St. Petersburg paradox

The classical resolution of the paradox involved the explicit introduction of a utility function, an expected utility hypothesis, and the presumption of diminishing marginal utility of money.

Daniel Bernoulli;

The determination of the value of an item must not be based on the price, but rather on the utility it yields ... There is no doubt that a gain of one thousand ducats is more significant to the pauper than to a rich man though both gain the same amount.

A common utility model, suggested by Daniel Bernoulli, is the logarithmic function U(w) = ln(w) (known as log utility). It is a function of the gambler's total wealth w, and the concept of diminishing marginal utility of money is built into it. The expected utility hypothesis posits that a utility function exists that provides a good criterion for real people's behavior; i.e. a function that returns a positive or negative value indicating if the wager is a good gamble. For each possible event, the change in utility ln(wealth after the event) − ln(wealth before the event) will be weighted by the probability of that event occurring. Let c be the cost charged to enter the game. The expected incremental utility of the lottery now converges to a finite value:

${\displaystyle \Delta E(U)=\sum _{k=1}^{+\infty }{\frac {1}{2^{k}}}\left[\ln \left(w+2^{k}-c\right)-\ln(w)\right]<+\infty \,.}$

This formula gives an implicit relationship between the gambler's wealth and how much he should be willing to pay (specifically, any c that gives a positive change in expected utility). For example, with natural log utility, a millionaire ($1,000,000) should be willing to pay up to $20.88, a person with $1,000 should pay up to $10.95, a person with $2 should borrow $1.35 and pay up to $3.35.

Before Daniel Bernoulli published, in 1728, a mathematician from Geneva, Gabriel Cramer, had already found parts of this idea (also motivated by the St. Petersburg paradox) in 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[9] that a square root function describing the diminishing marginal benefit of gains can resolve the problem. However, unlike Daniel Bernoulli, he did not consider the total wealth of a person, but only the gain by the lottery.

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. To this aim, we just need to change the game so that it gives even more rapidly increasing payoffs. 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.[10]

Recently, expected utility theory has been extended to arrive at more behavioral decision models. In some of these new theories, as in cumulative prospect theory, the St. Petersburg paradox again appears in certain cases, even when the utility function is concave, but not if it is bounded.[11]

Nicolas Bernoulli himself proposed an alternative idea for solving the paradox. He conjectured that people will neglect unlikely events.[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. Paul Weirich similarly wrote that risk aversion could solve the paradox. Weirich went on to write that increasing the prize actually decreases the chance of someone paying to play the game, stating "there is some number of birds in hand worth more than any number of birds in the bush".[12][13] However, this has been rejected by some theorists because, as they point out, some people enjoy the risk of gambling and because it is illogical to assume that increasing the prize will lead to more risks.

Cumulative prospect theory is one popular generalization of expected utility theory that can predict many behavioral regularities.[14] However, the overweighting of small probability events introduced in cumulative prospect theory may restore the St. Petersburg paradox. Cumulative prospect theory avoids the St. Petersburg paradox only when the power coefficient of the utility function is lower than the power coefficient of the probability weighting function.[15] Intuitively, the utility function must not simply be concave, but it must be concave relative to the probability weighting function to avoid the St. Petersburg paradox. One can argue that the formulas for the prospect theory are obtained in the region of less than $400.[14] This is not applicable for infinitely increasing sums in the St. Petersburg paradox.

The classical St. Petersburg game assumes that the casino or banker has infinite resources. This assumption has long been challenged as unrealistic.[6][7] Alexis Fontaine des Bertins pointed out in 1754 that the resources of any potential backer of the game are finite.[16] More importantly, the expected value of the game only grows logarithmically with the resources of the casino. 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.[17]

If the casino has finite resources, the game must end once those resources are exhausted.[7] 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). Then the maximum number of times the casino can play before it no longer can fully cover the next bet is L = ⌊log₂(W)⌋.[18][nb 1] Assuming the game ends when the casino can no longer cover the bet, the expected value E of the lottery then becomes:[18]

${\displaystyle {\begin{aligned}E&=\sum _{k=1}^{L}{\frac {1}{2^{k}}}\cdot 2^{k}=L\,.\end{aligned}}}$

The following table shows the expected value E of the game with various potential bankers and their bankroll W:

Note: Under game rules which specify that if the player wins more than the casino's bankroll they will be paid all the casino has, the additional expected value is less than it would be if the casino had enough funds to cover one more round, i.e. less than $1. For the player to win W he must be allowed to play round L+1. So the additional expected value is W/2^L+1.

The premise of infinite resources produces a variety of apparent paradoxes in economics. In the martingale betting system, a gambler betting on a tossed coin doubles his bet after every loss so that an eventual win would cover all losses; this system fails with any finite bankroll. The gambler's ruin concept shows that a persistent gambler who raises his bet to a fixed fraction of his bankroll when he wins, but does not reduce his bet when he loses, will eventually and inevitably go broke—even if the game has a positive expected value.

Buffon[17] argued that a theory of rational behavior must correspond to what a rational decision-maker would do in real life, and since reasonable people regularly ignore events that are unlikely enough, a rational decision-maker should also ignore such rare events.

As an estimate of the threshold of ignorability, he argued that, since a 56-year-old man ignores the possibility of dying in the next 24 hours, which has a probability of 1/10189 according to the mortality tables, events with less than 1/10,000 probability could be ignored. Assuming that, the St Petersburg game has an expected payoff of only ${\displaystyle \sum _{k=1}^{13}2^{k}{\frac {1}{2^{k}}}=13}$.

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 of an alternative could be sufficiently high to reject it even if its expectation were enormous.[24] Recently, some researchers have suggested to replace the expected value by the median as the fair value.[25][26]

An early resolution containing the essential mathematical arguments assuming multiplicative dynamics was put forward in 1870 by William Allen Whitworth.[27] An explicit link to the ergodicity problem was made by Peters in 2011.[28] These solutions are mathematically similar to using the Kelly criterion or logarithmic utility. General dynamics beyond the purely multiplicative case can correspond to non-logarithmic utility functions, as was pointed out by Carr and Cherubini in 2020.[29]

A solution involving sampling was offered by William Feller.[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". In this method, when the games of infinite number of times are possible, the expected value will be infinity, and in the case of finite, the expected value will be a much smaller value.

Paul Samuelson resolves the paradox[31] by arguing that, even if an entity had infinite resources, the game would never be offered. If the lottery represents an infinite expected gain to the player, then it also represents an infinite expected loss to the host. No one could be observed paying to play the game because it 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."

Many variants of the St Petersburg game are proposed to counter proposed solutions to the game.[13]

For example, the "Pasadena game":[32] let ${\displaystyle n}$ be the number of coin-flips; if ${\displaystyle n}$ is odd, the player gains units of ${\displaystyle {\frac {2^{n}}{n}}}$; else the player loses ${\displaystyle {\frac {2^{n}}{n}}}$ units of utility. The expected utility from the game is then ${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n}}=\ln 2}$. However, since the sum is not absolutely convergent, it may be rearranged to sum to any number, including positive or negative infinity. This suggests that the expected utility of the Pasadena game depends on the summation order, but standard decision theory does not provide a principled way to choose a summation order.