The St. Petersburg Paradox

Origin

Nicolaus Bernoulli, nephew of Jacob Bernoulli (1655-1705), wrote the St. Petersburg Paradox in 1713. After it was published, a Swiss mathematician named Gabrial Cramer sent him a solution, which Bernoulli did not think was adequate, so he sent it to his cousin, Daniel Bernoullli (1700-1782), a member of the Imperial Academy at St. Petersburg. Daniel published the problem in a memoir, making the problem much more popular and giving the problem its name.

Problem and Solutions

Nicolaus first posed a simple version of the problem and sent it to R. de P. Montmort who published it in Essay d'analyse sur le jeux de hazard (1713) as cited by Dutka (1988):

$A$ promises to give a crown to $B$ if with an ordinary die he gets six points on the first throw, two crowns if he gets the six on the second throw, three crowns if he gets this point on the third throw, four crowns, if he gets it on the fourth, and so on; $B$'s expectation is required (p. 18).

In other words, Nicolaus was asking how much would be reasonable for $B$ to pay $A$ to play the game.

Using knowledge of calculating expected value, one can find an appropriate solution: $$\frac{1}{6} \sum_{m=1}^{\infty} m\left ( \frac{5}{6}\right )^{m-1}=\frac{1}{6} \left(1-\frac{5}{6}\right)^{-2}=6$$

The next question Nicolaus posed is the one known as the St. Petersburg Paradox:

The same is required if $A$ promises $B$ to give the crowns in the progression $1, 2, 4, 8, 16, etc.$ [Premise 1] or $1, 3, 9, 27, etc.$ [Premise 2] or $1, 4, 9, 16, 25, etc.$ [Premise 3], or $1, 8, 27, 64, etc.$ [Premise 4] instead of $1, 2, 3, 4, 5, etc.$ as before. Although these problems are not difficult for the most part, yet you will find them very curious (Dutka, 1988, p. 19).

The third and fourth premises in this question can be solved using expected values and modern knowledge of convergent series (see Dutka, 1988, p. 19). The first premise of the second set of questions is the one discussed in the rest of this section as the St. Petersburg Paradox. An attempt to solve this the same way as other expectation problems is fruitless because it would be $$\frac{1}{6} \sum_{m=1}^{\infty} 2^{m-1}\left ( \frac{5}{6}\right )^{m-1}$$ which is an infinite series that does not converge. The paradox is that since over the long run, there is no limit to how much one would expect to win, it seems that any finite amount would be reasonable to pay to play this game, since it would be less than the infinite winnings expected from the game. However, Nicolaus argued that most people would not pay more than twenty crowns to play the game.

Cramer rephrased the question where the game was tossing a fair coin until a head appeared on the $m$th toss, at which time, the player was awarded $2^{m-1}$ crowns. Cramer's solution was based on the idea that at some point, the sums owed would not be able to be paid. For example, he limited the winnings to no more than 20 million crowns, saying that even if the amount due was more than that, there was no way to receive that much, so the amount won after the $24$th round was limited to $2^{24}=16,777,216$ crowns. Then the expected value was a sum of a finite and a converging infinite series:

$$\sum_{k=1}^{24} \frac{2^{k-1}}{2k}+\sum_{k=25}^{\infty} \frac{2^{24}}{2^k}=12+1=13$$

Thus Cramer suggested that the amount that should be paid to play the game was 13 crowns. Another of his approaches to the problem was to consider the moral value of the winnings where he explained, It is true that 100 millions yields more pleasure than 10 millions, but not ten times as much (Dutka, 1988, p. 22). He then proposed a square root relationship, which converges when part of an infinite series (For more detail see Dutka, 1988, p. 23).

Daniel Bernoulli argued that the expectation would differ for each pair of people playing the game. It would matter how much they were willing to risk based on how much wealth they already had. He then calculated the expectation after setting up a differential equation based on the financial situation of each participant (see Dutka, 1988, p. 27).

Nicolaus Bernoulli considered a different approach, arguing that no reasonable person would pay twenty crowns to play because there was such a small probability of winning anything more than twenty crowns that such a probability was negligible. This perspective is interesting because many people often spend similar amounts of money to participate in lotteries with even lower probabilities of winning.

Since Bernoulli's time, computer simulations have provided insight into possible outcomes of the random experiment and their frequencies. Graphs showing the results of simulations based on the infinite expectation of the original compared to both Daniel Bernoulli's and Cramer's solutions can be seen in Ceasar (1984).

Ultimately, there is still not a definite solution to the problem because the expected value is not finite. However, multiple practical solutions exist, based on reasonable assumptions for human decision making.