St. Petersburg paradox

Several resolutions to the paradox have been proposed, including the impossible amount of money a casino would need to continue the game indefinitely.

The problem was invented by Nicolas Bernoulli,[2] who stated it in a letter to Pierre Raymond de Montmort on September 9, 1713.

[5] A casino offers a game of chance for a single player in which a fair coin is tossed at each stage.

Considering nothing but the expected value of the net change in one's monetary wealth, one should therefore play the game at any price if offered the opportunity.

Yet, Daniel Bernoulli, after describing the game with an initial stake of one ducat, stated, "Although the standard calculation shows that the value of [the player's] expectation is infinitely great, it has ... to be admitted that any fairly reasonable man would sell his chance, with great pleasure, for twenty ducats.

"[5] Robert Martin quotes Ian Hacking as saying, "Few of us would pay even $25 to enter such a game", and he says most commentators would agree.

[6] The apparent paradox is the discrepancy between what people seem willing to pay to enter the game and the infinite expected value.

It is a function of the gambler's total wealth w, and the concept of diminishing marginal utility of money is built into it.

Before Daniel Bernoulli's 1738 publication, mathematician Gabriel Cramer from Geneva had already in 1728 found parts of this idea (also motivated by the St. Petersburg paradox), stating that the mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it.He demonstrated in a letter to Nicolas Bernoulli[7] that a square root function describing the diminishing marginal benefit of gains can resolve the problem.

This solution by Cramer and Bernoulli, however, is not completely satisfying, as the lottery can easily be changed in a way such that the paradox reappears.

For any unbounded utility function, one can find a lottery that allows for a variant of the St. Petersburg paradox, as was first pointed out by Menger.

[4] Since in the St. Petersburg lottery only unlikely events yield the high prizes that lead to an infinite expected value, this could resolve the paradox.

The idea of probability weighting resurfaced much later in the work on prospect theory by Daniel Kahneman and Amos Tversky.

[12] However, the overweighting of small probability events introduced in cumulative prospect theory may restore the St. Petersburg paradox.

[14][15] Alexis Fontaine des Bertins pointed out in 1754 that the resources of any potential backer of the game are finite.

As a result, the expected value of the game, even when played against a casino with the largest bankroll realistically conceivable, is quite modest.

In 1777, Georges-Louis Leclerc, Comte de Buffon calculated that after 29 rounds of play there would not be enough money in the Kingdom of France to cover the bet.

[15] Suppose the total resources (or maximum jackpot) of the casino are W dollars (more generally, W is measured in units of half the game's initial stake).

Various authors, including Jean le Rond d'Alembert and John Maynard Keynes, have rejected maximization of expectation (even of utility) as a proper rule of conduct.

[23][24] Keynes, in particular, insisted that the relative risk[clarification needed] of an alternative could be sufficiently high to reject it even if its expectation were enormous.

[25][26] An early resolution containing the essential mathematical arguments assuming multiplicative dynamics was put forward in 1870 by William Allen Whitworth.

General dynamics beyond the purely multiplicative case can correspond to non-logarithmic utility functions, as was pointed out by Carr and Cherubini in 2020.

[30] Intuitively Feller's answer is "to perform this game with a large number of people and calculate the expected value from the sample extraction".

Paul Samuelson resolves the paradox[31] by arguing that, even if an entity had infinite resources, the game would never be offered.

As Samuelson summarized the argument, "Paul will never be willing to give as much as Peter will demand for such a contract; and hence the indicated activity will take place at the equilibrium level of zero intensity."

One approach that is attracting much interest in solving the St Petersburg paradox is to use a parameter related to the cognitive aspect of a strategy.

From this point of view, the St. Petersburg paradox teaches us that an expected gain that tends to infinity does not always imply the presence of a cognitive and non-random strategy.

Consequently, from the decision-making point of view, we can create a hierarchy of values, in which knowledge turns out to be more important than expected gain.