- Birth of the concept of [effect
The St. Petersburg paradox is one of the paradoxes in decision theory. It is a paradox that arises when the expected value diverges in cases where a very large profit can be obtained with a very small probability.
- [St. Petersburg paradox - Wikipedia https://ja.wikipedia.org/wiki/%E3%82%B5%E3%83%B3%E3%82%AF%E3%83%88%E3%83%9A%E3%83%86%E3%83%AB%E3%83%96 %E3%83%AB%E3%82%AF%E3%81%AE%E3%83%91%E3%83%A9%E3%83%89%E3%83%83%E3%82%AF%E3%82%B9]
In 1738, Daniel Bernoulli, who lived in St. Petersburg, published in his article “A New Theory on Measuring Risk” in the journal “Transactions of the Petersburg Imperial Academy”. The purpose was to show that the classical “fairness” of expected values was not necessarily applicable in reality, and to develop a new theory of “utility” (Latin: emolumentum). - Some say it was Nicholas Bernoulli, son of Nicholas, but I’m not sure of the facts.
The St. Petersburg Paradox (Stanford Encyclopedia of Philosophy/Winter 2017 Edition)
Bernoulli avoided this paradox by defining “effect”, which could be called subjective value.
This is the same idea as diminishing marginal utility in current economics.
Bernoulli further stated that utility is the logarithm of the amount of money. The utility obtained by the logarithmic function is called “logarithmic utility. The utility of an increase in assets is inversely proportional to the total amount of assets
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This is called “Bernoulli’s rule”.
No matter how gradual the increase in utility, if the prize amount is set so that “utility doubles” for each backstop, the expected value of utility will still diverge. To prevent this completely, we need to consider that utility has an upper limit. That is, once a certain amount is exceeded, utility essentially cannot increase any further. The upper limit of utility means the state in which all desires that can be bought with money are satisfied, which is called the “bliss level.
- d’Alembert “The assumption that the game can continue indefinitely cannot be real” I can do endless rounds.
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