a quantity aptly called
equivocation
. It is a measure of ambiguity. In the case of an unreliable source (assuming you choose to consider that source as part of the communications channel), equivocation can be due to words that sound alike, typos, intentionally vague statements, mistakes, evasions, or lies. Equivocation describes the chance that a received message is wrong. Shannon showed that you must deduct equivocation from the channel capacity to get the information rate.

Kelly’s gambler must also take equivocation into account. He places bets according to his best informed estimates of the probabilities. When you believe that War Admiral has a 24 percent chance of winning, you should put 24 percent of your capital on War Admiral. This approach has come to be called “betting your beliefs.”

In the long run, “bet your beliefs” will earn you the maximum possible compound return—provided that your assessment of the odds is more accurate than the public’s.

You may still be wondering, why not just bet on the horse most likely to win? The quick answer is that the horse most likely to win might not win. Say you have a very accurate wire service and believe that Northern Dancer has a 99 percent chance of winning. You bet 99 percent of your money on Northern Dancer. But you keep the other 1 percent in your pocket.

There is a 1 percent chance that Northern Dancer
won’t
win. Should that happen, you’ll be left with only the pittance in your pocket. You would have done better to hedge your bets by wagering that pittance on all the other horses. You would be sure to win something, and possibly a lot. The bets on the horses you think will lose are a valuable “insurance policy.” When rare disaster strikes, you’ll be glad you had the insurance.

There is a poetic elegance to “bet your beliefs.” You play the happy fool. You ignore the odds on the tote board and bet on every horse according to your own private beliefs. Nothing could be more simple (-minded). Nothing achieves a better return on investment.

Those of less poetic mind will note that “bet your beliefs” is of little use at a real track. U.S. racetracks skim anywhere from 14 to 19 percent of the amount wagered. It’s 25 percent in Japan. That means that anyone who bets an entire bankroll on every race is giving 14 to 25 percent of that bankroll to the track each time out. It would take a phenomenally accurate stream of inside tips to overcome that.

Kelly describes an alternate and more useful version of the same basic system. I will give a slightly different formula from the one in Kelly’s 1956 article. It is easier to remember and can be used in many types of gambling situations. It is what gamblers now call the “Kelly formula.”

The Kelly formula says that you should wager this fraction of your bankroll on a favorable bet:

edge/odds

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.

Odds
means the public or tote-board odds. It measures the profit
if
you win. The odds will be something like 8 to 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 Secretariat are 5 to 1. Odds are a fraction—5 to 1 means 5/1 or 5. The 5 is all you need.

The wire service’s tips convince you that Secretariat actually has a 1-in-3 chance of winning. Then by betting $100 on Secretariat 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 Secretariat.

A couple of observations will help to make sense of this. First:

Edge is zero or negative when you have no private wire
. When you don’t have any “inside information,” you know nothing that anyone else doesn’t. Your edge will be zero (or really, negative with the track take). When edge is zero, the Kelly wager,
edge
/
odds
, is zero. Don’t bet.

Edge equals odds in a fixed horse race
. The most informative thing you can learn from a private wire is that the race has been fixed and that such and such a horse is certain to win. How much you can make on a fixed race depends on the odds. It’s better for the sure-to-win horse to have long odds. At odds of 30 to 1, a $100 wager will get you $3,000 profit. When a horse
has
to win, your edge and the public odds are the same thing (30 in this case). The Kelly formula is 30/30 or 100 percent. You stake everything you’ve got.

You do unless you suspect that people who fix horse races are not always trustworthy. “Equivocation” will reduce your estimated edge and should reduce your wager.

One of Kelly’s equations is as beautifully daring as
E
=
mc
²
. Kelly showed that

G
_max
= R

The
G
is the growth rate of the gambler’s money. It’s a way of stating the compound return rate on the bettor’s “investment.” The subscript
max
means that we’re talking about the maximum possible rate of return.

Kelly equates this optimal return to
R
, the information transmission rate in Shannon’s theory. The maximum rate of return is equal to the flow of “inside information.”

To many of Einstein’s contemporaries,
E
=
mc
²
made no sense. Matter and energy were totally different concepts. Kelly’s equation provokes similar mystification. Money equals information? How do you equate bits and bytes to dollars, yen, and euros?

Well, first of all, currency units don’t matter.
G
_max
describes a rate of return, as in a percentage gain per year, or so many basis points (a basis point is a hundredth of a percentage point of annual return). A 7 percent return is a 7 percent return in any currency.

The
R
is the information rate in bits per time unit. The time units have to be the same on both sides of the equation. When you measure return in percent per year, you need to measure information rate in bits per year, too.

Today, a racetrack tip is likely to come by mobile phone or Internet. These relatively high-bandwidth channels may use thousands or millions of bits just to say “Seabiscuit is a sure thing.” The tipster may fill more bandwidth with small talk.

Obviously, small talk does not add to the gambler’s potential gain. Nor does having a voice channel add anything, when the same information could be conveyed in fewer bits as a text message or something even more concise. Kelly’s equation sets only an upper limit on the profit you can obtain from a given bandwidth. This maximum will occur only when the winning horse is signaled in the fewest bits possible. Think of something more along the lines of the original wire services, with a messenger flashing the winner with a flash or no-flash code.

The most concise way of identifying one winning horse out of eight equally likely contenders is to use a three-bit code. There are eight 3-digit binary numbers (000, 001, 010, 011, 100, 101, 110, 111). Assign a number to each of the horses. Then you need 3 bits to identify the winning horse.

Were this 3-bit tip a sure thing, the bettor could wager his entire bankroll on the named horse. At a take-free track where all eight horses are judged equally likely to win, every dollar bet on the winning horse would return $8. Kelly’s bettor can increase his wealth by a factor of 8 every time he receives 3 bits of information. Notice that 8 = 23. The 3 is an exponent, and it determines how fast the gambler’s wealth compounds. This exponent is equal to the number of bits worth of inside tips received.