St. Petersburg paradox
In economics, the St. Petersburg paradox is a paradox related to probability theory and decision theory. It is based on a particular (theoretical) 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 criterion (which takes only the expected value into account) would recommend a course of action that no (real) rational person would be willing to take. Several resolutions are possible.
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 (Bernoulli 1738). However, the problem was invented by Daniel's cousin Nicolas Bernoulli who first stated it in a letter to Pierre Raymond de Montmort of September 9, 1713 (de Montmort 1713).
A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The pot starts at 1 dollar and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins whatever is in the pot. Thus the player wins 1 dollar if a tail appears on the first toss, 2 dollars if a head appears on the first toss and a tail on the second, 4 dollars if a head appears on the first two tosses and a tail on the third, 8 dollars if a head appears on the first three tosses and a tail on the fourth, and so on. In short, the player wins 2k−1 dollars if the coin is tossed k times until the first tail appears.
What would be a fair price to pay the casino for entering the game? To answer this we need to consider what would be the average payout: With probability 1/2, the player wins 1 dollar; with probability 1/4 the player wins 2 dollars; with probability 1/8 the player wins 4 dollars, and so on. The expected value is thus
Assuming the game can continue as long as the coin toss results in heads, in particular that the casino has unlimited resources, this sum grows without bound, and so the expected win for the player, at least in this idealized form, is an infinite amount of money. Considering nothing but the expectation value of the net change in one's monetary wealth, one should therefore play the game at any price if offered the opportunity. Yet, in published descriptions of the game, e.g., (Martin 2004), many people expressed disbelief in the result. Martin quotes Ian Hacking as saying "few of us would pay even $25 to enter such a game" and says most commentators would agree. The paradox is the discrepancy between what people seem willing to pay to enter the game and the infinite expected value.
Solutions of the paradox
There are different approaches for solving the paradox.
Expected utility theory
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 whose expected net change 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 utility of the lottery now converges to a finite value:
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 expected utility). For example, with log utility a millionaire should be willing to pay up to $10.94, a person with $1000 should pay up to $5.94, a person with $2 should pay up to $2, and a person with $0.60 should borrow $0.87 and pay up to $1.47.
Before Daniel Bernoulli published, in 1728, another Swiss mathematician, 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  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 the (even larger) payoff . 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).
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. 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.
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 probabilities 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 utility function is lower than the power coefficient of probability weighting function (Blavatskyy 2005). Intuitively, utility function must not simply be concave, but it must be concave relative to probability weighting function to avoid the St. Petersburg paradox.
Rejection of mathematical expectation
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 were its expectation enormous.
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. If the total resources (or total maximum jackpot) of the casino are W dollars, then L = 1 + floor(log2(W)) is the maximum number of times the casino can play before it no longer covers the next bet. The expected value E of the lottery then becomes:
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):
|Banker||Bankroll||Expected value of lottery|
|Bill Gates (2008)||$58,000,000,000||$18.84|
|U.S. GDP (2007)||$13.8 trillion||$22.78|
|World GDP (2007)||$54.3 trillion||$23.77|
Notes: The estimated net worth of Bill Gates is from Forbes. The GDP data are as estimated for 2007 by the International Monetary Fund, where one trillion dollars equals $1012 (one million times one million dollars). A “googolaire” is a hypothetical person worth a googol dollars ($10100).
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.
Despite the fact that this paradox is three centuries old, papers with new and fresh ideas on how to resolve it still pop up.
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)
Ole Peters thinks that the St. Petersburg paradox can be solved by using concepts and ideas from ergodic theory (Peters 2011a). In statistical mechanics it is a central problem to understand whether time averages resulting from a long observation of a single system are equivalent to expectation values. This is the case only for a very limited class of systems that are called "ergodic" there. For non-ergodic systems there is no general reason why expectation values should have any relevance.
Peters points out that computing the naive expected payout is mathematically equivalent to considering multiple outcomes of the same lottery in parallel universes. This is irrelevant to the individual considering whether to buy a ticket since he exists in only one universe and is unable to exchange resources with the others. It is therefore unclear why expected wealth should be a quantity whose maximization should lead to a sound decision theory. Indeed, the St. Petersburg paradox is only a paradox if one accepts the premise that rational actors seek to maximize their expected wealth. The classical resolution is to apply a utility function to the wealth, which reflects the notion that the "usefulness" of an amount of money depends on how much of it one already has, and then to maximise the expectation of this. The choice of utility function is often framed in terms of the individual's risk preferences and may vary between individuals: it therefore provides a somewhat arbitrary framework for the treatment of the problem.
An alternative premise, which is less arbitrary and makes fewer assumptions, is that the performance over time of an investment better characterises an investor's prospects and, therefore, better informs his investment decision. In this case, the passage of time is incorporated by identifying as the quantity of interest the average rate of exponential growth of the player's wealth in a single round of the lottery,
per round, where is the th (positive finite) payout and is the (non-zero) probability of receiving it. In the standard St. Petersburg lottery, and .
Although this is an expectation value of a growth rate, and may therefore be thought of in one sense as an average over parallel universes, it is in fact equivalent to the time average growth rate that would be obtained if repeated lotteries were played over time (Peters 2011a). While is identical to the rate of change of the expected logarithmic utility, it has been obtained without making any assumptions about the player's risk preferences or behaviour, other than that he is interested in the rate of growth of his wealth.
Under this paradigm, an individual with wealth should buy a ticket at a price provided
It should be noted that this strategy counsels against paying any amount of money for a ticket that admits the possibility of bankruptcy, i.e.
for any , since this generates a negatively divergent logarithm in the sum for which can be shown to dominate all other terms in the sum and guarantee that . If we assume the smallest payout is , then the individual will always be advised to decline the ticket at any price greater than
regardless of the payout structure of the lottery. The ticket price for which the expected growth rate falls to zero will be less than but may be greater than , indicating that borrowing money to purchase a ticket for more than one's wealth can be a sound decision. This would be the case, for example, where the smallest payout exceeds the player's current wealth, as it does in Menger's game.
It should also be noted in the above treatment that, contrary to Menger's analysis, no higher-paying lottery can generate a paradox which the time resolution - or, equivalently, Bernoulli's or Laplace's logarithmic resolutions - fail to resolve, since there is always a price at which the lottery should not be entered, even though for especially favourable lotteries this may be greater than one's worth.
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).
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