The St. Petersburg Paradox: Is MathWorld’s discussion confused?

MathWorld‘s page on the St. Petersburg paradox is misleading:

This paradox can clearly be resolved by making the distinction between the amount of the final payoff and the net amount won in the game. It is misleading to consider the payoff without taking into account the amount lost on previous bets, as can be shown as follows. At the time the player first wins (say, on the nth toss), he will have lost

\sum_{k=1}^{n-1}2^k=2^{n-2}
(3)

dollars. In this toss, however, he wins 2^n dollars. This means that the net gain for the player is a whopping $2, no matter how many tosses it takes to finally win. As expected, the large payoff after a long run of tails is exactly balanced by the large amount that the player has to invest.

The fundamental problem here is that the paradox is not mathematical in nature (unlike say this) and to treat it as such leads to confusion. The fundamental issue is not to explain how the unbounded expectation comes about, but how to resolve the unbounded expectation with the intuitive notion of fair value to pay to play the game. The attempt above to explain the unbounded expectation hinges on the idea that by the nth toss, 2^n-2 dollars have been “lost” by the player and  winning in the nth toss results in a net gain of only 2 dollars. This is fallacious because, the “payoff” given up by the player up until the n th toss is 2^{n-1}, i.e. what she could have earned by throwing an head on the (n-1) toss. The “net gain” from winning on the the nth toss is therefore 2^{n-1}. However, even this reasoning is wrong! The “final payoff” to the player by throwing a head on the nth toss is 2^{n} and the “net gain” is 2^{n} -c, where c is the amount paid by the player to enter the game. Assuming that the player has reached the nth toss of the St. Petersburg game, she has not won anything on any of the previous tosses and by the rules of the game “lost” only c to enter the game.

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The St. Petersburg Paradox: Cramer-Bernoulli Resolution

In my previous post, I referred to solutions of the St. Petersburg paradox. A more appropriate term for these would be resolutions, as more assumptions are required.  The oldest known resolution is attributed to the Swiss mathematician, Gabriel Cramer. In a letter to Nicolas Bernoulli, dated 21 May 1728, he proposed that the reason for the difference between what a reasonable man would pay for the St. Petersburg game (then not known by that name) and  the calculated expected value was due to his belief :

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 took this idea in two distinct directions. First, he proposed that there could be a limit beyond which any additional reward would not make the player any likelier to play. In his words:

This which renders the mathematical expectation infinite, this is the prodigious sum that I am able to receive, if Heads falls only very late, the 100th or 1000th toss. Now this sum, if I reason as a sensible man, is not more for me, does not make for more pleasure for me, does not engage me more to accept the game, than if it would be only 10 or 20 million coins. Suppose therefore that the total sum beyond 20 millions or (for more ease) beyond 16,777,216  coins, is equal to him or rather that I am never able to receive more than coins, however late comes Heads.

In the version of the game that Cramer was analysing, each payoff was half as much as the one in our game. His resolved fair value of 13 would correspond to 26 for our game (with a maximum chosen payoff of 33,554,432). In the post-script, he took the idea somewhat further and prefigured the work of Daniel Bernoulli on Marginal Utility.  He proposed that the utility of a payoff to the player only grows as the square root of the payoff. Under these conditions, the expected utility is bounded and the bound is given by \sum_{i=1}^\infty 2^{(i/2)} \cdot (1/2^i) = \genfrac{}{}{1pt}{}{{ \sqrt 2 }}{{ \sqrt 2 } - 1}.

Daniel Bernoulli published his resolution of the paradox in the Commentaries of the Imperial Academy of Science of  Saint Petersburg in 1738. This is the earliest known publication dealing with the paradox. In this, he proposed the first known formalization of the notion of marginal utility.  In his 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.

The utility, U, obtained as a result of earning a net reward, r, for a player with an existing wealth, w is \ln( w+ r ) - \ln( w ). Using this, the fair value, c, of the game to the player would be any value such that the expected utility, EU = \sum_{i=1}^\infty \frac{ \ln( w + 2^i - c ) - \ln( w ) }{ 2^i } < \infty, would be greater than 0.  The reasonableness of Bernoulli’s utility function is open to debate, but it is very commonly used.

The main problem with both of Cramer and Bernoulli’s resolutions is that they leave open the possibility of a paradox for similar games with more rapidly growing payoffs. To reintroduce the paradox of infinite expectation with Bernoulli’s utility function, one need just select a payoff of e^{2^i} at the ith coin toss. This paradox is commonly known as the Super St. Petersburg paradox. In fact, one can construct a payoff with infinite expected utility with any unbounded utility function.

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The St. Petersburg Paradox

The St. Petersburg paradox was first described by Daniel Bernoulli in the St. Petersburg Academy Proceedings (1738).  It involves computing the fair value of the following game (St. Petersburg game):

A fair coin is flipped until it comes up heads. The total number of flips, n, determines the prize, that equals 2^n.

It is relatively easy exercise to see that the expected value of the prize earned by a player of this game is \sum_{i=1}^\infty (1/2^i)\cdot 2^i = \infty. Typically, the expected value is the fair value of any game. Intuitively, however, this game does not seem to be worth that much. Therein lies the paradox. As should be clear, the nature of the paradox is philosophical, not mathematical. All solutions to the paradox, therefore, involve additional intuitively appealing assumptions. Some of these can be found in this article from The Stanford Encyclopedia of Philosophy. A more comprehensive set of solutions and objections can be found in the Wikipedia entry.

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