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.

The paradox

In a game of chance, you pay a fixed fee to enter, and then a fair coin will be tossed repeatedly until a "head" first appears. You win 1 cent if a head appears on the first toss, 2 cents if on the second, 4 cents if on the third, 8 cents if on the fourth, etc. It doubles with every toss. In short, you win

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2k−1 cents if the coin must be tossed k times.

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How much would you be willing to pay to enter the game?

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The probability that the first "head" occurs on the kth toss is:

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:p_k=operatorname{Pr}(mbox{first head on }kmbox{th toss})

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::=operatorname{Pr}(mbox{tail on 1st toss})cdot operatorname{Pr}(mbox{tail on 2nd toss})cdotsoperatorname{Pr}(mbox{head on }kmbox{th toss})

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::=rac{1}{2}cdotrac{1}{2}cdotsrac{1}{2}

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::=rac{1}{2^k}.

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How much can you expect to win, on average? With probability 1/2, you win 1 cent; with probability 1/4 you win 2 cents; with probability 1/8 you win 4 cents etc. The expected value is thus

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:E=rac{1}{2}cdot 1+rac{1}{4}cdot 2 + rac{1}{8}cdot 4 + cdots

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::=sum_{k=1}^infty p_k 2^{k-1} =sum_{k=1}^infty {1 over 2}=infty.

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(See summation notation for an explanation of the Σ notation.)

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This sum diverges to infinity; "on average" you can expect to win an infinite amount of money when playing this game.

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Yet, the probability that you win $10.24 or more (i.e., 210 cents) is less than one in a thousand. The probability that you win more than $1 is less than one in a hundred.

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According to traditional expected value theory, no matter how much you pay to enter (imagine paying $1 billion each time, and winning only a few cents on nearly all occasions when you have paid that fee for the privilege) you will come out ahead in the long run, the idea being that on the very rare occasions when a large payoff comes along, it will far more than repay all the hundreds of trillions of dollars you have paid to play.

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Decision theory applied naively without taking utility into account would suggest that any fee, no matter how high, would be worth paying for this opportunity. In practice, no reasonable person would pay more than a few cents to enter.

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