In 1713 Nikolaus Bernoulli (1687–1759), the famous Swiss mathematician, posed a question about the following game:

• Toss a coin.

• If it shows heads, you get $2 and the game is over. Otherwise you toss again.

• If the coin now shows heads, you get $4, and so on. Whenever you toss tails the prize is doubled.

• After n tosses the player gets $2ⁿ if heads appears for the first time.

After 30 tosses this is a sum of more than $1 billion—a gigantic prize. Now the question: How much would a gambler pay for the right to play this game? Most people would offer between $5 and $20, but is that reasonable? On the one hand, the chance of winning more than $4 is just 25 percent. On the other hand, the prize could be enormous because the probability of tossing a very long series of tails before that first toss of heads, although very small, is by no means zero. The huge prize that could be won in this case compensates for the very small probability of success. Nikolaus Bernoulli found that the expected prize is infinite! (The expected prize is calculated by multiplying all the possible prizes by the probability that they are obtained and adding the resulting numbers.)

And therein lies the paradox: If the expected prize is infinite, why is nobody willing to pay $100,000, $10,000, or even $1,000 to play the game?

The explanation of this mysterious behavior touches the areas of statistics, psychology, and economics. Two other Swiss mathematicians, Gabriel Cramer (1704–1742)

and Nikolaus’s cousin Daniel Bernoulli (1700–1782), suggested a solution. They postulated that $1 does not always carry the same “utility” for its owner. One dollar brings more utility to a beggar than to a millionaire. Whereas to the former, owning $1 can mean the difference between going to bed hungry at night or not, the latter would hardly notice the increase of his fortune by $1. Similarly, the second billion that one would win if the 31st toss showed tails wouldn’t carry the same utility as the first billion that one would receive already after 30 tosses. The utility of $2 billion just isn’t twice the utility of $1 billion.

Herein lies the explanation of the mystery. The crucial factor is the expected utility of the game (the utilities of the prizes multiplied by their probabilities), which is far less than the expected prize. Daniel Bernoulli’s treatise was published in the Commentaries of the Imperial Academy of Science of St. Petersburg, and this surprising insight was henceforth called the St. Petersburg paradox.

Around 1940 the idea of the utility function was taken up by two immigrants from Europe working at the Institute for Advanced Study in Princeton, New Jersey. John von Neumann (1903–1957), one of the outstanding mathematicians of the 20th century, was Jewish and had been forced to flee his native Hungary when the Nazis invaded the country. The economist Oskar Morgenstern (1902–1976) had left Austria because he loathed the National Socialists.

In Princeton the two immigrants worked together on what they thought would be a short paper on the theory of games. But the treatise kept growing. When it finally appeared in 1944 under the title Theory of Games and Economic Behavior, it had attained a length of 600 pages. This pioneering work was to have a profound influence on the further development of economics. In the book Bernoulli and Cramer’s utility function served as an axiom to describe the behavior of the proverbial “economic man.” However, it was soon noticed that in situations with very low probabilities and very high amounts of money, test candidates often made decisions that contravened the pos-

tulated axiom. The economists remained unfazed. They insisted that the theory was correct and that many people were simply acting irrationally.

Despite its shortcomings, utility theory has had a far-reaching consequence. The explanation that Bernoulli and Cramer offered for the St. Petersburg paradox formed the theoretical basis of the insurance business. The existence of a utility function means that most people prefer having $98 in cash to gambling in a lottery where they could win $70 or $130, each with a chance of 50 percent—even though the lottery has the higher expected prize of $100. The difference of $2 is the premium most of us would be willing to pay for insurance against the uncertainty. That many people buy insurance to avoid risk yet at the same time spend money on lottery tickets in order to take risks is another paradox, one that is still awaiting an explanation.