St. Petersburg paradox

In probability theory and decision theory the St. Petersburg paradox is a paradox that exhibits a random variable whose value is probably very small, and yet has an infinite expected value. This poses a situation where decision theory may superficially appear to recommend a course of action that no rational person would be willing to take. That appearance evaporates when utilities are taken into account.

Proposed solutions

Familiarity with the paradox leads to a deeper understanding of a variety of issues in economics and decision theory.

Related Topics:
Paradox - Economics - Decision theory

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Bernoulli's original insight was the diminishing marginal utility of money. For example, 9 trillion dollars is not much more useful than 900 billion dollars, despite being ten times as large. Therefore, a one-in-900,000,000,000 chance of earning 900,000,000,000 cents is not worth even the one cent that naive decision theory says that it is.

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A way around that solution is to change the game so that it offers a quantity of utility (enough money, lifespan, knowledge, etc., arranged so that each prize is worth twice as much as the last) rather than money. In this case, the game should be worth an infinite amount. Possibly, however, there is an upper bound to the amount of utility that a person can have.

Related Topics:
Utility - Upper bound

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The following issues also need to be considered:

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• No person has the time and money necessary to play over the long run, or even a good approximation of it;
• There is a real chance of breaking the bank. It is unlikely that the "house" can afford $90,000,000, let alone 9 trillion dollars.
• If the payout is capped at $0.01, the expected value of the game is $0.01; capped at $10.24, the game is worth $0.06; at $85,900,000, $0.18; at 11 ¼ trillion dollars, $0.26.
• Risk aversion;
• The gestalt of factors that are not simply represented in mathematical models but which provide human decision-making with its context.