The Saint Petersburg paradox

Description from the Stanford Encyclopedia of Philosophy. Credits Daniel Bernoulli with formulating the problem in 1738. Actually, his cousin, Nicholas Bernoulli, proposed it in 1713 in a letter to Pierre Rémond de Montmort who put it in his book Essay d'analyse sur les jeux de hazard that same year (see Montmort and the Grand Jan). See also Bernoulli and the St. Petersburg paradox. While in Saint Petersburg, Daniel wrote his 1738 paper about the problem. Hence the name. Daniel's analysis centered on the idea of diminishing marginal utility, which seems to have been used for the first time in this paper.

Correspondence concerning the Petersburg Game has excerpts from the letters between Nicolas Bernoulli, Montmort, Cramer, and Daniel Bernoulli about the game.

Pascal's wager is related to the Saint Petersburg paradox. How?

The following passage is from Daniel Bernoulli's paper, "Specimen theoriae novae de mensura sortis", which appeared in 1738 in Commentarii Academiae Scientiarum Imperialis Petropolitanae [Papers of the Imperial Academy of Sciences in Petersburg] 5 175-192. The translation, "Exposition of a new theory of the measurement of risk" appeared in Econometrica 22 (1954) 123-136.



My most honorable cousin the celebrated Nicolaus Bernoulli, Professor utriusque iuris at the University of Basle, once submitted five problems to the highly distinguished mathematician Montmort. These problems are reproduced in the work L'analyse sur les jeux de hazard de M. de Montmort, p. 402. The last of these problems runs as follows: Peter tosses a coin and continues to do so until it should land "heads" when it comes to the ground. He agrees to give Paul one ducat if he gets "heads" on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled. Suppose we seek to determine the value of Paul's expectation. My aforementioned cousin discussed this problem in a letter to me asking for my opinion. Although the standard calculation shows that the value of Paul's expectation is infinitely great, it has, he said, 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.
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After having read this paper to the Society I sent a copy to the aforementioned Mr. Nicolas Bernoulli, to obtain his opinion of my proposed solution to the difficulty he had indicated. In a letter to me written in 1732 he declared that he was in no way dissatisfied with my proposition on the evaluation of risky propositions when applied to the case of a man who is to evaluate his own prosepcts. However, he thinks that the case is different if a third person, somewhat in the position of a judge, is to evaluate the prospects of any participant in a game in accord with equity and justice. I myself have discussed this problem in Section 2. Then this distinguished scholar informed me that the celebrated mathematician, Cramer, had developed a theory on the same subject several years before I produced my paper. Indeed I have found his theory so similar to mine that it seems miraculous that we independently reached such close agreement on this sort of subject. Therefore it seems worth quoting the words with which the celebrated Cramer himself first described his theory in his letter of 1728 to my cousin. His words are as follows:

"Perhaps I am mistaken, but I believe that I have solved the extraordinary problem which you submitted to M. de Montmort, in your letter of September 9, 1713, (problem 5, page 402). For the sake of simplicity I shall assume that A tosses a coin into the air and B commits himself to give A one ducat ... "

For the beginning of this paper, see Exposition of a new theory.

The Bernoulli family tree. See also the link "Bernoulli family tree" near the end of the page Johann Bernoulli, which has no pictures but is a little more complete and easier to understand.