St. Petersburg paradox (original) (raw)

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Paradox involving a game with repeated coin flipping

Portrait of Nicolas Bernoulli (1723)

The St. Petersburg paradox or St. Petersburg lottery[1] is a paradox involving the game of flipping a coin where the expected payoff of the lottery game is infinite but nevertheless seems to be worth only a very small amount to the participants. The St. Petersburg paradox is a situation where a naïve decision criterion that 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, 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.[3][4] However, the paradox takes its name from its analysis by Nicolas' cousin Daniel Bernoulli, one-time resident of Saint Petersburg, who in 1738 published his thoughts about the problem in the Commentaries of the Imperial Academy of Science of Saint Petersburg.[5]

The St. Petersburg game

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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 tails appears. The first time heads appears, the game ends and the player wins whatever is the current stake. Thus the player wins 2 dollars if heads appears on the first toss, 4 dollars if tails appears on the first toss and heads on the second, 8 dollars if tails appears on the first two tosses and heads on the third, and so on. Mathematically, the player wins 2 k + 1 {\displaystyle 2^{k+1}} {\displaystyle 2^{k+1}} dollars, where k {\displaystyle k} {\displaystyle k} is the number of consecutive tails tosses.[5] 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 tails and, in particular, that the casino has unlimited resources, the expected value is thus

E = 1 2 ⋅ 2 + 1 4 ⋅ 4 + 1 8 ⋅ 8 + 1 16 ⋅ 16 + ⋯ = 1 + 1 + 1 + 1 + ⋯ = ∞ . {\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}}} {\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 so the expected win is an infinite amount of money.

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.[5]

Several approaches have been proposed for solving the paradox.

Expected utility theory

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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:

Δ E ( U ) = ∑ k = 1 + ∞ 1 2 k [ ln ⁡ ( w + 2 k − c ) − ln ⁡ ( w ) ] < + ∞ . {\displaystyle \Delta E(U)=\sum _{k=1}^{+\infty }{\frac {1}{2^{k}}}\left[\ln \left(w+2^{k}-c\right)-\ln(w)\right]<+\infty \,.} {\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,apersonwith20.88, a person with 20.88,apersonwith1,000 should pay up to 10.95,apersonwith10.95, a person with 10.95,apersonwith2 should borrow 1.35andpayupto1.35 and pay up to 1.35andpayupto3.35.

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.[8]

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.[9]

Probability weighting

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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".[10][11] 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.[12] 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.[13] 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.[12] This is not applicable for infinitely increasing sums in the St. Petersburg paradox.

Finite St. Petersburg lotteries

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The classical St. Petersburg game assumes that the casino or banker has infinite resources. This assumption has long been challenged as unrealistic.[14][15] 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.[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). 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]

E = ∑ k = 1 L 1 2 k ⋅ 2 k = L . {\displaystyle {\begin{aligned}E&=\sum _{k=1}^{L}{\frac {1}{2^{k}}}\cdot 2^{k}=L\,.\end{aligned}}} {\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
Millionaire 1,050,000∣1,050,000 1,050,000∣20
Billionaire 1,075,000,000∣1,075,000,000 1,075,000,000∣30
Elon Musk (Apr 2022)[19] 265,000,000,000∣265,000,000,000 265,000,000,000∣38
U.S. GDP (2020)[20] 20.8[trillion](/wiki/Trillion20.8 [trillion](/wiki/Trillion%5F%28short%5Fscale%29 "Trillion (short scale)") 20.8[trillion](/wiki/Trillion44
World GDP (2020)[20] 83.8trillion∣83.8 trillion 83.8trillion∣46
Billion-billionaire[21] 1018∣1018 1018∣59
Atoms in the universe[22] ~$1080 $266
Googolionaire 10100∣10100 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. For the player to win W he must be allowed to play round L+1. So the additional expected value is W/2L+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.

Ignore events with small probability

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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 had a probability of 1/10189 according to the mortality tables of the day, events with less than 1/10,000 probability could be ignored. Assuming that, the St Petersburg game has an expected payoff of only ∑ k = 1 13 2 k 1 2 k = 13 {\displaystyle \sum _{k=1}^{13}2^{k}{\frac {1}{2^{k}}}=13} {\displaystyle \sum _{k=1}^{13}2^{k}{\frac {1}{2^{k}}}=13}.

Rejection of mathematical expectation

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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.[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]

Although this paradox is three centuries old, new arguments have still been introduced in recent years.

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.[11]

For example, the "Pasadena game":[32] let n {\displaystyle n} {\displaystyle n} be the number of coin-flips; if n {\displaystyle n} {\displaystyle n} is odd, the player gains units of 2 n n {\displaystyle {\frac {2^{n}}{n}}} {\displaystyle {\frac {2^{n}}{n}}}; else the player loses 2 n n {\displaystyle {\frac {2^{n}}{n}}} {\displaystyle {\frac {2^{n}}{n}}} units of utility. The expected utility from the game is then ∑ n = 1 ∞ ( − 1 ) n + 1 n = ln ⁡ 2 {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=\ln 2} {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{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.

  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. ^ a b de Montmort, Pierre Remond (1713). Essay d'analyse sur les jeux de hazard [_Essays on the analysis of games of chance_] (Reprinted in 2006) (in French) (Second ed.). Providence, Rhode Island: American Mathematical Society. ISBN 978-0-8218-3781-8. Translated by Pulskamp, Richard J (January 1, 2013). "Correspondence of Nicolas Bernoulli concerning the St. Petersburg Game" (PDF). Archived from the original (PDF) on April 14, 2021. Retrieved July 22, 2010.

  5. ^ a b c d Bernoulli, Daniel; originally published in 1738 ("Specimen Theorize Naval de Mensura Sortis", "Commentarii Academiae Scientiarum Imperialis Petropolitanae"); translated by Dr. Louise Sommer (January 1954). "Exposition of a New Theory on the Measurement of Risk". Econometrica. 22 (1): 22–36. doi:10.2307/1909829. JSTOR 1909829. Retrieved May 30, 2006.{{[cite journal](/wiki/Template:Cite%5Fjournal "Template:Cite journal")}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)

  6. ^ Martin, Robert (Fall 2004). "The St. Petersburg Paradox". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy. Stanford, California: Stanford University. ISSN 1095-5054. Retrieved May 30, 2006.

  7. ^ Xavier University Computer Science. correspondence_petersburg_game.pdf Nicolas Bernoulli Archived May 1, 2015, at the Wayback Machine

  8. ^ Menger, Karl (August 1934). "Das Unsicherheitsmoment in der Wertlehre Betrachtungen im Anschluß an das sogenannte Petersburger Spiel" [The element of uncertainty in value theory: Reflections on the so-called St Petersburg game]. Zeitschrift für Nationalökonomie (in German). 5 (4): 459–485. doi:10.1007/BF01311578. ISSN 0931-8658. S2CID 151290589.

  9. ^ Rieger, Marc Oliver; Wang, Mei (August 2006). "Cumulative prospect theory and the St. Petersburg paradox" (PDF). Economic Theory. 28 (3): 665–679. doi:10.1007/s00199-005-0641-6. hdl:20.500.11850/32060. ISSN 0938-2259. S2CID 790082. (Publicly accessible, older version. Archived June 4, 2006, at the Wayback Machine)

  10. ^ Martin, R. M. "The St. Petersburg Paradox". Stanford Library. Stanford University.

  11. ^ a b Peterson, Martin (July 30, 2019) [July 30, 2019]. "The St. Petersburg Paradox". In Edward N. Zalta (ed.). Stanford Encyclopedia of Philosophy (Fall 2020 ed.). Retrieved March 24, 2021.

  12. ^ a b Tversky, Amos; Kahneman (1992). "Advances in prospect theory: Cumulative representation of uncertainty". Journal of Risk and Uncertainty. 5 (4): 297–323. doi:10.1007/bf00122574. S2CID 8456150.

  13. ^ Blavatskyy, Pavlo (April 2005). "Back to the St. Petersburg Paradox?" (PDF). Management Science. 51 (4): 677–678. doi:10.1287/mnsc.1040.0352.

  14. ^ Peterson, Martin (2011). "A New Twist to the St. Petersburg Paradox". Journal of Philosophy 108 (12):697–699.

  15. ^ a b Jeffrey, Richard C. (1990). The Logic of Decision (2 ed.). University of Chicago Press. pp. 154. ISBN 9780226395821. [O]ur rebuttal of the St. Petersburg paradox consists in the remark that anyone who offers to let the agent play the St. Petersburg game is a liar for he is pretending to have an indefinitely large bank.

  16. ^ Fontaine, Alexix (1764). "Solution d'un problème sur les jeux de hasard" [Solution to a problem about gambling games]. Mémoires donnés à l'Académie Royale des Sciences: 429–431. cited in Dutka, 1988

  17. ^ a b Buffon, G. L. L. (1777). "Essai d'Arithmétique Morale". Supplements a l'Histoire Naturelle. IV: 46–14. Reprinted in Oeuvres Philosophiques de Buffon, Paris, 1906, cited in Dutka, 1988

  18. ^ a b Dutka, Jacques (1988). "On the St. Petersburg Paradox". Archive for History of Exact Sciences. 39 (1): 13–39. doi:10.1007/BF00329984. JSTOR 41133842. S2CID 121413446. Retrieved March 23, 2021.

  19. ^ 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.

  20. ^ a b The GDP data are as estimated for 2020 by the International Monetary Fund.

  21. ^ Jeffery 1983, p.155, noting that no banker could cover such a sum because "there is not that much money in the world".

  22. ^ "Notable Properties of Specific Numbers (page 19) at MROB".

  23. ^ d'Alembert, Jean le Rond; Opuscules mathématiques (1768), vol. iv, p. 284-5.

  24. ^ a b Keynes, John Maynard; A Treatise on Probability (1921), Pt IV Ch XXVI §9.

  25. ^ Hayden, B.; Platt, M. (2009). "The mean, the median, and the St. Petersburg paradox". Judgment and Decision Making. 4 (4): 256–272. doi:10.1017/S1930297500003831. PMC 3811154. PMID 24179560.

  26. ^ 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. S2CID 199124414.

  27. ^ Whitworth, William Allen (1870). Choice and Chance (2 ed.). London: Deighton Bell.

  28. ^ Peters, Ole (2011a). "The time resolution of the St Petersburg paradox". Philosophical Transactions of the Royal Society. 369 (1956): 4913–4931. arXiv:1011.4404. Bibcode:2011RSPTA.369.4913P. doi:10.1098/rsta.2011.0065. PMC 3270388. PMID 22042904.

  29. ^ Carr, Peter; Cherubini, Umberto (2020). "Generalized Compounding and Growth Optimal Portfolios: Reconciling Kelly and Samuelson". SSRN. doi:10.2139/ssrn.3529729. S2CID 219384143.

  30. ^ Feller, William (1968). An Introduction to Probability Theory and its Applications Volume I, II. Wiley. ISBN 978-0471257080.

  31. ^ Samuelson, Paul (January 1960). "The St. Petersburg Paradox as a Divergent Double Limit". International Economic Review. 1 (1): 31–37. doi:10.2307/2525406. JSTOR 2525406.

  32. ^ Nover, H. (April 1, 2004). "Vexing Expectations". Mind. 113 (450): 237–249. doi:10.1093/mind/113.450.237. ISSN 0026-4423.

  33. ^ The notation ⌊X⌋ indicates the floor function, the largest integer less than or equal to X.