The St. Petersburg paradox is typically framed in terms of gambles on the outcome of fair coin tosses.

The St. Petersburg paradox or St. Petersburg lottery[1] is a paradox related to probability and decision theory in economics. It is based on a theoretical lottery game that leads to a random variable with infinite expected value (i.e., infinite expected payoff) but nevertheless seems to be worth only a very small amount to the participants. The St. Petersburg paradox is a situation where a naive decision criterion which takes only the expected value into account predicts a course of action that presumably no actual person would be willing to take. Several resolutions to the paradox have been proposed.

The paradox takes its name from its analysis by Daniel Bernoulli, one-time resident of the eponymous Russian city, who published his arguments in the Commentaries of the Imperial Academy of Science of Saint Petersburg (Bernoulli 1738). However, the problem was invented by Daniel's cousin, Nicolas Bernoulli,[2] who first stated it in a letter to Pierre Raymond de Montmort on September 9, 1713 (de Montmort 1713).[3]

The St. Petersburg game

A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The initial stake begins at 2 dollars and is doubled every time heads appears. The first time tails appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if tails appears on the first toss, 4 dollars if heads appears on the first toss and tails on the second, 8 dollars if heads appears on the first two tosses and tails on the third, and so on. Mathematically, the player wins ${\displaystyle 2^{k+1}}$ dollars, where ${\displaystyle k}$ is the number of consecutive head tosses. What would be a fair price to pay the casino for entering the game?

To answer this, one needs to consider what would be the expected payout at each stage: with probability 1/2, the player wins 2 dollars; with probability 1/4 the player wins 4 dollars; with probability 1/8 the player wins 8 dollars, and so on. Assuming the game can continue as long as the coin toss results in heads and, in particular, that the casino has unlimited resources, the expected value is thus

{\displaystyle {\begin{aligned}E&={\frac {1}{2}}\cdot 2+{\frac {1}{4}}\cdot 4+{\frac {1}{8}}\cdot 8+{\frac {1}{16}}\cdot 16+\cdots \\&=1+1+1+1+\cdots \\&=\infty \,.\end{aligned}}}

This sum grows without bound, and so the expected win is an infinite amount of money.[4]

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 says most commentators would agree.[6] The paradox is the discrepancy between what people seem willing to pay to enter the game and the infinite expected value.[5] Solutions Several approaches have been proposed for solving the paradox. Expected utility theory 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. In Daniel Bernoulli's own words: 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 Bernoulli himself, 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 the sign of whose expected net change from accepting the gamble is a good criterion for real people's behavior. 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[7] 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 (Menger 1934).

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 (Rieger & Wang 2006).

Probability weighting

Nicolas Bernoulli himself proposed an alternative idea for solving the paradox. He conjectured that people will neglect unlikely events (de Montmort 1713). 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.

Cumulative prospect theory is one popular generalization of expected utility theory that can predict many behavioral regularities (Tversky & Kahneman 1992). 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 (Blavatskyy 2005). 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 (Tversky & Kahneman 1992). This is not applicable for infinitely increasing sums in the St. Petersburg paradox. Finite St. Petersburg lotteries The classical St. Petersburg game assumes that the casino or banker has infinite resources. This assumption has long been challenged as unrealistic.[8][9] Alexis Fontaine des Bertins pointed out in 1754 that the resources of any potential backer of the game are finite.[10] 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.[11] If the casino has finite resources, the game must end once those resources are exhausted.[9] 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 = floor(log2(W)).[12] Assuming the game ends when the casino can no longer cover the bet, the expected value E of the lottery then becomes:[13] {\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: Banker Bankroll Expected value of one game Millionaire1,050,000 $20 Billionaire$1,075,000,000 $30 Jeff Bezos (Jan. 2021)[14]$179,000,000,000 $37 U.S. GDP (2020)[15]$20.8 trillion $44 World GDP (2020)[15]$83.8 trillion $46 Billion-billionaire[16]$1018 $59 Googolionaire$10100 $332 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.

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 a persistent gambler will go broke, even if the game provides a positive expected value, and no betting system can avoid this inevitability.

Rejection of mathematical expectation

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. 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.[citation needed] Recently, some researchers have suggested to replace the expected value by the median as the fair value.[17][18]

Recent discussions

Although this paradox is three centuries old, new arguments are still being introduced.

Feller

A solution involving sampling was offered by William Feller.[19] 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.

Samuelson

Samuelson resolves the paradox 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 Paul Samuelson describes 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.

Further discussions

Marginal utility and philosophical view

The St. Petersburg paradox and the theory of marginal utility have been highly disputed in the past. For a discussion from the point of view of a philosopher, see Martin (2004).

Heuristic parameters and risks

Recently some authors suggested using heuristic parameters [20] (e.g. assessing the possible gains without neglecting the risks of the Saint Petersburg lottery) because of the highly stochastic context of this game (Cappiello 2016). The expected output should therefore be assessed in the limited period where we can likely make our choices and, besides the non-ergodic features (Peters 2011a), considering some inappropriate consequences we could attribute to the expected value (Feller 1968).

Notes and references

Citations
1. ^ Weiss, Michael D. (1987). Conceptual foundations of risk theory. U.S. Dept. of Agriculture, Economic Research Service. p. 36.
2. ^ Plous, Scott (January 1, 1993). "Chapter 7". The psychology of decision-making. McGraw-Hill Education. ISBN 978-0070504776.
3. ^ Eves, Howard (1990). An Introduction To The History of Mathematics (6th ed.). Brooks/Cole – Thomson Learning. p. 427.
4. ^ Peterson 2019
5. ^ a b Bernoulli 1738, cited in Dutka 1988, p. 14
6. ^ (Martin 2004).
7. ^ Xavier University Computer Science. correspondence_petersburg_game.pdf – Nicolas Bernoulli
8. ^ Peterson 2019, Section 3
9. ^ a b Jeffery 1983, p.154, "Our rebuttal of the St. Petersburg paradox consists in the remark that anyone who offers to let the agent play the Saint Petersburg game is a liar for he is pretending to have an indefinitely large bank."
10. ^ Fontaine 1764, cited in Dutka 1988, p. 31
11. ^ Buffon 1777, cited in Dutka 1988, p. 31
12. ^ Dutka 1988, p. 33 (Eq. 6-2)
13. ^ Dutka 1988, p. 31 (Eq. 5-5)
14. ^ Klebnikov, Sergei (January 11, 2021). "Elon Musk Falls To Second Richest Person In The World After His Fortune Drops Nearly \$14 Billion In One Day". Forbes. Retrieved March 25, 2021.
15. ^ a b The GDP data are as estimated for 2020 by the International Monetary Fund.
16. ^ Jeffery 1983, p.155, noting that no banker could cover such a sum because "there is no that much money in the world."
17. ^ Hayden, B.; Platt, M. (2009). "The mean, the median, and the St. Petersburg paradox". Judgment and Decision Making. 4 (4): 256–272. PMC 3811154. PMID 24179560.
18. ^ Okabe, T.; Nii, M.; Yoshimura, J. (2019). "The median-based resolution of the St. Petersburg paradox". Physics Letters A. 383 (26): 125838. Bibcode:2019PhLA..38325838O. doi:10.1016/j.physleta.2019.125838.
19. ^ Feller, William. An Introduction to Probability Theory and its Applications Volume I.
20. ^ "Decision making and Saint Petersburg Paradox: focusing on heuristic parameters, considering the non-ergodic context and the gambling risks" (PDF). Rivista Italiana di Economia Demografia e Statistica. 70 (4): 147–158. 2016.
Works cited
• Buffon, G. L. L. (1777). "Essai d'Arithmétique Motale". Supplements a l'Histoire Naturelle , T. IV: 46–14. Reprinted in ‘’Oeuvres Philosophiques de Buffon,’’ Paris, 1906, cited in Dutka, 1988
• Feller, William. An Introduction to Probability Theory and its Applications Volume I.
• Fontaine, Alexix (1764). "Solution d'un problème sur les jeux de hasard". Mémoires donnés à l'Académie Royale des Sciences: 429–431. cited in Dutka, 1988
• Jeffrey, Richard C. (1983). The Logic of Decision (2 ed.). Chicago: University of Chicago Press.