Shannon understood that the Differential Analyzer was two machines in one. It was a mechanical computer regulated by an electrical computer. Thinking about the machine convinced Shannon that electrical circuits could compute more efficiently than mechanical linkages. Shannon envisioned an ideal computer in which numbers would be represented by states of electrical circuits. There would be nothing to lubricate and a lot less to break.

He had slot machines hidden in back rooms. These were popular with children, and Zwillman’s mob made sure the youngsters had little stools to help them reach the handles.

In real conversations, we are always trying to outguess each other. We have a continuously updated sense of where the conversation is going, of what is likely to be said next, and what would be a complete non sequitur. The closer two people are (personally and culturally), the easier this game of anticipation is. A long-married couple can finish each other’s sentences. Teen best friends can be in hysterics over a three-character text message. It would be unwise to rely on verbal shorthand when speaking to a complete stranger or someone who doesn’t share your cultural reference points. Nor would the laconic approach work, even with a spouse, when communicating a message that can’t be anticipated. Assuming you wanted your spouse to bring home Shamu, you wouldn’t just say, “Pick up Shamu!” You would need a good explanation. The more improbable the message, the less “compressible” it is, and the more bandwidth it requires. This is Shannon’s point: the essence of a message is its improbability.

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.

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. (Our radio stations still do this.)

Given a favorable betting opportunity, the Kelly system promises maximum profit and protection against ruin.

The Kelly criterion is meaningful only when gambling profits are reinvested.

In the late 1950s, Shannon began an intensive study of the stock market that was motivated both by intellectual curiosity and desire for gain. He filled three library shelves with something like a hundred books on economics and investing. The titles included Adam Smith’s The Wealth of Nations, John von Neumann and Oskar Morgenstern’s Theory of Games and Economic Behavior, and Paul Samuelson’s Economics, as well as books with a more practical focus on investment. One book Shannon singled out as a favorite was Fred Schwed’s wry classic, Where Are the Customers’ Yachts?

Shannon was not the first great scientific mind to suppose that his talents extended to the stock market. Carl Friedrich Gauss, often rated the greatest mathematician of all time, played the market. On a salary of 1,000 thalers a year, Euler left an estate of 170,587 thalers in cash and securities. Nothing is known of Gauss’s investment methods. On the other hand, Isaac Newton lost some 20,000 pounds investing in the South Sea Trading Company. Newton’s loss would be something like $ 3.6 million in today’s terms. Said Newton: “I can calculate the motions of heavenly bodies, but not the madness of people.”

Sharpe offered a counterargument. Divide the world into “active” investors and “passive” investors, Sharpe said. A passive investor is defined as anyone sensible enough to realize you can’t beat the market. The passive investor puts all his money into a market portfolio of every stock in existence (roughly, an “index fund”). An active investor is anyone who suffers from the delusion that he can beat the market. The active investor puts his money into anything except a market portfolio. By Sharpe’s terminology, an active investor need not trade “actively.” A retired teacher who has two shares of AT& T in the bottom of her dresser drawer counts as an active investor. She is operating on the assumption that AT& T is a better stock to own than a total market index fund. Active investors include anyone who tries to pick “good” stocks and shun “bad” ones, or who hires someone else to do that by putting money in an actively managed mutual fund or investment partnership. Who does better, Sharpe asked: the active investors or the passive investors? Collectively, the world’s investors own 100 percent of all the world’s stock. (Nothing is owned by extraterrestrials!) That means that the average return of all the world’s investors— before you factor in management expenses, brokerage fees, and taxes— has to be identical to the average return of the stock market as a whole. It can’t be otherwise. Even more clearly, the average return of just the passive investors is equal to the average stock market return. That’s because these investors keep their money in index funds or portfolios that match the return of the whole market. Subtract the return of the passive investors from the total. This leaves the return of the active investors. Since the passive investors have exactly the same return as the whole, it follows that the active investors, as a group, must also have the same average return as the whole market. This leads to a surprising conclusion. Collectively, active investors must do no better or worse (before fees and taxes) than the passive investors. Some active investors do better than others, as we all know. Every active investor hopes to do better than the others. One thing’s for sure. Everyone can’t do “better than average.” Active investing is therefore a zero-sum game. The only way for one active investor to do better than average is for another active investor to do worse than average. You can’t squirm out of this conclusion by imagining that the active investors’ profits come at the expense of those wimpy passive investors who settle for average return. The average return of the passive investors is exactly the same as that of the active investors, for the reason just outlined. Now factor in expenses. The passive investors have little or no brokerage fees, management fees, or capital gains taxes (they rarely have to sell). The expenses of the active traders vary. We’re using that term for everyone from day traders and hedge fund partners to people who buy and hold a few shares of stock. For the most part, active investors will be paying a percent or two in fees and more in commissions and taxes. (Hedge fund investors pay much more in fees when the fund does well.) This is something like 2 percent of capital, per year, and must be deducted from the return. Two percent is no trifle. In the twentieth century, the average stock market return was something like 5 percent more than the risk-free rate. Yet an active investor has to earn about two percentage points more than average just to keep up with the passive investors. Do some active investors do that? Absolutely. They’re the smart or lucky few who fall at the upper end of the spectrum of returns. The majority of active investors do not achieve that break-even point. Most people who think they can beat the market do worse than the market. This is an irrefutable conclusion, Sharpe said, and it is not based on fancy economic theorizing. It follows from the laws of arithmetic.

Bernoulli’s thesis was that risky ventures should be evaluated by the geometric mean of outcomes. (...) The geometric mean is almost always less than the arithmetic mean. (The exception is when all the averaged values are identical. Then the two kinds of mean are the same.) This means that the geometric mean is a more conservative way of valuing risky propositions. Bernoulli believed that this conservatism better reflects people’s distaste for risk.

As Mark Twain wrote, “A banker is a fellow who lends you his umbrella when the sun is shining but wants it back the moment it rains.”

Warren Buffett marveled at how “ten or 15 guys with an average IQ of maybe 170” could get themselves “into a position where they can lose all their money.” That was much the sentiment of Daniel Bernoulli, way back in 1738, when he wrote: “A man who risks his entire fortune acts like a simpleton, however great may be the possible gain.”

Thorp found that LTCM had based some of its models on a mere four years of data. In that short period, the spread between junk bonds and treasuries hovered in the range of 3 to 4 percentage points. The fund essentially bet that the spread would not greatly exceed this range. But as recently as 1990, the spread had topped 9 percent. “People think that if things are bounded in a certain historical range, there’s necessity or causality here,” Thorp explained. Of course, there’s not. In 1998, when the spread widened suddenly to 6 percent, “they said this was a one-in-a-million-year event. A year or two later, it got wider, and two years after that, it got wider yet.”

A decade rarely passes without a market event that some respected economist claims, with a straight face, to be a perfect storm, a ten-sigma event, or a catastrophe so fantastically improbable that it should not have been expected to occur in the entire history of the universe from the big bang onward. In a world where financial models can be so incredibly wrong, the extreme downside caution of Kelly betting is hardly out of place. For reasons mathematical, psychological, and sociological, it is a good idea to use a money management system that is relatively forgiving of estimation errors.

The core of John Kelly’s philosophy of risk can be stated without math. It is that even unlikely events must come to pass eventually. Therefore, anyone who accepts small risks of losing everything will lose everything, sooner or later. The ultimate compound return rate is acutely sensitive to fat tails.

Warren Buffett himself said that Singleton had the best operating and capital deployment record in American business. It is at least conceivable that Shannon’s judgments played a supporting role in that success.

In a 1984 speech, Buffett asked his listeners to imagine that all 215 million Americans pair off and bet a dollar on the outcome of a coin toss. The one who calls the toss incorrectly is eliminated and pays his dollar to the one who was correct. The next day, the winners pair off and play the same game with each other, each now betting $ 2. Losers are eliminated and that day’s winners end up with $ 4. The game continues with a new toss at doubled stakes each day. After twenty tosses, 215 people will be left in the game. Each will have over a million dollars. According to Buffett, some of these people will write books on their methods: How I Turned a Dollar into a Million in Twenty Days Working Thirty Seconds a Morning. Some will badger ivory-tower economists who say it can’t be done: “If it can’t be done, why are there 215 of us?” “Then some business school professor will probably be rude enough to bring up the fact that if 215 million orangutans had engaged in a similar exercise, the result would be the same— 215 egotistical orangutans with 20 straight winning flips.”

To Ziemba, the races are an instructive model of the securities markets. It is the same fallible humans who set prices for technology stocks and show bets. Both sets of speculators are motivated by desire for gain. This does not guarantee perfect market efficiency.