|
In probability theory and decision theory the St. Petersburg paradox describes a particular lottery game (sometimes called St. Petersburg Lottery) that leads to a random variable with infinite expected value, i.e. infinite expected payoff, but would nevertheless be considered to be worth only a very small amount of money. The St. Petersburg paradox is a classical situation where a naïve decision theory (which takes only the expected value into account) would recommend a course of action that no (real) rational person would be willing to take. The paradox can be resolved when the decision model is refined via the notion of marginal utility, by taking into account the finite resources of the participants, or by noting that one simply cannot buy that which is not sold (and sellers would not produce a lottery whose expected loss to them were unacceptable). It has been suggested that this article or section be merged with Probability axioms. ...
Decision theory is an interdisciplinary area of study, related to and of interest to practitioners in mathematics, statistics, economics, philosophy, management and psychology. ...
Look up paradox in Wiktionary, the free dictionary. ...
A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...
In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...
In epistemology and in its broadest sense, rationalism is any view appealing to reason as a source of knowledge or justification (Lacey, 286). ...
In economics, marginal utility is the additional utility (satisfaction or benefit) that a consumer derives from an additional unit of a commodity or service. ...
The paradox is named from Daniel Bernoulli's presentation of the problem and his solution, published in 1738 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 who first stated it in a letter to Pierre Raymond de Montmort of 9 September 1713. [1] Daniel Bernoulli Daniel Bernoulli (Groningen, January 29, 1700 – Basel, March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland. ...
Events February 4 - Court Jew Joseph Suss Oppenheimer is executed in Württenberg April 15 - Premiere in London of Serse, an Italian opera by George Frideric Handel. ...
Saint Petersburg (Russian: Санкт-Петербу́рг, English transliteration: Sankt-Peterburg), colloquially known as Питер (transliterated Piter), formerly known as Leningrad (Ленингра́д, 1924–1991) and...
Nicolaus Bernoulli (* October 21, 1687 in Basel, †November 29, 1759 in Basel), sometimes also written Nicolas or Nikolas, was a Swiss mathematician; he was the nephew of Jacob and Johann Bernoulli. ...
Pierre Raymond de Montmort was born in Paris on Oct. ...
September 9 is the 252nd day of the year (253rd in leap years). ...
// Events April 11 - War of the Spanish Succession: Treaty of Utrecht June 23 - French residents of Acadia given one year to declare allegiance to Britain or leave Nova Scotia Canada first Orrery built by George Graham Ongoing events Great Northern War (1700-1721) War of the Spanish Succession (1702-1713...
The paradox In a game of chance, you pay a fixed fee to enter, and then a fair coin will be tossed repeatedly until a "tail" first appears, ending the game. The "pot" starts at 1 dollar and is doubled every time a "head" appears. You win whatever is in the pot after the game ends. Thus you win 1 dollar if a tail appears on the first toss, 2 dollars if on the second, 4 dollars if on the third, 8 dollars if on the fourth, etc. In short, you win 2k−1 dollars if the coin is tossed k times until the first tail appears. (In the original introduction, this game was set in a hypothetical casino in St. Petersburg, hence the name of the paradox.) A game of chance is a game whose outcome is strongly influenced by some randomizing device, and upon which contestants frequently wager money. ...
Coin flipping or coin tossing is the practice of throwing a coin in the air to resolve a dispute between two parties or otherwise choose between two alternatives. ...
This article or section does not cite its references or sources. ...
Saint Petersburg (Russian: Санкт-Петербу́рг, English transliteration: Sankt-Peterburg), colloquially known as Питер (transliterated Piter), formerly known as Leningrad (Ленингра́д, 1924–1991) and...
How much would you be willing to pay to enter the game?
The probability that the first "tail" occurs on the kth toss is: Probability is the extent to which something is likely to happen or be the case[1]. Probability theory is used extensively in areas such as statistics, mathematics, science, philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. ...
 -
 -
 -
 How much can you expect to win, on average? With probability 1/2, you win 1 dollar; with probability 1/4 you win 2 dollars; with probability 1/8 you win 4 dollars etc. The expected value is thus In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...
 -
 -
 (Σ denotes the summation, see Sigma notation.) This sum diverges to infinity; "on average" you can expect to win an infinite amount of money when playing this game. Addition is one of the basic operations of arithmetic. ...
In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. ...
Yet, the probability that you win $1024 or more (i.e., 210 dollars) is less than one in a thousand.
According to traditional expected value theory, under this analysis of the game with the assumption that the casino has infinite resources, no matter how much you pay to enter (imagine paying $1 billion each time, and winning only a few dollars on nearly all occasions when you have paid that fee for the privilege) you will come out ahead in the long run, the idea being that on the very rare occasions when a large payoff comes along, it will far more than repay however much money you have paid to play. In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...
A decision theory using only this expected value would therefore suggest that any fee, no matter how high, would be worth paying for this opportunity. Published descriptions of the paradox, e.g. (Martin, 2004), generally express disbelief that real people would, in fact, pay large sums to enter this game. Martin quotes Ian Hacking as saying "few of us would pay even $25 to enter such a game" and says most commentators would agree. Decision theory is an interdisciplinary area of study, related to and of interest to practitioners in mathematics, statistics, economics, philosophy, management and psychology. ...
Solutions of the paradox There are different approaches 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 economics, utility is a measure of the relative happiness or satisfaction (gratification) gained by consuming different bundles of goods and services. ...
The expected utility hypothesis is the hypothesis in economics that the utility of an agent facing uncertainty is calculated by considering utility in each possible state and constructing a weighted average. ...
In economics, marginal utility is the additional utility (satisfaction or benefit) that a consumer derives from an additional unit of a commodity or service. ...
In 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.
Using a utility function, e.g., as suggested by Bernoulli himself, the logarithmic function u(x) = ln(x) (known as “log utility”[2]), the expected utility of the lottery (for simplicity assuming an initial wealth of zero) becomes finite: The ducat (IPA: ) is a gold coin that was used as a trade currency throughout Europe before World War I. Its weight is 3. ...
Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...
-
 (This particular utility function suggests that the lottery is as useful as 2 dollars.)
Before Daniel Bernoulli published, in 1728, another Swiss mathematician, Gabriel Cramer, found already parts of this idea (also motivated by the St. Petersburg Paradox) in stating that Events Astronomical aberration discovered by the astronomer James Bradley Swedish academy of sciences founded at Uppsala The founding of the University of Havana (Universidad de la Habana), Cubas most well-established university. ...
Gabriel Cramer Gabriel Cramer (July 31, 1704 - January 4, 1752) was a Swiss mathematician, born in Geneva. ...
- 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[3] 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.
The solution by Cramer and Bernoulli, however, is not yet completely satisfying, since the lottery can easily be changed in a way that the paradox reappears: To this aim, we just need to change the game so that it gives the (even larger) payoff  . Again, the game should be worth an infinite amount. More generally, one can find a lottery that allows for a variant of the St. Petersburg paradox for every unbounded utility function, as was first pointed out by (Menger, 1934).
There are basically two ways of solving this generalized paradox, which is sometimes called the Super St. Petersburg paradox:
- We can take into account that a casino would only offer lotteries with a finite expected value. Under this restriction, it has been proved that the St. Petersburg paradox disappears as long as the utility function is concave, which translates into the assumption that people are (at least for high stakes) risk averse [Compare (Arrow, 1974)].
- It is possible to assume an upper bound to the utility function. This does not mean that the utility function needs to be constant at some point, an example would be u(x) = 1 − e − x.
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 and Wang, 2006). In geometry, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions. ...
Risk aversion is a concept in economics and finance theory explaining the behaviour of consumers and investors under uncertainty. ...
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ...
Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel winner Daniel Kahneman, was an important figure in the development of behavioral finance and economics and continues to write extensively in the field. ...
Cumulative Prospect Theory is a model for descriptive decisions under risk which has been introduced by Amos Tversky and Daniel Kahneman in 1992 (Tversky, Kahneman, 1992). ...
Probability weighting 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. However, their experiments indicated that, very much to the contrary, people tend to overweight small probability events. Therefore the proposed solution by Nicolas Bernoulli is nowadays not considered to be satisfactory. Prospect theory was developed by Daniel Kahneman and Amos Tversky in 1979 as a psychologically realistic alternative to expected utility theory. ...
Daniel Kahneman Daniel Kahneman (born March 5, 1934 in Tel Aviv, in the then British Mandate of Palestine, now in Israel), is a key pioneer and theorist of behavioral finance, which integrates economics and cognitive science to explain seemingly irrational risk management behavior in human beings. ...
Amos Tversky (March 16, 1937 - June 2, 1996) was a pioneer of cognitive science, a longtime collaborator of Daniel Kahneman, and a key figure in the discovery of systematic human cognitive bias and handling of risk. ...
Finite St. Petersburg lotteries The classical St. Petersburg lottery assumes the casino has infinite resources. This assumption is often criticized as unrealistic, particularly in connection with the paradox, which involves the reactions of ordinary people to the lottery. Of course, the resources of an actual casino (or any other potential backer of the lottery) are finite. More importantly, the expected value of the lottery only grows logarithmically with the resources of the casino. As a result, the expected value of the lottery, even when played against a casino with the largest resources realistically conceivable, is quite modest. This can be seen from a consideration of the finite variant of the St. Petersburg lottery:
If the total resources of the casino are W dollars, then the expected value of the lottery becomes
-
 where L = 1 + floor(log2(W)). L is the maximum number of times the casino can play before it can no longer cover the next bet. The function log2(W) is the base-2 logarithm of W, which can be computed as log(W)/log(2) in any other base. The floor function gives the greatest integer less than or equal to its argument. The logarithm function becomes infinite as its argument becomes infinite, but does so very, very slowly. This logarithmic growth is the inverse behavior of exponential growth. Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...
The floor and fractional part functions In mathematics, the floor function of a real number x, denoted or floor(x), is the largest integer less than or equal to x (formally, ). For example, floor(2. ...
A graph of logarithmic growth In mathematics, logarithmic growth describes a phenomenon that whose size or cost can be described as a logarithm function of some input. ...
In mathematics, a quantity that grows exponentially is one whose growth rate is always proportional to its current size. ...
 A typical graph of average winnings over one course of a St. Petersburg Paradox lottery shows how occasional large payoffs lead to an overall very slow rise in average winnings. After 20,000 gameplays in this simulation the average winning per lottery was just under 8 dollars. The graph encapsulates the paradox of the lottery: The overall upward slope in the average winnings graph shows that average winnings tend upward to infinity, but the slowness of the rise in average winnings (a rise that becomes yet slower as gameplay progresses) indicates that a tremendously huge number of lottery plays will be required to reach average winnings of even modest size. The following table shows the expected value of the game with various potential backers and their bankroll: Image File history File links Download high resolution version (829x559, 8 KB) Summary Graph of average value per play over approximately 20,000 game plays of the St. ...
Image File history File links Download high resolution version (829x559, 8 KB) Summary Graph of average value per play over approximately 20,000 game plays of the St. ...
| Backer | Bankroll | Expected value of lottery | | Friendly game | $64 | $3.50 | | Millionaire | $1,050,000 | $10.50 | | Billionaire | $1,075,000,000 | $15.50 | | Bill Gates | $51,000,000,000 (2005) | $18.00 | | U.S. GDP | $11.7 trillion (2004) | $22.00 | | World GDP | $40.9 trillion (2004) | $23.00 | | Googolnaire | $10100 | $166.50 | Note: the slightly higher bankrolls for "millionaire" and "billionaire" allow a final round of play at those levels; otherwise for each, the maximum payout would be half as much and the expected value would be $0.50 less. A mansion on Diamond Head Road in Honolulu near Diamond Head State Park. ...
A billionaire is a person who has a net worth of at least one billion units of currency, such as United States Dollars (USD), Pounds or Euros. ...
For other persons named Bill Gates, see Bill Gates (disambiguation). ...
Countries by nominal GDP. Source: IMF (2005) This article includes a list of countries of the world sorted by their gross domestic product (GDP), the value of all final goods and services produced within a nation in a given year. ...
Countries by nominal GDP. Source: IMF (2005) This article includes a list of countries of the world sorted by their gross domestic product (GDP), the value of all final goods and services produced within a nation in a given year. ...
A “Googolnaire” is a hypothetical person worth a googol dollars ($10100). There are believed to be far fewer than a googol atoms in the observable universe, so even if each atom were worth one dollar, no one could be that rich and thus the value of the game can never get as high as $170 (assuming wealth in the form of physical assets, excluding for example, electronically held funds, which do not necessarily need to be materialised). Look up googol in Wiktionary, the free dictionary. ...
The observable Universe is a term used in cosmology to describe a ball-shaped region of space surrounding the Earth that is close enough that we might observe objects in it. ...
An average person might not find the lottery worth even the modest amounts in the above table, arguably showing that the naive decision model of the expected return causes the same problems as for the infinite lottery, however the possible discrepancy between theory and reality is far less dramatic.
The assumption of infinite resources can produce other apparent paradoxes in economics. See martingale (roulette system) and gambler's ruin. A separate article treats the topic of martingales in probability theory. ...
The basic meaning of gamblers ruin is a gamblers loss of the last of his bank of gambling money and consequent inability to continue gambling. ...
Iterated St. Petersburg lottery Players may assign a higher value to the game when the lottery is repeatedly played. This can be seen by simulating a typical series of lotteries and accumulating the returns, compare the illustration (right). In game theory, a repeated game (or iterated game) is an extensive form game which consists in some number of repetitions of some base game (called a stage game). ...
Further discussions The St. Petersburg paradox and the theory of marginal utility have been highly disputed in the past. For an interesting (but not always sound) contribution from the point of view of a philosopher, see (Martin, 2004).
See also In mathematics, a quantity that grows exponentially is one whose growth rate is always proportional to its current size. ...
The basic meaning of gamblers ruin is a gamblers loss of the last of his bank of gambling money and consequent inability to continue gambling. ...
A separate article treats the topic of martingales in probability theory. ...
References
Works cited ^ Arrow, Kenneth J. (February 1974). "The use of unbounded utility functions in expected-utility maximization: Response" (PDF). Quarterly Journal of Economics 88 (1): 136–138. Handle: RePEc:tpr:qjecon:v:88:y:1974:i:1:p:136-38. Kenneth Arrow Kenneth Joseph Arrow (born August 23, 1921) is an American economist, winner of the Bank of Sweden Prize in Economic Sciences in 1972. ...
^ Bernoulli, Daniel; Originally published in 1738; translated by Dr. Lousie Sommer. (January 1954). "Exposition of a New Theory on the Measurement of Risk". Econometrica 22 (1): 22–36. Retrieved on 2006-05-30. Daniel Bernoulli Daniel Bernoulli (Groningen, January 29, 1700 – Basel, March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland. ...
For the Manfred Mann album, see 2006 (album). ...
May 30 is the 150th day of the year in the Gregorian calendar (151st in leap years). ...
^ Martin, Robert. (July 26, 2004). "The St. Petersburg Paradox". The Stanford Encyclopedia of Philosophy (Fall 2004 Edition). Ed. Edward N. Zalta. Stanford, California: Stanford University. ISSN 1095-5054. Retrieved on 2006-05-30. Stanford is a census-designated place (CDP) located in Santa Clara County, California. ...
ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ...
For the Manfred Mann album, see 2006 (album). ...
May 30 is the 150th day of the year in the Gregorian calendar (151st in leap years). ...
^ Menger, Karl (August 1934). "Das Unsicherheitsmoment in der Wertlehre Betrachtungen im Anschluß an das sogenannte Petersburger Spiel". Zeitschrift für Nationalökonomie 5 (4): 459–485. DOI:10.1007/BF01311578. ISSN 0931-8658 (Paper) ISSN 1617-7134 (Online). Karl Menger Karl Menger (Vienna, Austria, January 13, 1902 – Highland Park, Illinois, USA, October 5, 1985) was a mathematician of great scope and depth. ...
A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ...
ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ...
ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ...
^ Rieger, Marc Oliver; Mei Wang (August 2006). "Cumulative prospect theory and the St. Petersburg paradox". Economic Theory 28 (3): 665–679. DOI:10.1007/s00199-005-0641-6. ISSN 0938-2259 (Paper) ISSN 1432-0479 (Online). A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ...
ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ...
ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ...
An older, publicly accessible version of the above paper may be found here:
For the Manfred Mann album, see 2006 (album). ...
May 30 is the 150th day of the year in the Gregorian calendar (151st in leap years). ...
Bibliography - Aumann, Robert J. (April 1977). "The St. Petersburg paradox: A discussion of some recent comments.". Journal of Economic Theory 14 (2): 443–445. DOI:10.1016/0022-0531(77)90143-0.
Israel Robert John Aumann (ישר×ל ×ומן) (born June 8, 1930) is an Israeli mathematician and a member of the United States National Academy of Sciences. ...
A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ...
The New School for Social Research is the graduate division of The New School. ...
For the Manfred Mann album, see 2006 (album). ...
May 30 is the 150th day of the year in the Gregorian calendar (151st in leap years). ...
External links |
Feeds |
Contact