St. Petersburg paradox

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 Daniel 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.

A common utility model, suggested by Bernoulli himself, is the logarithmic function U(w) = ln(w) (known as log utility). It is a function of the gambler's total wealth w, and the concept of diminishing marginal utility of money is built into it. The expected utility hypothesis posits that a utility function exists the sign of whose expected net change from accepting the gamble is a good criterion for real people's behavior. For each possible event, the change in utility ln(wealth after the event) − ln(wealth before the event) will be weighted by the probability of that event occurring. Let c be the cost charged to enter the game. The expected incremental utility of the lottery now converges to a finite value:

${\displaystyle \Delta E(U)=\sum _{k=1}^{+\infty }{\frac {1}{2^{k}}}\left[\ln \left(w+2^{k}-c\right)-\ln(w)\right]<+\infty \,.}$

This formula gives an implicit relationship between the gambler's wealth and how much he should be willing to pay to play (specifically, any c that gives a positive change in expected utility). For example, with natural log utility, a millionaire ($1,000,000) should be willing to pay up to $20.88, a person with $1,000 should pay up to $10.95, a person with $2 should borrow $1.35 and pay up to $3.35.

Before Daniel Bernoulli published, in 1728, a mathematician from Geneva, Gabriel Cramer, had already found 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[7] 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.

This solution by Cramer and Bernoulli, however, is not completely satisfying, since the lottery can easily be changed in a way such that the paradox reappears. To this aim, we just need to change the game so that it gives even more rapidly increasing payoffs. For any unbounded utility function, one can find a lottery that allows for a variant of the St. Petersburg paradox, as was first pointed out by Menger (Menger 1934).

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 (Rieger & Wang 2006).

Nicolas Bernoulli himself proposed an alternative idea for solving the paradox. He conjectured that people will neglect unlikely events (de Montmort 1713). 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.

Cumulative prospect theory is one popular generalization of expected utility theory that can predict many behavioral regularities (Tversky & Kahneman 1992). However, the overweighting of small probability events introduced in cumulative prospect theory may restore the St. Petersburg paradox. Cumulative prospect theory avoids the St. Petersburg paradox only when the power coefficient of the utility function is lower than the power coefficient of the probability weighting function (Blavatskyy 2005). Intuitively, the utility function must not simply be concave, but it must be concave relative to the probability weighting function to avoid the St. Petersburg paradox. One can argue that the formulas for the prospect theory are obtained in the region of less than $400 (Tversky & Kahneman 1992). This is not applicable for infinitely increasing sums in the St. Petersburg paradox.

Various authors, including Jean le Rond d'Alembert and John Maynard Keynes, have rejected maximization of expectation (even of utility) as a proper rule of conduct. Keynes, in particular, insisted that the relative risk of an alternative could be sufficiently high to reject it even if its expectation were enormous.

The classical St. Petersburg lottery assumes that the casino has infinite resources. This assumption is 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. If the total resources (or total maximum jackpot) of the casino are W dollars, then L = floor(log₂(W)) is the maximum number of times the casino can play before it no longer fully covers the next bet. The expected value E of the lottery then becomes:

${\displaystyle {\begin{aligned}E&=\sum _{k=1}^{L}{\frac {1}{2^{k}}}\cdot 2^{k}+\sum _{k=L+1}^{+\infty }{\frac {1}{2^{k}}}\cdot W\\&=L+{\frac {W}{2^{L}}}\,.\end{aligned}}}$

The following table shows the expected value E of the game with various potential bankers and their bankroll W (with the assumption that if you win more than the bankroll you will be paid what the bank has):

A rational person might not find the lottery worth even the modest amounts in the above table, suggesting that the naive decision model of the expected return causes essentially the same problems as for the infinite lottery. Even so, the possible discrepancy between theory and reality is far less dramatic.

The premise of infinite resources produces a variety of paradoxes in economics. In the martingale betting system, a gambler betting on a tossed coin doubles his bet after every loss, so that an eventual win would cover all losses; this system fails with any finite bankroll. The gambler's ruin concept shows a persistent gambler will go broke, even if the game provides a positive expected value, and no betting system can avoid this inevitability.

A mathematically correct solution involving sampling was offered by William Feller.[10] In order to understand Feller's answer correctly, sufficient knowledge about probability theory and statistics is necessary, but it can be understood intuitively "to perform this game with a large number of people and calculate the expected value from the sample extraction". In this method, when the games of infinite number of times are possible, the expected value will be infinity, and in the case of finite, the expected value will be a much smaller value.

Samuelson resolves 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." (Samuelson 1960)

The St. Petersburg paradox and the theory of marginal utility have been highly disputed in the past. For a discussion from the point of view of a philosopher, see (Martin 2004).

Recently some authors suggested using heuristic parameters [11] (e.g. assessing the possible gains without neglecting the risks of the Saint Petersburg lottery) because of the highly stochastic context of this game (Cappiello 2016). The expected output should therefore be assessed in the limited period where we can likely make our choices and, besides the non-ergodic features (Peters 2011a), considering some inappropriate consequences we could attribute to the expected value (Feller 1968).

• Aumann, Robert J. (April 1977). "The St. Petersburg paradox: A discussion of some recent comments". Journal of Economic Theory. 14 (2): 443–445. doi:10.1016/0022-0531(77)90143-0.
• Feller, William (1968). An Introduction to Probability Theory and its Applications Volume I, II. Wiley. ISBN 978-0471257080.
• Durand, David (September 1957). "Growth Stocks and the Petersburg Paradox". The Journal of Finance. 12 (3): 348–363. doi:10.2307/2976852. JSTOR 2976852.

• "Bernoulli and the St. Petersburg Paradox". The History of Economic Thought. The New School for Social Research, New York. Archived from the original on June 18, 2006. Retrieved May 30, 2006.

• Haigh, John (1999). Taking Chances. Oxford, UK: Oxford University Press. pp. 330. ISBN 978-0198526636.(Chapter 4)
• Sen, P.K.; Singer, J.M. (1993). Large Sample Methods in Statistics. An Introduction with Applications. New York: Springer. ISBN 978-0412042218.