St. Petersburg Paradox
The St. Petersburg paradox is a theoretical game first proposed by Nicolas Bernoulli, in which you pretend that you are a player in a casino playing a special coin toss game.
The casino starts with a guaranteed payout to you of $2. The game proceeds using a fair coin, tossed in succession until it flips a tails. After each flip where the coin is heads, the casino doubles the pot. So, if a tails appears right at the first toss, you get $2. If a tails does not arrive until the second toss, you win $4. If a tails arrives on the third toss, you win $8, and so on.
The challenge is that you have to pay some amount of money to be allowed to play this game. If you were the player and were told to act completely rationally, considering only the expected payout, and the casino places no limits on the maximum payout, what is the maximum amount you should be willing to pay to play this game?
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This is a counter-intuitive paradox, like many other common misconceptions . The correct answer is that you should be willing to pay any finite amount of money, yes even $1,000,000,000 (presuming you have that much), to play this game. This is because of the average payout for players of this game is ∞ .
Here's how that works: Tails landing on the first toss has a 1/2 odds of occurring, on the second but not the first toss has a 1/2 of 1/2, or 1/4 odds of occurring, on the third toss, but not the first or second tosses is 1/8, and so on. But the pot rises equally. So the average payout in 1/2 the cases is $2, in 1/4 the cases is $4, in 1/8 the cases is $8, and so on. This can be expressed as:
P = ( 1 / 2 × $ 2 ) + ( 1 / 4 × $ 4 ) + ( 1 / 8 × $ 8 ) + . . . or
P = $ 1 + $ 1 + $ 1 + . . . . = ∞
However, this paradox fails to take into account the utility of the game or behavioral economics . The problem states to only consider the expected value, but this leads you to taking a risk (for instance paying $1,000,000,000) that no actual person would accept. Economists are still debating what the right price to pay is, given utility, and a decision that considers more that the average payout.