Paradox 1… St. Petersburg

In a game of chance, you pay a fixed fee to enter, and then a fair coin will be tossed repeatedly until a tail first appears, ending the game. The pot starts at 1 dollar and is doubled every time a head appears. You win whatever is in the pot after the game ends. Thus you win 1 dollar if a tail appears on the first toss, 2 dollars if on the second, 4 dollars if on the third, 8 dollars if on the fourth, etc. In short, you win 2^(k−1) dollars if the coin is tossed k times until the first tail appears.

What would be a fair price to pay for entering the game? To answer this we need to consider what would be the average payout: With probability 1/2, you win 1 dollar; with probability 1/4 you win 2 dollars; with probability 1/8 you win 4 dollars etc. The expected value is thus,

dox1.jpg

This sum diverges to infinity, and so the expected win for the player of this game, at least in its idealized form, in which the casino has unlimited resources, is an infinite amount of money. This means that the player should almost surely come out ahead in the long run, no matter how much they pay to enter; while a large payoff comes along very rarely, when it eventually does it will typically far more than repay however much money they have already paid to play. According to the usual treatment of deciding when it is advantageous and therefore rational to play, you should therefore play the game at any price if offered the opportunity. Yet, in published descriptions of the paradox, 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.

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