Quick Answer: The St. Petersburg paradox is a theoretical game where you flip a coin until it lands tails. The prize doubles with every consecutive heads. Mathematically, the expected payout of the game is infinite. Yet, no rational person would pay a large amount (e.g., $1,000) to play it. This paradox birthed the concept of "Expected Utility."
The Rules of the Game
- A casino offers you a game: a fair coin is flipped until it comes up tails.
- If it lands tails on the 1st flip, you win $2.
- If it lands heads then tails (2nd flip), you win $4.
- If it lands heads, heads, tails (3rd flip), you win $8.
- The prize doubles for every consecutive heads. How much would you pay to play this game?
The Mathematics: Infinite Expected Value
To find the Expected Value (EV), we multiply the payout by the probability for each round and sum them up:
- Round 1: 1/2 probability × $2 payout = $1 expected value
- Round 2: 1/4 probability × $4 payout = $1 expected value
- Round 3: 1/8 probability × $8 payout = $1 expected value
- Round N: (1/2)^N probability × $2^N payout = $1 expected value
Since there is no limit to how many times the coin can land heads, there are an infinite number of rounds. The EV is $1 + $1 + $1 + ... = Infinity. Mathematically, you should be willing to pay all the money in the world to play this game once.
The Paradox and the Solution
If the EV is infinite, why would most people hesitate to pay even $20 to play? (There's a 75% chance you win $4 or less). The mathematician Daniel Bernoulli solved this in 1738 by introducing Expected Utility. He proposed that the value of money diminishes. To a billionaire, an extra $1,000 means very little. To a starving person, it is life-changing. Therefore, a 1-in-a-million chance to win a billion dollars has infinite mathematical value, but finite utility value.