Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street by William Poundstone

Author:William Poundstone [Poundstone, William]
Language: eng
Format: epub, mobi
ISBN: 9780809045990
Google: 9YDlaUfmhkgC
Publisher: Farrar, Straus and Giroux
Published: 2006-09-19T07:00:00+00:00

Although the standard calculation shows that the value of Paul’s expectation is infinitely great, it has…to be admitted that any fairly reasonable man would sell his chance, with great pleasure, for twenty ducats. The accepted method of calculation does, indeed, value Paul’s prospects at infinity although no one would be willing to purchase it at a moderately high price.

Daniel published these words in Latin. The wager has come to be known as the “St. Petersburg wager” or “St. Petersburg paradox.” It has provoked sporadic interest ever since. A mention in John Maynard Keynes’s 1921 Treatise on Probability made it part of the mental furniture of nearly every twentieth-century economist. Bernoulli’s wager makes an appearance in von Neumann and Morgenstern’s Theory of Games and Economic Behavior and in papers by Kenneth Arrow, Milton Friedman, and Paul Samuelson.

The paradox can be resolved easily by noting that Peter would have to possess infinite wealth to make good on the game’s potential payouts. No one has infinite wealth. Therefore most of the terms of the infinite series are irrelevant. A minuscule chance of winning a quadrillion dollars is not worth what you might compute. It’s worth practically nothing because no one has a quadrillion dollars to award.

Suppose a casino offered this wager with winnings capped at a billion dollars. How much would the wager be worth then? A lot less! Assume prizes start with a dollar. Normally, the prize for heads on the 31st toss would be $1,073,741,824. The most reasonable course for the casino would be to halt the game at 30 tosses and award the billion dollars to anyone who has gotten 30 tails. The expected value of this truncated game is a measly $15.93.

That’s a lot more reasonable. The wager is not worth infinity, just a few dollars. This explanation of the puzzle is as good as any hardheaded realist could ask for. Yet philosophers, mathematicians—even economists—have rarely accepted this solution. Most take the position that we can pretend that Peter possesses infinite wealth. Isn’t it still ridiculous to say that Paul should be willing to pay any amount to play the game?

Daniel Bernoulli thought so. He proposed a different solution that was highly influential for future economic thought. Bernoulli drew a distinction between money and the value people place on money. To a billionaire, $1,000 is pocket change. To a starving beggar, $1,000 may be a fortune. The value of a financial gain (or loss) depends on the wealth of the person it affects.

You’re probably saying to yourself that you already knew that. Well okay, Bernoulli’s real contribution was to coin a word. The word has been translated into English as “utility.” It describes this subjective value people place on money. Bernoulli claimed that people instinctively act to achieve the greatest possible utility—not necessarily the greatest number of dollars or ducats. “The value of an item must not be based on its price,” Bernoulli wrote, “but rather on the utility it yields. The price of the item is dependent only

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