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1. Gambling For an Infinite Amount of Money

The question:

Peter tosses a coin. If it lands heads, Paul gets two dollars. If it lands tails Peter tosses again and if it lands heads on the second toss, Paul gets four dollars. If it lands tails again, and falls on heads only on the third throw, Paul gets eight dollars. And so on…. If the first time it lands heads is on the n-th toss, Paul gets 2^n dollars.

Should Paul be willing to pay ten dollars to participate in the lottery? A hundred? A thousand?

Yes, yes and yes!

Paul should be willing to pay even ten thousand dollars and more to participate.

In the middle of the seventeenth century, the philosopher Blaise Pascal in Paris, and the judge Pierre de Fermat in Toulouse, in southern France, concluded that the expected value of an uncertain event is computed by multiplying the potential values with the probabilities of their occurrence. If a gamble has winning odds of 10 percent and the prize is 80\$, the expected payout is 8\$ (0.10 x 80). If there are several prize possibilities, the expected payout is the sum of all possible outcomes, multiplied by their respective probabilities. So, if, in addition to the 10 percent chance of winning 80\$, there is also a second prize – a twenty percent chance of a 30\$ payout – the expected winnings would be \$14 (= 0.10 x 80 + 0.20 x 30). A gambler should be willing to pay any amount up to \$14 to participate in the gamble.

Hence, the expected win of the toin cossing game must be calculated as follows: the chances of the coin landing heads on the first throw, and Paul receiving two ducats, is ½. The probability of the coin landing heads only on the second throw, and Paul receiving four ducats, is ¼, the probability that heads will appear only on the third throw, and Paul receiving eight ducats, is 1/8, and so on. If, say, the coin lands on tails nine times in a row, and heads appears on the tenth throw, the payout would be 1024 ducats. The probability of such a series of throws is 1/1024.

Now, since the expected payout is the sum of the individual payouts (2, 4, 8, …, 1024, …) multiplied by the probabilities (½, ¼, 1/8, …, 1/1024, …) the expected win is

(2 x ½) + (4 x ¼) + (8 x 1/8) + … + (1024 x 1/1024) +… = 1 + 1 + 1 + …+ 1 + …

Wow! Since there is a real, if minute chance that many, many tails are thrown before the first head appears, the series never ends. To compute the expected value, infinitely many 1’s must be summed. The expected win amounts, shockingly, to infinity! Accordingly, a gambler should be willing to pay any amount to participate in this lottery.

History:

The mathematician Nikolaus Bernoulli, scion of the renowned family of scientists in Basle, Switzerland, described this problem in 1713. In the Spring of 1728, the 24-year old Gabriel Cramer sent a letter to Bernoulli. There must be a difference, he wrote, between the mathematical calculations and what he calls the vulgar estimate. “Mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they make of it.”

It was a far-ranging insight. In modern parlance, ‘the ‘vulgar estimate’ is called the ducat’s ‘utility’. Since one ducat allows a beggar to obtain some food, whereas a rich heir would hardly notice its addition to his fortune, the utility of one ducat is much greater to a pauper than it is to a millionaire. Hence, the utility of money diminishes with increasing wealth. Granted, the additional ducat increases the millionaire’s fortune from 1,000,000 to 1,000,001, but the difference is hardly noticable. Hence, the utility of each additional ducat should be less than the utility of the previous one (See Figure 1.)

Cramer and Nikolaus Bernoulli did not see eye to eye on the problem. The question of who was right was of deep theoretical interest and in 1730 Nikolaus called upon his nephew, Daniel Bernoulli, himself a mathematician of note, to be the judge. To Nikolaus’s chagrin, Daniel sided with Cramer.

Since he taught in St. Petersburg in Russia, Daniel submitted his reasoning to the Proceedings of the city’s Imperial Academy of Sciences and it was published in 1738, in Latin, under the title Exposition of a new theory on the measurement of risk.

Dénouement:

Paul’s expected gain really is infinite. Then why are most people unwilling to pay more than just a few ducats to participate in this lottery?

To illustrate his point, Cramer suggested the square root of wealth as an indicator of utility. This utility keeps growing, but it does so at a diminishing rate. Since the square root of 1 is 1 and the square root of 2 is 1.41…, the second Ducat has a utility of about 41 percent of the first Ducat. And the ten million and first Ducat would be valued at about 0.016 percent of the very first Ducat’s utility. With the new scheme, Cramer re-computed the lottery’s expected win as follows:

½ x √2 + ¼ x √4 + 1/8­ x √8 + … = 2.41…

This is what Paul should be willing to pay for the lottery ticket!

Technical supplement:

Bernoulli maintained that Paul simply disregards minute probabilities and sets all probabilities beyond, say, (½)20 to zero. He then calculates the expected win as

½ x 2 + ¼ x 4 + 1/8­ x 8 + … + (½)20 x 220 + 0 + 0 + 0 + …. = 20

Paul would be justified in disregarding the later terms of the series, not because the probabilities are minute, as Bernoulli claimed, but because the winnings to be paid out become too large. Peter may not be rich enough to pay over a million Ducats if the first Heads appears only later than 20 throws. Thus, it is no wonder that Paul values any winnings beyond 220 ducats as worthless.

Both Cramer’s 2.41 ducats and Bernoulli’s 20 are reasonable amounts, and nobody of good sense would be willing to pay a thousand, or ten thousand ducats.

In the dispute that followed, Daniel sided with Cramer. He suggested that utility of wealth follows the logarithmic function. It is just as reasonable a suggestion as Cramer’s, since that function always rises but an additional ducat always provides less utility than the previous one.

***

What Daniel Bernoulli and Gabriel Cramer proposed was nothing less than the basic principle on which all economic behavior is built: the fact that an additional ducat gives less utility to a rich man than to a pauper entails that people in general are averse to risk. Risk aversion is the reason for the existence of the multi-trillion-dollar insurance industry. (See the chapter on Insurance and Gambling///) It also implies that investors expect higher returns for riskier assets, that returns on bonds are lower than on stock, that borrowing rates are higher when a jobless person takes out a loan than when a well-established professional does, to name just a few examples.

Figure 1:

Utility curve