22:
3627:
3617:
460:, 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
1193:. 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.
362:
711:
function is lower than the power coefficient of the probability weighting function. 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
45:
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
678:
Nicolas
Bernoulli himself proposed an alternative idea for solving the paradox. He conjectured that people will neglect unlikely events. 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
658:
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
85:
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
1204:
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
706:
that can predict many behavioral regularities. 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
1005:
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
654:
He demonstrated in a letter to
Nicolas Bernoulli 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.
1412:, 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.
1048:
Buffon 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.
199:
695:
worth more than any number of birds in the bush". 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.
1205:
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."
866:
386:, stated, "Although the standard calculation shows that the value of expectation is infinitely great, it has ... to be admitted that any fairly reasonable man would sell his chance, with great pleasure, for twenty ducats." Robert Martin quotes
390:
as saying, "Few of us would pay even $ 25 to enter such a game", and he says most commentators would agree. The apparent paradox is the discrepancy between what people seem willing to pay to enter the game and the infinite expected value.
623:
691:. 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
1735:(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].
1035:
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
2137:
1406:
1125:
2024:
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.
1052:
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
780:
204:
728:
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,
357:{\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}}}
1609:; originally published in 1738 ("Specimen Theorize Naval de Mensura Sortis", "Commentarii Academiae Scientiarum Imperialis Petropolitanae"); translated by Dr. Louise Sommer (January 1954).
1173:
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.
1322:
1288:
1149:
of an alternative could be sufficiently high to reject it even if its expectation were enormous. Recently, some researchers have suggested to replace the expected value by the
190:
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
775:
378:
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,
1031:, 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
117:
426:
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
635:($ 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.
743:
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
1254:
1234:
137:
1716:
1567:
473:
735:
If the casino has finite resources, the game must end once those resources are exhausted. Suppose the total resources (or maximum jackpot) of the casino are
46:
take. Several resolutions to the paradox have been proposed, including the impossible amount of money a casino would need to continue the game indefinitely.
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692:
720:
The classical St. Petersburg game assumes that the casino or banker has infinite resources. This assumption has long been challenged as unrealistic.
1657:
2335:
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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
724:
pointed out in 1754 that the resources of any potential backer of the game are finite. More importantly, the expected value of the game only
2643:"Decision Making And Saint Petersburg Paradox: Focusing On Heuristic Parameters, Considering The Non-Ergodic Context And The Gambling Risks"
712:
theory are obtained in the region of less than $ 400. This is not applicable for infinitely increasing sums in the St. Petersburg paradox.
650:
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.
2893:
1841:
1327:
3063:
627:
This formula gives an implicit relationship between the gambler's wealth and how much he should be willing to pay (specifically, any
1143:, have rejected maximization of expectation (even of utility) as a proper rule of conduct. Keynes, in particular, insisted that the
1056:, events with less than 1/10,000 probability could be ignored. Assuming that, the St Petersburg game has an expected payoff of only
2398:
Carr, Peter; Cherubini, Umberto (2020). "Generalized
Compounding and Growth Optimal Portfolios: Reconciling Kelly and Samuelson".
670:, the St. Petersburg paradox again appears in certain cases, even when the utility function is concave, but not if it is bounded.
3211:
1161:
An early resolution containing the essential mathematical arguments assuming multiplicative dynamics was put forward in 1870 by
1059:
2036:
Fontaine, Alexix (1764). "Solution d'un problème sur les jeux de hasard" [Solution to a problem about gambling games].
729:
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1737:
1557:
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1859:
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be the cost charged to enter the game. The expected incremental utility of the lottery now converges to a finite value:
3671:
3272:
2901:
2865:
2716:
2441:
1503:
951:
934:
416:
2177:
Jeffery 1983, p.155, noting that no banker could cover such a sum because "there is not that much money in the world".
3241:
3078:
3003:
2017:
639:
3497:
1713:
1574:
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calculated that after 29 rounds of play there would not be enough money in the
Kingdom of France to cover the bet.
3661:
3651:
1912:
1409:
3681:
2835:
2586:
139:
is the number of consecutive tails tosses. What would be a fair price to pay the casino for entering the game?
2138:"Elon Musk Falls To Second Richest Person In The World After His Fortune Drops Nearly $ 14 Billion In One Day"
1169:
to the ergodicity problem was made by Peters in 2011. These solutions are mathematically similar to using the
3490:
2459:
1881:
1549:
1676:
659:
find a lottery that allows for a variant of the St. Petersburg paradox, as was first pointed out by Menger.
3314:
1956:
2009:
3656:
3144:
2165:
1910:
Tversky, Amos; Kahneman (1992). "Advances in prospect theory: Cumulative representation of uncertainty".
1442:
1028:
721:
412:
1293:
1259:
1181:
Although this paradox is three centuries old, new arguments have still been introduced in recent years.
142:
To answer this, one needs to consider what would be the expected payout at each stage: with probability
3266:
3206:
2620:
2271:
Okabe, T.; Nii, M.; Yoshimura, J. (2019). "The median-based resolution of the St. Petersburg paradox".
1537:
1136:
699:
667:
54:
3261:
861:{\displaystyle {\begin{aligned}E&=\sum _{k=1}^{L}{\frac {1}{2^{k}}}\cdot 2^{k}=L\,.\end{aligned}}}
646:, had already found parts of this idea (also motivated by the St. Petersburg paradox) in stating that
3334:
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1793:
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1213:
Many variants of the St
Petersburg game are proposed to counter proposed solutions to the game.
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703:
2708:
2578:
1027:
The premise of infinite resources produces a variety of apparent paradoxes in economics. In the
3676:
3324:
2670:
941:
57:
on
September 9, 1713. However, the paradox takes its name from its analysis by Nicolas' cousin
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89:
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that gives a positive change in expected utility). For example, with natural log utility, a
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3582:
3572:
3537:
3485:
3414:
3329:
3201:
3196:
3119:
3114:
3043:
2833:(March 1977). "St. Petersburg Paradoxes: Defanged, Dissected, and Historically Described".
2794:
2354:
2280:
1684:
1166:
50:
2457:
Samuelson, Paul (January 1960). "The St. Petersburg
Paradox as a Divergent Double Limit".
1785:
8:
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3319:
3283:
3216:
3083:
3053:
3048:
3018:
2983:
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1457:
1447:
1140:
765:. Assuming the game ends when the casino can no longer cover the bet, the expected value
2798:
2358:
2284:
618:{\displaystyle \Delta E(U)=\sum _{k=1}^{+\infty }{\frac {1}{2^{k}}}\left<+\infty \,.}
3630:
3587:
3567:
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1762:
1632:
1427:
1239:
1219:
725:
663:
122:
430:
is more significant to the pauper than to a rich man though both gain the same amount.
3620:
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3542:
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3439:
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3339:
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2013:
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3354:
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2822:
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2629:
2595:
2579:"The use of unbounded utility functions in expected-utility maximization: Response"
2507:
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2403:
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2243:
2235:
2095:
1965:
1933:
1921:
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1802:
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1624:
1610:
1606:
1548:] (Reprinted in 2006) (in French) (Second ed.). Providence, Rhode Island:
1422:
975:
379:
367:
62:
58:
2618:(April 1977). "The St. Petersburg paradox: A discussion of some recent comments".
1832:
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2875:
2775:
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2003:
1999:
1845:
1720:
1437:
1170:
684:
680:
407:
The classical resolution of the paradox involved the explicit introduction of a
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2511:
2429:
1201:
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78:
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1824:
1758:
1692:
1145:
82:
2773:
Peters, Ole; Gell-Mann, Murray (2016). "Evaluating gambles using dynamics".
2668:
Durand, David (September 1957). "Growth Stocks and the Petersburg Paradox".
2200:
d'Alembert, Jean le Rond; Opuscules mathématiques (1768), vol. iv, p. 284-5.
3186:
3159:
3154:
2814:
2384:
2366:
2257:
2187:
1969:
1615:
688:
1948:
21:
3364:
2407:
2212:
Keynes, John Maynard; A Treatise on Probability (1921), Pt IV Ch XXVI §9.
1732:
632:
387:
2879:
2107:
2083:
1568:"Correspondence of Nicolas Bernoulli concerning the St. Petersburg Game"
2848:
2691:
2607:
2480:
2434:
An Introduction to Probability Theory and its Applications Volume I, II
2099:
1925:
1750:
1636:
1053:
41:
involving the game of flipping a coin where the expected payoff of the
2806:
1982:
Peterson, Martin (2011). "A New Twist to the St. Petersburg Paradox".
662:
Recently, expected utility theory has been extended to arrive at more
2858:
Large Sample Methods in Statistics. An Introduction with Applications
920:
435:
2932:
2683:
2599:
2472:
1628:
1781:
1483:. U.S. Dept. of Agriculture, Economic Research Service. p. 36.
67:
Commentaries of the Imperial Academy of Science of Saint Petersburg
2789:
2764:
2349:
1401:{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=\ln 2}
462:
ln(wealth after the event) − ln(wealth before the event)
2964:
708:
638:
Before Daniel Bernoulli published, in 1728, a mathematician from
464:
will be weighted by the probability of that event occurring. Let
408:
42:
38:
1150:
988:
399:
Several approaches have been proposed for solving the paradox.
875:
of the game with various potential bankers and their bankroll
434:
A common utility model, suggested by Daniel Bernoulli, is the
65:, who in 1738 published his thoughts about the problem in the
1324:
units of utility. The expected utility from the game is then
427:
383:
1523:(6th ed.). Brooks/Cole – Thomson Learning. p. 427.
3028:
1786:"Cumulative prospect theory and the St. Petersburg paradox"
679:
probability weighting resurfaced much later in the work on
2188:"Notable Properties of Specific Numbers (page 19) at MROB"
1120:{\displaystyle \sum _{k=1}^{13}2^{k}{\frac {1}{2^{k}}}=13}
382:, after describing the game with an initial stake of one
2055:
Buffon, G. L. L. (1777). "Essai d'Arithmétique Motale".
1130:
1611:"Exposition of a New Theory on the Measurement of Risk"
1043:
2758:
Peters, Ole (October 2011b). "Menger 1934 revisited".
2224:"The mean, the median, and the St. Petersburg paradox"
2881:
A history of the mathematical theory of probabilities
1330:
1296:
1262:
1242:
1222:
1062:
778:
476:
202:
125:
92:
2648:
Rivista italiana di economia demografia e statistica
2008:(2 ed.). University of Chicago Press. pp.
1714:
correspondence_petersburg_game.pdf Nicolas Bernoulli
370:
so the expected win is an infinite amount of money.
16:
Paradox involving a game with repeated coin flipping
2731:(2 ed.). Chicago: University of Chicago Press.
2700:
2331:"The time resolution of the St Petersburg paradox"
1400:
1316:
1282:
1248:
1228:
1119:
860:
715:
617:
456:). It is a function of the gambler's total wealth
356:
131:
111:
2270:
3643:
2707:. Oxford, UK: Oxford University Press. pp.
2038:Mémoires donnés à l'Académie Royale des Sciences
2928:Online simulation of the St. Petersburg lottery
2772:
2336:Philosophical Transactions of the Royal Society
1909:
2164:The GDP data are as estimated for 2020 by the
158:, the player wins 2 dollars; with probability
2948:
2397:
1994:
1992:
1521:An Introduction To The History of Mathematics
1494:Plous, Scott (January 1, 1993). "Chapter 7".
1189:A solution involving sampling was offered by
871:The following table shows the expected value
2749:] (in French) (Second ed.). Paris:
2556:, the largest integer less than or equal to
1656:: CS1 maint: multiple names: authors list (
174:the player wins 4 dollars; with probability
2322:
2221:
1536:
3616:
2955:
2941:
2894:"Bernoulli and the St. Petersburg Paradox"
1989:
1946:
1779:
1660:) CS1 maint: numeric names: authors list (
1410:since the sum is not absolutely convergent
402:
72:
2874:
2855:
2829:
2788:
2763:
2640:
2456:
2374:
2348:
2313:
2247:
2135:
1814:
1605:
1546:Essays on the analysis of games of chance
850:
611:
346:
2035:
1879:
1565:
673:
20:
2757:
2735:
2726:
2328:
1998:
1681:The Stanford Encyclopedia of Philosophy
1566:Pulskamp, Richard J (January 1, 2013).
666:. In some of these new theories, as in
3644:
2667:
2614:
2428:
2160:
2158:
2136:Klebnikov, Sergei (January 11, 2021).
2054:
1731:
1674:
1542:Essay d'analyse sur les jeux de hazard
1216:For example, the "Pasadena game": let
1017:. So the additional expected value is
730:Georges-Louis Leclerc, Comte de Buffon
2962:
2936:
2698:
2573:
2493:
2208:
2206:
2088:Archive for History of Exact Sciences
2081:
2077:
2075:
2050:
2048:
1949:"Back to the St. Petersburg Paradox?"
1905:
1903:
1875:
1873:
1493:
1480:Conceptual foundations of risk theory
1476:
1176:
1131:Rejection of mathematical expectation
2318:(2 ed.). London: Deighton Bell.
1712:Xavier University Computer Science.
1601:
1599:
1597:
1595:
1532:
1530:
1518:
1044:Ignore events with small probability
25:Portrait of Nicolas Bernoulli (1723)
2742:Théorie analytique des probabilités
2155:
2069:, Paris, 1906, cited in Dutka, 1988
1886:Stanford Encyclopedia of Philosophy
1880:Peterson, Martin (July 30, 2019) .
1839:Publicly accessible, older version.
13:
3273:What the Tortoise Said to Achilles
2902:The New School for Social Research
2747:Analytical theory of probabilities
2567:
2422:
2203:
2072:
2057:Supplements a l'Histoire Naturelle
2045:
1976:
1900:
1870:
1857:
1347:
1317:{\displaystyle {\frac {2^{n}}{n}}}
1283:{\displaystyle {\frac {2^{n}}{n}}}
1256:is odd, the player gains units of
608:
514:
477:
343:
14:
3693:
2921:
2314:Whitworth, William Allen (1870).
1592:
1527:
1496:The psychology of decision-making
1010:he must be allowed to play round
702:is one popular generalization of
3626:
3625:
3615:
2856:Sen, P.K.; Singer, J.M. (1993).
2067:Oeuvres Philosophiques de Buffon
1947:Blavatskyy, Pavlo (April 2005).
1738:Zeitschrift für Nationalökonomie
1236:be the number of coin-flips; if
2898:The History of Economic Thought
2536:
2487:
2450:
2391:
2307:
2264:
2215:
2194:
2180:
2171:
2129:
2084:"On the St. Petersburg Paradox"
2029:
1940:
1913:Journal of Risk and Uncertainty
1851:
716:Finite St. Petersburg lotteries
53:, who stated it in a letter to
2836:Journal of Economic Literature
2587:Quarterly Journal of Economics
2293:10.1016/j.physleta.2019.125838
2222:Hayden, B.; Platt, M. (2009).
1773:
1725:
1706:
1668:
1512:
1487:
1470:
1365:
1355:
594:
588:
489:
483:
373:
1:
2460:International Economic Review
1679:. In Zalta, Edward N. (ed.).
1550:American Mathematical Society
1464:
1156:
769:of the lottery then becomes:
81:for a single player in which
3667:Probability theory paradoxes
2727:Jeffrey, Richard C. (1983).
2634:10.1016/0022-0531(77)90143-0
2228:Judgment and Decision Making
1884:. In Edward N. Zalta (ed.).
1882:"The St. Petersburg Paradox"
1860:"The St. Petersburg Paradox"
1677:"The St. Petersburg Paradox"
1675:Martin, Robert (Fall 2004).
1196:
417:diminishing marginal utility
394:
49:The problem was invented by
7:
2641:Cappiello, Antonio (2016).
2494:Nover, H. (April 1, 2004).
2166:International Monetary Fund
1443:Martingale (betting system)
1415:
1208:
1135:Various authors, including
722:Alexis Fontaine des Bertins
413:expected utility hypothesis
10:
3700:
2904:, New York. Archived from
2663:. RePEc:ite:iteeco:160406.
2621:Journal of Economic Theory
1538:de Montmort, Pierre Remond
1477:Weiss, Michael D. (1987).
987:
974:
963:
950:
933:
919:
908:
897:
700:Cumulative prospect theory
668:cumulative prospect theory
664:behavioral decision models
55:Pierre Raymond de Montmort
3672:Decision-making paradoxes
3611:
3463:
3292:
3092:
2971:
2240:10.1017/S1930297500003831
1807:10.1007/s00199-005-0641-6
1498:. McGraw-Hill Education.
1184:
1029:martingale betting system
739:dollars (more generally,
415:, and the presumption of
2529:
2512:10.1093/mind/113.450.237
1290:; else the player loses
3192:Paradoxes of set theory
2082:Dutka, Jacques (1988).
1687:: Stanford University.
1163:William Allen Whitworth
1137:Jean le Rond d'Alembert
704:expected utility theory
403:Expected utility theory
112:{\displaystyle 2^{k+1}}
73:The St. Petersburg game
61:, one-time resident of
3662:Mathematical economics
3652:Paradoxes in economics
2860:. New York: Springer.
2671:The Journal of Finance
2367:10.1098/rsta.2011.0065
1970:10.1287/mnsc.1040.0352
1866:. Stanford University.
1402:
1351:
1318:
1284:
1250:
1230:
1121:
1083:
862:
813:
652:
619:
518:
432:
358:
133:
113:
35:St. Petersburg lottery
31:St. Petersburg paradox
26:
3682:Paradoxes of infinity
2737:Laplace, Pierre Simon
2729:The Logic of Decision
2496:"Vexing Expectations"
2329:Peters, Ole (2011a).
2005:The Logic of Decision
1984:Journal of Philosophy
1844:June 4, 2006, at the
1780:Rieger, Marc Oliver;
1519:Eves, Howard (1990).
1453:Two envelopes problem
1403:
1331:
1319:
1285:
1251:
1231:
1122:
1063:
976:Atoms in the universe
863:
793:
726:grows logarithmically
674:Probability weighting
648:
620:
495:
424:
359:
134:
114:
83:a fair coin is tossed
24:
3558:Kavka's toxin puzzle
3330:Income and fertility
2699:Haigh, John (1999).
2408:10.2139/ssrn.3529729
2042:cited in Dutka, 1988
1888:(Fall 2020 ed.)
1719:May 1, 2015, at the
1685:Stanford, California
1328:
1294:
1260:
1240:
1220:
1060:
776:
474:
436:logarithmic function
200:
123:
90:
3217:Temperature paradox
3140:Free choice paradox
3004:Fitch's knowability
2799:2016Chaos..26b3103P
2359:2011RSPTA.369.4913P
2343:(1956): 4913–4931.
2285:2019PhLA..38325838O
2000:Jeffrey, Richard C.
1153:as the fair value.
1141:John Maynard Keynes
964:Billion-billionaire
368:grows without bound
3657:Behavioral finance
3593:Prisoner's dilemma
3279:Heat death paradox
3267:Unexpected hanging
3232:Chicken or the egg
2886:Macmillan & Co
2100:10.1007/BF00329984
1957:Management Science
1926:10.1007/bf00122574
1816:20.500.11850/32060
1751:10.1007/BF01311578
1428:Exponential growth
1398:
1314:
1280:
1246:
1226:
1177:Recent discussions
1117:
858:
856:
615:
422:Daniel Bernoulli;
354:
352:
129:
109:
77:A casino offers a
27:
3639:
3638:
3310:Arrow information
2807:10.1063/1.4940236
2616:Aumann, Robert J.
2577:(February 1974).
2575:Arrow, Kenneth J.
2316:Choice and Chance
2273:Physics Letters A
1986:108 (12):697–699.
1607:Bernoulli, Daniel
1580:on April 14, 2021
1559:978-0-8218-3781-8
1384:
1312:
1278:
1249:{\displaystyle n}
1229:{\displaystyle n}
1109:
1000:
999:
926:$ 265,000,000,000
829:
534:
282:
263:
244:
225:
132:{\displaystyle k}
51:Nicolas Bernoulli
3689:
3629:
3628:
3619:
3618:
3430:Service recovery
3284:Olbers's paradox
2984:Buridan's bridge
2957:
2950:
2943:
2934:
2933:
2917:
2915:
2913:
2908:on June 18, 2006
2889:
2876:Todhunter, Isaac
2871:
2852:
2826:
2792:
2769:
2767:
2754:
2732:
2722:
2706:
2695:
2664:
2637:
2611:
2583:
2561:
2551:
2550:
2546:
2540:
2524:
2523:
2506:(450): 237–249.
2491:
2485:
2484:
2454:
2448:
2447:
2426:
2420:
2419:
2395:
2389:
2388:
2378:
2352:
2326:
2320:
2319:
2311:
2305:
2304:
2268:
2262:
2261:
2251:
2219:
2213:
2210:
2201:
2198:
2192:
2191:
2184:
2178:
2175:
2169:
2162:
2153:
2152:
2150:
2148:
2133:
2127:
2126:
2124:
2122:
2079:
2070:
2064:
2052:
2043:
2041:
2033:
2027:
2026:
1996:
1987:
1980:
1974:
1973:
1953:
1944:
1938:
1937:
1907:
1898:
1897:
1895:
1893:
1877:
1868:
1867:
1864:Stanford Library
1855:
1849:
1836:
1818:
1790:
1777:
1771:
1770:
1729:
1723:
1710:
1704:
1703:
1701:
1699:
1672:
1666:
1665:
1655:
1647:
1645:
1643:
1603:
1590:
1589:
1587:
1585:
1579:
1573:. Archived from
1572:
1563:
1534:
1525:
1524:
1516:
1510:
1509:
1491:
1485:
1484:
1474:
1458:Zeno's paradoxes
1448:Pascal's mugging
1423:Ellsberg paradox
1407:
1405:
1404:
1399:
1385:
1380:
1379:
1378:
1353:
1350:
1345:
1323:
1321:
1320:
1315:
1313:
1308:
1307:
1298:
1289:
1287:
1286:
1281:
1279:
1274:
1273:
1264:
1255:
1253:
1252:
1247:
1235:
1233:
1232:
1227:
1126:
1124:
1123:
1118:
1110:
1108:
1107:
1095:
1093:
1092:
1082:
1077:
1054:mortality tables
1023:
1016:
1009:
952:World GDP (2020)
882:
881:
867:
865:
864:
859:
857:
843:
842:
830:
828:
827:
815:
812:
807:
764:
763:
751:
630:
624:
622:
621:
616:
601:
597:
578:
574:
567:
566:
535:
533:
532:
520:
517:
509:
467:
463:
459:
451:
409:utility function
380:Daniel Bernoulli
363:
361:
360:
355:
353:
336:
299:
283:
275:
264:
256:
245:
237:
226:
218:
189:
187:
186:
183:
180:
173:
171:
170:
167:
164:
157:
155:
154:
151:
148:
138:
136:
135:
130:
118:
116:
115:
110:
108:
107:
63:Saint Petersburg
59:Daniel Bernoulli
3699:
3698:
3692:
3691:
3690:
3688:
3687:
3686:
3642:
3641:
3640:
3635:
3607:
3518:Decision-making
3464:Decision theory
3459:
3288:
3212:Hilbert's Hotel
3145:Grelling–Nelson
3088:
2967:
2961:
2924:
2911:
2909:
2892:
2868:
2831:Samuelson, Paul
2719:
2684:10.2307/2976852
2600:10.2307/1881800
2581:
2570:
2568:Further reading
2565:
2564:
2548:
2544:
2543:
2541:
2537:
2532:
2527:
2492:
2488:
2473:10.2307/2525406
2455:
2451:
2444:
2430:Feller, William
2427:
2423:
2396:
2392:
2327:
2323:
2312:
2308:
2269:
2265:
2220:
2216:
2211:
2204:
2199:
2195:
2186:
2185:
2181:
2176:
2172:
2163:
2156:
2146:
2144:
2134:
2130:
2120:
2118:
2080:
2073:
2053:
2046:
2034:
2030:
2020:
1997:
1990:
1981:
1977:
1951:
1945:
1941:
1908:
1901:
1891:
1889:
1878:
1871:
1856:
1852:
1846:Wayback Machine
1794:Economic Theory
1788:
1784:(August 2006).
1778:
1774:
1730:
1726:
1721:Wayback Machine
1711:
1707:
1697:
1695:
1673:
1669:
1649:
1648:
1641:
1639:
1629:10.2307/1909829
1604:
1593:
1583:
1581:
1577:
1570:
1560:
1535:
1528:
1517:
1513:
1506:
1492:
1488:
1475:
1471:
1467:
1462:
1438:Kelly criterion
1418:
1368:
1364:
1354:
1352:
1346:
1335:
1329:
1326:
1325:
1303:
1299:
1297:
1295:
1292:
1291:
1269:
1265:
1263:
1261:
1258:
1257:
1241:
1238:
1237:
1221:
1218:
1217:
1211:
1199:
1187:
1179:
1171:Kelly criterion
1159:
1133:
1103:
1099:
1094:
1088:
1084:
1078:
1067:
1061:
1058:
1057:
1046:
1021:
1018:
1014:
1011:
1007:
956:$ 83.8 trillion
912:$ 1,075,000,000
892:
891:Expected value
855:
854:
838:
834:
823:
819:
814:
808:
797:
786:
779:
777:
774:
773:
761:
759:
755:
749:
747:
744:
718:
685:Daniel Kahneman
681:prospect theory
676:
628:
625:
562:
558:
551:
547:
540:
536:
528:
524:
519:
510:
499:
475:
472:
471:
465:
461:
457:
449:
445:
441:
438:
405:
397:
376:
364:
351:
350:
334:
333:
297:
296:
274:
255:
236:
217:
210:
203:
201:
198:
197:
184:
181:
178:
177:
175:
168:
165:
162:
161:
159:
152:
149:
146:
145:
143:
124:
121:
120:
119:dollars, where
97:
93:
91:
88:
87:
75:
17:
12:
11:
5:
3697:
3696:
3685:
3684:
3679:
3674:
3669:
3664:
3659:
3654:
3637:
3636:
3634:
3633:
3623:
3612:
3609:
3608:
3606:
3605:
3600:
3595:
3590:
3585:
3580:
3575:
3570:
3565:
3560:
3555:
3550:
3545:
3540:
3535:
3530:
3525:
3520:
3515:
3510:
3505:
3500:
3495:
3494:
3493:
3488:
3483:
3473:
3467:
3465:
3461:
3460:
3458:
3457:
3452:
3447:
3442:
3437:
3435:St. Petersburg
3432:
3427:
3422:
3417:
3412:
3407:
3402:
3397:
3392:
3387:
3382:
3377:
3372:
3367:
3362:
3357:
3352:
3347:
3342:
3337:
3332:
3327:
3322:
3317:
3312:
3307:
3302:
3296:
3294:
3290:
3289:
3287:
3286:
3281:
3276:
3269:
3264:
3259:
3254:
3249:
3244:
3239:
3234:
3229:
3224:
3219:
3214:
3209:
3204:
3199:
3194:
3189:
3184:
3183:
3182:
3177:
3172:
3167:
3162:
3152:
3147:
3142:
3137:
3132:
3127:
3122:
3117:
3112:
3107:
3102:
3096:
3094:
3090:
3089:
3087:
3086:
3081:
3076:
3071:
3066:
3064:Rule-following
3061:
3056:
3051:
3046:
3041:
3036:
3031:
3026:
3021:
3016:
3011:
3006:
3001:
2996:
2991:
2989:Dream argument
2986:
2981:
2975:
2973:
2969:
2968:
2960:
2959:
2952:
2945:
2937:
2931:
2930:
2923:
2922:External links
2920:
2919:
2918:
2890:
2872:
2867:978-0412042218
2866:
2853:
2827:
2770:
2755:
2733:
2724:
2718:978-0198526636
2717:
2703:Taking Chances
2696:
2678:(3): 348–363.
2665:
2655:(4): 147–158.
2638:
2628:(2): 443–445.
2612:
2594:(1): 136–138.
2569:
2566:
2563:
2562:
2554:floor function
2552:indicates the
2542:The notation
2534:
2533:
2531:
2528:
2526:
2525:
2486:
2449:
2443:978-0471257080
2442:
2421:
2390:
2321:
2306:
2279:(26): 125838.
2263:
2234:(4): 256–272.
2214:
2202:
2193:
2179:
2170:
2154:
2128:
2071:
2044:
2028:
2018:
1988:
1975:
1964:(4): 677–678.
1939:
1920:(4): 297–323.
1899:
1869:
1858:Martin, R. M.
1850:
1801:(3): 665–679.
1772:
1745:(4): 459–485.
1724:
1705:
1667:
1591:
1564:Translated by
1558:
1526:
1511:
1505:978-0070504776
1504:
1486:
1468:
1466:
1463:
1461:
1460:
1455:
1450:
1445:
1440:
1435:
1433:Gambler's ruin
1430:
1425:
1419:
1417:
1414:
1397:
1394:
1391:
1388:
1383:
1377:
1374:
1371:
1367:
1363:
1360:
1357:
1349:
1344:
1341:
1338:
1334:
1311:
1306:
1302:
1277:
1272:
1268:
1245:
1225:
1210:
1207:
1202:Paul Samuelson
1198:
1195:
1191:William Feller
1186:
1183:
1178:
1175:
1158:
1155:
1132:
1129:
1116:
1113:
1106:
1102:
1098:
1091:
1087:
1081:
1076:
1073:
1070:
1066:
1045:
1042:
1038:expected value
1033:gambler's ruin
1019:
1012:
998:
997:
994:
991:
985:
984:
981:
978:
972:
971:
968:
965:
961:
960:
957:
954:
948:
947:
944:
938:
931:
930:
927:
924:
917:
916:
913:
910:
906:
905:
902:
899:
895:
894:
889:
886:
869:
868:
853:
849:
846:
841:
837:
833:
826:
822:
818:
811:
806:
803:
800:
796:
792:
789:
787:
785:
782:
781:
757:
753:
745:
717:
714:
675:
672:
644:Gabriel Cramer
614:
610:
607:
604:
600:
596:
593:
590:
587:
584:
581:
577:
573:
570:
565:
561:
557:
554:
550:
546:
543:
539:
531:
527:
523:
516:
513:
508:
505:
502:
498:
494:
491:
488:
485:
482:
479:
470:
447:
443:
439:
404:
401:
396:
393:
375:
372:
349:
345:
342:
339:
337:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
300:
298:
295:
292:
289:
286:
281:
278:
273:
270:
267:
262:
259:
254:
251:
248:
243:
240:
235:
232:
229:
224:
221:
216:
213:
211:
209:
206:
205:
196:
192:expected value
128:
106:
103:
100:
96:
79:game of chance
74:
71:
15:
9:
6:
4:
3:
2:
3695:
3694:
3683:
3680:
3678:
3677:Coin flipping
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3658:
3655:
3653:
3650:
3649:
3647:
3632:
3624:
3622:
3614:
3613:
3610:
3604:
3601:
3599:
3596:
3594:
3591:
3589:
3586:
3584:
3581:
3579:
3576:
3574:
3571:
3569:
3566:
3564:
3563:Morton's fork
3561:
3559:
3556:
3554:
3551:
3549:
3546:
3544:
3541:
3539:
3536:
3534:
3531:
3529:
3526:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3503:Buridan's ass
3501:
3499:
3496:
3492:
3489:
3487:
3484:
3482:
3479:
3478:
3477:
3476:Apportionment
3474:
3472:
3469:
3468:
3466:
3462:
3456:
3453:
3451:
3448:
3446:
3443:
3441:
3438:
3436:
3433:
3431:
3428:
3426:
3423:
3421:
3418:
3416:
3413:
3411:
3408:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3388:
3386:
3383:
3381:
3378:
3376:
3373:
3371:
3368:
3366:
3363:
3361:
3358:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3338:
3336:
3335:Downs–Thomson
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3297:
3295:
3291:
3285:
3282:
3280:
3277:
3274:
3270:
3268:
3265:
3263:
3260:
3258:
3255:
3253:
3252:Plato's beard
3250:
3248:
3245:
3243:
3240:
3238:
3235:
3233:
3230:
3228:
3225:
3223:
3220:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3181:
3178:
3176:
3173:
3171:
3168:
3166:
3163:
3161:
3158:
3157:
3156:
3153:
3151:
3150:Kleene–Rosser
3148:
3146:
3143:
3141:
3138:
3136:
3133:
3131:
3128:
3126:
3123:
3121:
3118:
3116:
3113:
3111:
3108:
3106:
3103:
3101:
3098:
3097:
3095:
3091:
3085:
3082:
3080:
3077:
3075:
3074:Theseus' ship
3072:
3070:
3067:
3065:
3062:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3035:
3034:Mere addition
3032:
3030:
3027:
3025:
3022:
3020:
3017:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2985:
2982:
2980:
2977:
2976:
2974:
2972:Philosophical
2970:
2966:
2958:
2953:
2951:
2946:
2944:
2939:
2938:
2935:
2929:
2926:
2925:
2907:
2903:
2899:
2895:
2891:
2887:
2883:
2882:
2877:
2873:
2869:
2863:
2859:
2854:
2850:
2846:
2842:
2838:
2837:
2832:
2828:
2824:
2820:
2816:
2812:
2808:
2804:
2800:
2796:
2791:
2786:
2783:(2): 023103.
2782:
2778:
2777:
2771:
2766:
2761:
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2065:Reprinted in
2062:
2058:
2051:
2049:
2039:
2032:
2025:
2021:
2019:9780226395821
2015:
2011:
2007:
2006:
2001:
1995:
1993:
1985:
1979:
1971:
1967:
1963:
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1950:
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1935:
1931:
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1192:
1182:
1174:
1172:
1168:
1167:explicit link
1164:
1154:
1152:
1148:
1147:
1146:relative risk
1142:
1138:
1128:
1114:
1111:
1104:
1100:
1096:
1089:
1085:
1079:
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1025:
1004:
995:
992:
990:
989:Googolionaire
986:
982:
979:
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973:
969:
966:
962:
958:
955:
953:
949:
945:
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790:
788:
783:
772:
771:
770:
768:
742:
738:
733:
731:
727:
723:
713:
710:
705:
701:
697:
694:
693:birds in hand
690:
686:
682:
671:
669:
665:
660:
656:
651:
647:
645:
641:
636:
634:
612:
605:
602:
598:
591:
585:
582:
579:
575:
571:
568:
563:
559:
555:
552:
548:
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541:
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511:
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486:
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455:
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84:
80:
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68:
64:
60:
56:
52:
47:
44:
40:
36:
32:
23:
19:
3583:Preparedness
3434:
3415:Productivity
3395:Mandeville's
3187:Opposite Day
3115:Burali-Forti
3110:Bhartrhari's
2910:. Retrieved
2906:the original
2897:
2880:
2857:
2843:(1): 24–55.
2840:
2834:
2780:
2774:
2751:Ve. Courcier
2746:
2741:
2728:
2702:
2675:
2669:
2652:
2646:
2625:
2619:
2591:
2585:
2557:
2538:
2503:
2499:
2489:
2467:(1): 31–37.
2464:
2458:
2452:
2433:
2424:
2399:
2393:
2340:
2334:
2324:
2315:
2309:
2276:
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2266:
2231:
2227:
2217:
2196:
2182:
2173:
2145:. Retrieved
2141:
2131:
2119:. Retrieved
2094:(1): 13–39.
2091:
2087:
2066:
2060:
2056:
2037:
2031:
2023:
2004:
1983:
1978:
1961:
1955:
1942:
1917:
1911:
1890:. Retrieved
1885:
1863:
1853:
1798:
1792:
1775:
1742:
1736:
1733:Menger, Karl
1727:
1708:
1696:. Retrieved
1680:
1670:
1652:cite journal
1640:. Retrieved
1623:(1): 22–36.
1620:
1616:Econometrica
1614:
1582:. Retrieved
1575:the original
1545:
1541:
1520:
1514:
1495:
1489:
1479:
1472:
1215:
1212:
1200:
1188:
1180:
1160:
1144:
1134:
1051:
1047:
1026:
1002:
1001:
893:of one game
876:
872:
870:
766:
740:
736:
734:
719:
698:
689:Amos Tversky
677:
661:
657:
653:
649:
637:
626:
453:
433:
425:
421:
406:
398:
377:
365:
141:
76:
66:
48:
34:
30:
28:
18:
3513:Condorcet's
3365:Giffen good
3325:Competition
3079:White horse
3054:Omnipotence
2723:(Chapter 4)
1408:. However,
909:Billionaire
901:$ 1,050,000
898:Millionaire
633:millionaire
454:log utility
388:Ian Hacking
374:The paradox
3646:Categories
3588:Prevention
3578:Parrondo's
3568:Navigation
3553:Inventor's
3548:Hedgehog's
3508:Chainstore
3491:Population
3486:New states
3420:Prosperity
3400:Mayfield's
3242:Entailment
3222:Barbershop
3135:Epimenides
2040:: 429–431.
1465:References
1157:Ergodicity
923:(Apr 2022)
452:(known as
419:of money.
3603:Willpower
3598:Tolerance
3573:Newcomb's
3538:Fredkin's
3425:Scitovsky
3345:Edgeworth
3340:Easterlin
3305:Antitrust
3202:Russell's
3197:Richard's
3170:Pinocchio
3125:Crocodile
3044:Newcomb's
3014:Goodman's
3009:Free will
2994:Epicurean
2965:paradoxes
2790:1405.0585
2765:1110.1578
2661:0035-6832
2520:0026-4423
2436:. Wiley.
2416:219384143
2350:1011.4404
2301:199124414
2147:March 25,
2121:March 23,
2116:121413446
1892:March 24,
1825:0938-2259
1782:Wang, Mei
1767:151290589
1759:0931-8658
1693:1095-5054
1393:
1359:−
1348:∞
1333:∑
1197:Samuelson
1065:∑
921:Elon Musk
832:⋅
795:∑
609:∞
586:
580:−
569:−
545:
515:∞
497:∑
478:Δ
395:Solutions
366:This sum
344:∞
331:⋯
294:⋯
285:⋅
266:⋅
247:⋅
228:⋅
3631:Category
3528:Ellsberg
3380:Leontief
3360:Gibson's
3355:European
3350:Ellsberg
3320:Braess's
3315:Bertrand
3293:Economic
3227:Catch-22
3207:Socratic
3049:Nihilism
3019:Hedonism
2979:Analysis
2963:Notable
2878:(1865).
2815:26931584
2739:(1814).
2432:(1968).
2385:22042904
2258:24179560
2108:41133842
2063:: 46–14.
2002:(1990).
1842:Archived
1717:Archived
1584:July 22,
1540:(1713).
1416:See also
1209:Variants
942:trillion
935:U.S. GDP
888:Bankroll
194:is thus
3533:Fenno's
3498:Arrow's
3481:Alabama
3471:Abilene
3450:Tullock
3405:Metzler
3247:Lottery
3237:Drinker
3180:Yablo's
3175:Quine's
3130:Curry's
3093:Logical
3069:Sorites
3059:Preface
3039:Moore's
3024:Liberal
2999:Fiction
2912:May 30,
2849:2722712
2823:9726238
2795:Bibcode
2692:2976852
2608:1881800
2481:2525406
2376:3270388
2355:Bibcode
2281:Bibcode
2249:3811154
1934:8456150
1698:May 30,
1642:May 30,
1637:1909829
940:$ 20.8
709:utility
446:) = ln(
188:
176:
172:
160:
156:
144:
43:lottery
39:paradox
3440:Thrift
3410:Plenty
3385:Lerner
3375:Jevons
3370:Icarus
3300:Allais
3262:Ross's
3100:Barber
3084:Zeno's
3029:Meno's
2864:
2847:
2821:
2813:
2715:
2690:
2659:
2606:
2518:
2479:
2440:
2414:
2383:
2373:
2299:
2256:
2246:
2142:Forbes
2114:
2106:
2016:
1932:
1833:790082
1831:
1823:
1765:
1757:
1691:
1635:
1556:
1502:
1185:Feller
1151:median
996:$ 332
983:$ 266
937:(2020)
885:Banker
640:Geneva
428:ducats
3543:Green
3523:Downs
3455:Value
3390:Lucas
3257:Raven
3165:No-no
3120:Court
3105:Berry
2845:JSTOR
2819:S2CID
2785:arXiv
2776:Chaos
2760:arXiv
2745:[
2688:JSTOR
2604:JSTOR
2582:(PDF)
2530:Notes
2477:JSTOR
2412:S2CID
2345:arXiv
2297:S2CID
2112:S2CID
2104:JSTOR
1952:(PDF)
1930:S2CID
1829:S2CID
1789:(PDF)
1763:S2CID
1633:JSTOR
1578:(PDF)
1571:(PDF)
1544:[
1165:. An
1003:Note:
980:~$ 10
970:$ 59
959:$ 46
946:$ 44
929:$ 38
915:$ 30
904:$ 20
411:, an
384:ducat
37:is a
3621:List
3445:Toil
3160:Card
3155:Liar
2914:2006
2862:ISBN
2811:PMID
2713:ISBN
2657:ISSN
2516:ISSN
2500:Mind
2438:ISBN
2400:SSRN
2381:PMID
2254:PMID
2149:2021
2123:2021
2014:ISBN
1894:2021
1821:ISSN
1755:ISSN
1700:2006
1689:ISSN
1662:link
1658:link
1644:2006
1586:2010
1554:ISBN
1500:ISBN
1139:and
993:$ 10
967:$ 10
687:and
603:<
29:The
2803:doi
2709:330
2680:doi
2630:doi
2596:doi
2508:doi
2504:113
2469:doi
2404:doi
2371:PMC
2363:doi
2341:369
2289:doi
2277:383
2244:PMC
2236:doi
2096:doi
2010:154
1966:doi
1922:doi
1811:hdl
1803:doi
1747:doi
1625:doi
752:log
683:by
33:or
3648::
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