St. Petersburg paradox

In probability theory and decision theory the St. Petersburg paradox is a paradox that exhibits a random variable whose value is probably very small, and yet has an infinite expected value. This poses a situation where decision theory may superficially appear to recommend a course of action that no rational person would be willing to take. That appearance evaporates when utilities are taken into account. The paradox is named from Daniel Bernoulli's original solution, published in 1738 in the Commentaries of the Imperial Academy of Science of Saint Petersburg.

Contents

1 The paradox 2 Proposed solutions 3 See also 4 External link 5 Reference

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 "head" first appears. You win 1 cent if a head appears on the first toss, 2 cents if on the second, 4 cents if on the third, 8 cents if on the fourth, etc. It doubles with every toss. In short, you win 2^k−1 cents if the coin must be tossed k times.

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

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

<math>p_k=\operatorname{Pr}(\mbox{first head on }k\mbox{th toss})<math>

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

<math>=\frac{1}{2}\cdot\frac{1}{2}\cdots\frac{1}{2}<math>

<math>=\frac{1}{2^k}.<math>

How much can you expect to win, on average? With probability 1/2, you win 1 cent; with probability 1/4 you win 2 cents; with probability 1/8 you win 4 cents etc. The expected value is thus

<math>E=\frac{1}{2}\cdot 1+\frac{1}{4}\cdot 2 + \frac{1}{8}\cdot 4 + \cdots<math>

<math>=\sum_{k=1}^\infty p_k 2^{k-1} =\sum_{k=1}^\infty {1 \over 2}=\infty.<math>

(See summation notation for an explanation of the Σ 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 $10.24 or more (i.e., 2¹⁰ cents) is less than one in a thousand. The probability that you win more than $1 is less than one in a hundred.

According to traditional expected value theory, no matter how much you pay to enter (imagine paying $1 billion each time, and winning only a few cents 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 all the hundreds of trillions of dollars you have paid to play.

Decision theory applied naively without taking utility into account would 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 cents to enter.

Proposed solutions

Familiarity with the paradox leads to a deeper understanding of a variety of issues in economics and decision theory.

Bernoulli's original insight was the diminishing marginal utility of money. For example, 9 trillion dollars is not much more useful than 900 billion dollars, despite being ten times as large. Therefore, a one-in-900,000,000,000 chance of earning 900,000,000,000 cents is not worth even the one cent that naive decision theory says that it is.

A way around that solution is to change the game so that it offers a quantity of utility (enough money, lifespan, knowledge, etc., arranged so that each prize is worth twice as much as the last) rather than money. In this case, the game should be worth an infinite amount. Possibly, however, there is an upper bound to the amount of utility that a person can have.

The following issues also need to be considered:

• No person has the time and money necessary to play over the long run, or even a good approximation of it;
• There is a real chance of breaking the bank. It is unlikely that the "house" can afford $90,000,000, let alone 9 trillion dollars.
• If the payout is capped at $0.01, the expected value of the game is $0.01; capped at $10.24, the game is worth $0.06; at $85,900,000, $0.18; at 11 ¼ trillion dollars, $0.26.
• Risk aversion;
• The gestalt of factors that are not simply represented in mathematical models but which provide human decision-making with its context.

See also

• Gambler's fallacy

External link

For a fuller treatment see:

• St Petersburg Paradox - Stanford Encyclopaedia of Philosophy (http://plato.stanford.edu/entries/paradox-stpetersburg/)

Reference

A translation of Bernoulli's original presentation is found in:

• Bernoulli, Daniel: 1738, "Exposition of a New Theory on the Measurement of Risk", Econometrica vol 22 (1954), pp23-36.de:Sankt-Petersburg-Paradoxon

es:Paradoja de San Petersburgo fr:Paradoxe de Saint-Pétersbourg