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 wins $4. If a tails arrives on the third toss, you win $8, and so on.
The challenge is that you have 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 place no limits on the maximum payout, what is the maximum amount you should be willing to pay to play this game?