Saint Petersburg Paradox

The Saint Petersburg Paradox (also Saint Petersburg Lottery ) describes a paradox in a game of chance . The random variable here has an infinite expected value , which is equivalent to an infinitely large expected payoff. Even so, starting the game seems to be worth only a small amount of money. The St. Petersburg Paradox is a classic situation in which a naive decision theory that only uses the expected value as a criterion would recommend a decision that no (real) rational person would make. The paradox can be solved by refining the decision model by using a utility function or by looking at finite variants of the lottery.

The paradox got its name from Daniel Bernoulli's presentation of the problem and its solution, which he published in the Commentarii Academiae Scientiarum Imperialis Petropolitanae ( Saint Petersburg ) in 1738 . Nikolaus I Bernoulli mentioned the problem as early as 1713 in an exchange of letters with Pierre Rémond de Montmort . In the original depiction, this story takes place in a hypothetical casino in Saint Petersburg, hence the name of the paradox.

The paradox

In a game of chance for which a participation fee is charged, a fair coin is tossed until the first "head" falls. This ends the game. The profit depends on the total number of coins tossed. If it was only one, the player receives 1 euro. With two tosses (ie one “tails”, one “heads”) he receives 2 euros, with three tosses 4 euros, with four tosses 8 euros and the amount doubles with each subsequent throw. So you win euros if the coin has been tossed twice. ${\ displaystyle 2 ^ {k-1}}$ ${\ displaystyle k}$

What amount of money would you want to pay to play this game?

Let be the probability that tails will fall on the -th coin toss and the probability that heads will fall on the -th coin toss. You get to the th roll exactly when you have thrown a number before . So the probability is that the first time "heads" will fall on the th coin toss: ${\ displaystyle P (Z_ {i})}$ ${\ displaystyle i}$ ${\ displaystyle P (K_ {i})}$ ${\ displaystyle i}$ ${\ displaystyle k}$ ${\ displaystyle k-1}$ ${\ displaystyle k}$

{\ displaystyle p_ {k} = P (Z_ {1}) \ cdot P (Z_ {2}) \ dotsm P (Z_ {k-1}) \ cdot P (K_ {k}) = {\ frac {1 } {2}} \ cdot {\ frac {1} {2}} \ dotsm {\ frac {1} {2}} = {\ frac {1} {2 ^ {k}}}.}

How much can you expect to win on average? With probability 1/2 the profit is 1 euro, with probability 1/4 it is 2 euros, with probability 1/8 it is 4 euros etc. The expected value is therefore

{\ displaystyle E = {\ frac {1} {2}} \ cdot 1 + {\ frac {1} {4}} \ cdot 2 + {\ frac {1} {8}} \ cdot 4+ \ dotsb = \ sum _ {k = 1} ^ {\ infty} {\ frac {1} {2 ^ {k}}} \ cdot 2 ^ {k-1} = \ sum _ {k = 1} ^ {\ infty} {1 \ over 2} = \ infty.}

This sum diverges towards infinity, that is, on average one expects an infinitely high profit.

However, the probability is e.g. B. To win 512 euros or more, very small, namely just 1: 512 (1: 1024 for at least 1024 euros).

According to a decision theory based on expected value, one should therefore accept any entry fee. Of course, this contradicts an actual decision and also seems to be irrational, since you usually only win a few euros. This apparently paradoxical discrepancy led to the name Saint Petersburg Paradox .

Solutions to the paradox

There are several approaches to solving this paradox.

Expected utility theory

Economists use this paradox to demonstrate concepts in decision theory. The paradox is solved here by the naive decision theory, based on the expected value by the (sensible) expected utility is replaced (Expected Utility Theory).

This theory of the diminishing marginal utility of money was already recognized by Bernoulli. The main idea here is that an amount of money is valued differently : For example, the relative difference in the (subjective) usefulness of 2 trillion euros to 1 trillion euros is certainly smaller than the corresponding difference between 1 trillion euros and no money at all. So the relationship between monetary value and utility is non-linear. If you generalize this idea, you have a 1: 100,000,000,000 chance of winning 100,000,000,000 euros, an expected value of one euro, but it does not necessarily have to be worth one euro.

If we now use a utility function, such as the logarithm function proposed by Bernoulli , then the Saint Petersburg lottery has a finite value: ${\ displaystyle u (x) = \ ln (x)}$

{\ displaystyle \ operatorname {E} [U] = \ sum _ {k = 1} ^ {\ infty} p_ {k} u (2 ^ {k-1}) = \ sum _ {k = 1} ^ { \ infty} {\ ln (2 ^ {k-1}) \ over {2 ^ {k}}} = \ ln (2) <\ infty.}

In Bernoulli's own words:

"The determination of the value of a thing should not be based on its price, but on the benefit it brings ... Undoubtedly, a gain of 1,000 ducats is more important for a beggar than for a wealthy, although both receive the same amount."

However, this solution is still not entirely satisfactory, as the lottery can be changed in a way that the paradox will reappear: for this we just have to change the lottery so that the payouts are, then the value of the lottery is calculated using the logarithmic Utility functions, again infinite. ${\ displaystyle e ^ {2 ^ {k}}}$

In general, for every unbounded utility function, one can find a variant of the Saint Petersburg paradox that delivers an infinite value, as the Austrian mathematician Karl Menger was the first to notice.

There are now two main ways of resolving this new paradox, sometimes called the Super Saint Petersburg Paradox :

One can take into account that a casino would only offer lotteries with a finite expected value. Under this assumption it can be shown that the paradox disappears if the utility function is concave , which means that one assumes a risk aversion (at least for large amounts of money).

One can assume that the utility function has an upper bound. This does not mean that the utility function has to be constant above a certain value. Consider as an example . ${\ displaystyle u (x) = 1-e ^ {- x}}$

In recent years, the expected utility theory has been expanded to include decision models that quantitatively describe the real behavior of test persons better. In some of these new theories, such as the Cumulative Prospect Theory , the Saint Petersburg Paradox also appears in some cases when the utility function is concave and the expectation is finite, but not when the utility function is bounded.

Final Saint Petersburg lottery

In the classic variant of the Saint Petersburg lottery, the casino has unlimited cash reserves. So there is no win that the casino cannot pay out and the game can go on for as long as you want.

If, on the other hand, one assumes a real casino with a capital of , then the casino cannot pay out more than a maximum profit. If the player reaches the resulting limit of coin flips, the winnings are paid out to him at this point and the game is aborted. The casino sets this limit in advance. ${\ displaystyle K}$ ${\ displaystyle N}$ ${\ displaystyle N}$

One now receives a finite expectation value. The formula is used for the calculation

{\ displaystyle E = \ sum _ {k = 1} ^ {N} p_ {k} 2 ^ {k-1} + 2 ^ {N-1} \ sum _ {k = N + 1} ^ {\ infty } p_ {k} = \ sum _ {k = 1} ^ {N} {1 \ over 2} + 2 ^ {N-1} \ left (1- \ left (1- {1 \ over {2 ^ { N}}} \ right) \ right) \ = \ {N + 1 \ over 2},}

with . ${\ displaystyle N = 1 + \ lfloor \ log _ {2} (K) \ rfloor}$

The following table shows the expected values of the Saint Petersburg finite lottery for different casino types:

Casino Fund K	Max. Game length N	Expected value E
100 €	7th	4 €	Game among friends
€ 100 million	27	14 €	(normal) casino
€ 18 trillion	44	€ 22.50	EU GDP 2009

Web links

The New School for Social Research, New York, " The history of economic thought " (in English)
Martin, Robert (2004), “ The St. Petersburg Paradox ”. In The Stanford Encyclopedia of Philosophy (Fall 2004 Edition), Edward N. Zalta (ed.) (In English)
Online St. Petersburg lottery
Daniel Bernoulli (1738), Theoriae Novae De Mensura Sortis , translated as “Exposition of a New Theory on the Measurement of Risk”, Econometrica Vol. 22 (1954), pp. 23–36 (version on the Internet: http://www.math.fau.edu/richman/Ideas/daniel.htm )
Robert Aumann , “The St. Petersburg paradox: A discussion of some recent comments”, Journal of Economic Theory , 1977, Vol. 14, pp. 443-445

Individual evidence

↑ For an overview see z. B. Hens, Thorsten and Rieger, Marc Oliver (2016): Financial Economics: A Concise Introduction to Classical and Behavioral Finance , Springer-Verlag, Chapter 2.
^ Karl Menger (1934): The moment of uncertainty in the theory of values , Z. Nationalokon. , Vol. 51, pp. 459-485.
↑ See Arrow, Kenneth J. (1974), "The use of unbounded utility functions in expected-utility maximization: Response", Quarterly Journal of Economics , Vol. 88, pp. 136-138.
^ Rieger, Marc Oliver and Wang, Mei (2006), “Cumulative prospect theory and the St. Petersburg paradox”, Economic Theory , Vol. 28, issue 3, pp. 665-679.

[1] For an overview see z. B. Hens, Thorsten and Rieger, Marc Oliver (2016): Financial Economics: A Concise Introduction to Classical and Behavioral Finance , Springer-Verlag, Chapter 2.

[2] Karl Menger (1934): The moment of uncertainty in the theory of values , Z. Nationalokon. , Vol. 51, pp. 459-485.

[3] See Arrow, Kenneth J. (1974), "The use of unbounded utility functions in expected-utility maximization: Response", Quarterly Journal of Economics , Vol. 88, pp. 136-138.

[4] Rieger, Marc Oliver and Wang, Mei (2006), “Cumulative prospect theory and the St. Petersburg paradox”, Economic Theory , Vol. 28, issue 3, pp. 665-679.