St. Petersburg paradox

St. Petersburg paradox

In probability theory and decision theory the St. Petersburg paradox describes that a certain lottery game (sometimes called St. Petersburg Lottery) 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 naive 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 or by taking into account the finite resources of the participants. The paradox is named from Daniel Bernoullis original presentation of the problem and his solution, published in 1738 in the Commentaries of the Imperial Academy of Science of Saint Petersburg.

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 ails 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 must be 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 ail occurs on the kth toss is:

 
 
 
 
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

 
 
(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 as utility function, e.g., as suggested by Bernoulli himself, the logarithmic function u(x)=ln(x), the expected utility of the lottery becomes finite:

 
In Bernoullis 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.
This solution, 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 e2k. 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 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 [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 descriptive 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].

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.

 
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 illionaire 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.

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 gamblers ruin.

Iterated St. Petersburg lottery
The value of the game obviously changes when the lottery is repeatedly played. This can already be seen from 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 [Martin, 2004].

See also
Exponential growth
Gamblers ruin
Martingale (roulette system)