The Saint Petersburg paradox
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.
concerning the Petersburg Game has excerpts from the letters
between Nicolas Bernoulli, Montmort, Cramer, and Daniel Bernoulli
about the game.
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
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.
For the beginning of this paper, see
Exposition of a new theory.
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 ... "
Bernoulli family tree. See also the link "Bernoulli family tree" near the end of the page
Bernoulli, which has no pictures but is a little more complete and easier to understand.