Paradosso di San Pietroburgo

Nella teoria della probabilità e nella teoria delle decisioni, il paradosso di San Pietroburgo descrive un particolare gioco d'azzardo basato su una variabile casuale con valore atteso infinito, cioè con una vincita media di valore infinito. Ciononostante, ragionevolmente, si considera adeguata solo una minima somma, da pagare per partecipare al gioco.

Il paradosso di SanPietroburgo è la classica situazione in cui una ingenua applicazione della teoria delle decisioni (che tiene conto solo del guadagno atteso) suggerisce una linea di condotta che nessuna persona rogionevole si sentirebbe di adottare. Il paradosso si risolve raffinando il modello di decisione e prendendo in considerazione il concetto di utilità marginale e il fatto che le risorse dei partecipanti sono limitate (non infinite).

Il paradosso prende il nome dalla presentazione del problema da parte di Daniel Bernoulli, nel 1738 in the Commentaries of the Imperial Academy of Science of Saint Petersburg. Comunque il problema fu inventato dal cugino di Daniel, Nicolas Bernoulli, che per primo lo enunciò in una lettera a Pierre Raymond de Montmort fin dal 9 Settembre 1713. [1]

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 "tails" 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 2^k−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.)

How much would you be willing to pay to enter the game?

The probability that the first "tail" occurs on the kth toss is:

$p_{k}=\operatorname {Pr} ({\mbox{first tail on }}k{\mbox{th toss}})$

$=\operatorname {Pr} ({\mbox{head on 1st toss}})\cdot \operatorname {Pr} ({\mbox{head on 2nd toss}})\cdots \operatorname {Pr} ({\mbox{tail on }}k{\mbox{th toss}})$

$={\frac {1}{2}}\cdot {\frac {1}{2}}\cdots {\frac {1}{2}}$

$={\frac {1}{2^{k}}}.$

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

$E={\frac {1}{2}}\cdot 1+{\frac {1}{4}}\cdot 2+{\frac {1}{8}}\cdot 4+{\frac {1}{16}}\cdot 8+\cdots$

$={\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{2}}+\cdots$

$=\sum _{k=1}^{\infty }{1 \over 2}=\infty .$

(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.

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, 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.

A naive decision theory using only this expected value would therefore suggest that any fee, no matter how high, would be worth paying for this opportunity. In practice, no reasonable person would pay more than a few dollars to enter. This seemingly paradoxical difference led to the name St. Petersburg paradox.

Solutions of the paradox

There are different approaches for solving the "paradox".

Expected utility theory

Economists use the paradox to illuminate a variety of issues in economics and decision theory. The paradox is thereby solved by replacing the naive decision theory (expected value) by the more reasonable Expected Utility Theory.

This diminishing marginal utility of money was already an insight of Bernoulli. The main idea is that twice the money does not need to be twice as good: For example, 2 trillion dollars are not much more useful than 1 trillion dollars, despite being twice the amount. Generalizing this idea, a one-in-100,000,000,000 chance of earning 100,000,000,000 dollars has an expected value of 1, but it is still not worth even this one dollar.

Using a utility function, e.g., as suggested by Bernoulli himself, the logarithmic function u(x)=ln(x), the expected utility of the lottery (for simplicity assuming an initial wealth of zero) becomes finite:

$EU=\sum _{k=1}^{\infty }p_{k}u(2^{k-1})=\sum _{k=1}^{\infty }{\ln(2^{k-1}) \over {2^{k}}}<\infty .$

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."

A couple of years before Daniel Bernoulli, in 1728, another Swiss mathematician, Gabriel Cramer, found already 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 [2] 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. It is interesting to notice that in considering only gains, rather than the final wealth, he anticipated the key idea of the framing effect, found 250 years later, which nowadays plays an important role in Behavioral finance.

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 e^{$2^{k}$}. 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 Template:Ref harvard.

There are basically two ways of solving this new 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 Template:Ref harvard].

• 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 Template:Ref harvard.

Probability weighting

Nicolas Bernoulli himself proposed an alternative idea for solving the paradox. He conjectured that people will neglect unlikely events[3]. 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 demonstrated 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 valid.

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

$E=\sum _{k=1}^{L}p_{k}2^{k-1}=\sum _{k=1}^{L}{1 \over 2}={L \over 2},$

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.

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:

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 "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).

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.

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).

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 Template:Ref harvard.

See also

• Exponential growth
• Gambler's ruin
• Martingale (roulette system)

References

Works cited

Template:Note label Kenneth J. Arrow, The use of unbounded utility functions in expected-utility maximization: Response, in Quarterly Journal of Economics, vol. 88, n. 1, February 1974, pp. 136–138, Handle: RePEc:tpr:qjecon:v:88:y:1974:i:1:p:136-38.

Template:Note label Daniel Bernoulli, Originally published in 1738; translated by Dr. Lousie Sommer., Exposition of a New Theory on the Measurement of Risk, in Econometrica, vol. 22, n. 1, January 1954, pp. 22–36. URL consultato il 30 maggio 2006.

Template:Note label<cite id="CITEREFISSN 1095-5054" class="citation libro" style="font-style:normal"> Robert Martin, The St. Petersburg Paradox, in Edward N. Zalta (a cura di), The Stanford Encyclopedia of Philosophy, Fall 2004 Edition, Stanford, California, Stanford University, July 26, 2004. URL consultato il 30 maggio 2006.

Template:Note label Karl Menger, Das Unsicherheitsmoment in der Wertlehre Betrachtungen im Anschluß an das sogenannte Petersburger Spiel, in Zeitschrift für Nationalökonomie, vol. 5, n. 4, August 1934, pp. 459–485, DOI:10.1007/BF01311578, ISSN 0931-8658 (Paper) ISSN 1617-7134 (Online).

Template:Note label Marc Oliver Rieger, Mei Wang, Cumulative prospect theory and the St. Petersburg paradox, in Economic Theory, vol. 28, n. 3, August 2006, pp. 665–679, DOI:10.1007/s00199-005-0641-6, ISSN 0938-2259 (Paper) ISSN 1432-0479 (Online).

An older, publicly accessible version of the above paper may be found here:

• Rieger, Marc Oliver and Wang, Mei, Cumulative Prospect Theory and the St.Petersburg Paradox (PDF), Sonderforschungsbereich 504 Publications Number 04-28, July 2004. URL consultato il 30 maggio 2006.

Bibliography

• Robert J. Aumann, The St. Petersburg paradox: A discussion of some recent comments., in Journal of Economic Theory, vol. 14, n. 2, April 1977, pp. 443–445, DOI:10.1016/0022-0531(77)90143-0.

• David Durand, Growth Stocks and the Petersburg Paradox., in The Journal of Finance, vol. 12, n. 3, September 1957, pp. 348–363.

• Bernoulli and the St. Petersburg Paradox, in The History of Economic Thought, The New School for Social Research, New York. URL consultato il 30 maggio 2006.

External links

• Online simulation of the St. Petersburg lottery