St. Petersburg paradox

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 or by taking into account the finite resources of the participants. Some economists claim that the paradox is resolved 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).

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. 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. 

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&minus;1 dollars if the coin is tossed k times until the first tail appears.

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

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


 * $$p_k=\operatorname{prob}(\mbox{first tail on }k\mbox{th toss})$$


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


 * $$=\frac{1}{2}\cdot\frac{1}{2}\cdot ...\cdot\frac{1}{2}$$


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

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 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.

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., 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.

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 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”), the expected utility of the lottery (for simplicity assuming an initial wealth of zero) becomes finite:
 * $$EU=\sum_{k=1}^\infty p_k\cdot u(2^{k-1}) =\sum_{k=1}^\infty {\ln(2^{k-1}) \over {2^k}} = \ln 2 = u(2) < \infty$$

(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
 * 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 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 $$e^{2^k}$$. 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.

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 ].


 * 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.

Probability weighting
Nicolas Bernoulli himself proposed an alternative idea for solving the paradox. He conjectured that people will neglect unlikely events. 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.

One can't buy what isn't sold
Some economists attempt to resolve the paradox by arguing that, even if an entity had infinite resources, the game would never be offered. If the lottery represents an infinite expected gain to the player, then it also represents an infinite expected loss to the host. No one could be observed paying to play the game because it would never be offered. As Paul Samuelson describes the argument:
 * Paul will never be willing to give as much as Peter will demand for such a contract; and hence the indicated activity will take place at the equilibrium level of zero intensity.

Finite St. Petersburg lotteries
The classical St. Petersburg lottery assumes that 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 maximum payoff and therefore the maximum number of rounds is "capped", and the expected value of the lottery becomes
 * $$E =\sum_{k=1}^L p_k 2^{k-1}+2^{L-1}\sum_{k=L+1}^\infty p_k$$
 * $$=\sum_{k=1}^L{1 \over 2}+2^{L-1}(1-(1-{1 \over {2^L}}))={L+1 \over 2},$$

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.



The following table shows the expected value of the game with various potential backers and their bankroll:

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.