St. Petersburg paradox: a thought experiment about probability theory

This paradox, also known as "St. Petersburg Game", has a very important place in probability theory.
st. petersburg paradox; A thought experiment discussed with the belief that there is a deficiency in decision making theory was developed by a mathematician named daniel bernoulli.
the game is a coin toss game and ends at the first post
If the article comes in the first shot, the player earns $ 2, if it comes in the second shot, $ 4, if it comes in the third shot, eight dollars ... The interestingness of the game is the expected value of this game is infinite dollars. because the probability of writing in the first shot and the probability of making $ 2 is 1/2, the probability of the article coming in the second shot, not in the first shot, and the probability of winning $ 4, etc .. in this case the expected value of the game:

bd = (1/2) * 2 + (1/4) * 4 + (1/8) * 8 + (1/16) * 16 +…
bd = 1 + 1 + 1 + 1 +… ..

It will be.
the argument that this thought experiment creates a paradox asks the question
While people have to agree to pay huge amounts of entry fees to play such a game, why don't they?
the answers to this question are roughly based on two concepts
According to the first one, people's attitudes towards risk are not taken into account in the question. The expected value of this gambling (game) may be infinite, but people choose not to play this game because they are at risk-free. The utility function for a risk-avoiding person [for the definition of risk-attitudes within the framework of the utility theory (see risk likelihood)] may show a risk-avoidance feature that does not make it rational to play this gamble. For example, the amount of money x, u (x) is a human utility function, and can be characterized by u (x) = log (x). then the benefit that this person expects from this game: (expected utility)

eu = (1/2) * u (2) + (1/4) * u (4) +…
eu = (1/2) * log2 + (1/4) * log4 + (1/8) * log8 +….

and this value is not infinite, converging to log (4). this equals the benefit of a $ 4 risk-free one. so this person will only pay $ 4 as an entry fee to this game.
However, in response to this view, which states that it brings a solution to the paradox, the same person is asked the question of what will happen if it is promised that he will win the same game for the first time, not $ 2, $ 10 ^ 2, if he comes in the second, $ 10 ^ 4, etc. In this case, the expected benefit of the same person from this game will be eu = (1/2) log100 + (1/4) log10000 +… = 1 + 1 + 1… = infinite, and this time this person will consider an infinite amount as an entry fee to this gamble. must. it can only be said that people take more risks than big gambling attitudes towards this view; For example, many people can afford to flip a coin for $ 10, but they cannot afford to flip a coin for $ 1000; In this case, his attitude towards risk has changed in the direction of risk-dislike. this is only a determination, it is not able to save the decision making theory much more than the dead end, because it is based on a basis that people's utility functions are variable.
the problem with the game is that it has infinitely few possibilities, but has an infinite number of results (such as the post coming in the 500th shot)
cutting it out somewhere will fix the problem. Parallel to this, it can be said that there is not much difference between $ 1 billion and $ 10 billion for man (in fact, this has to do with risk avoidance and diminishing marginal utility concepts, not a very different perspective). It is obvious that the casino will not allow an infinite number of shots for a person who knows that the casino has 1 billion dollars in the safe and knows that the casino can pay the maximum amount. the casino will allow a maximum of 30 shots (2 ^ 30 = 1 billion). In this case, the expected value of the game, which we have calculated infinitely above, will drop to $ 30. Even a person with a neutral attitude to risk gives this gambler a maximum of $ 30.

although this explanation seems to solve the situation, it was subject to the objection "but we were talking about a situation where infinite money can be paid, this solution changes the problem".


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