Imagine a casino offering this game. A fair coin is tossed until it comes up heads. If heads appears on the first toss, you win 2 coins. If it first appears on the second toss, you win 4. On the third, 8, and the payout keeps doubling for every extra toss it takes. The longer the run of tails, the more you win.
How much should you be willing to pay to play once? By the standard rule of expected value, you weigh each prize by its probability and add them up. There is a one-half chance of winning 2, a one-quarter chance of winning 4, a one-eighth chance of winning 8, and so on. Work out each term and you get one half plus one half plus one half, forever. The expected payout is infinite.
Taken at face value, that means you should be willing to hand over every coin you own, your house and your car, for a single go. And yet almost nobody would pay even 20 coins to play, because you can see with your own eyes that you will most likely walk away with a small sum. The chance of a huge win is real but vanishingly rare. This clash between the infinite theory and the sensible instinct is the St. Petersburg paradox.
It has a distinguished pedigree. The game was first posed by Nicolas Bernoulli in 1713, in a letter to a fellow mathematician, and it was his cousin Daniel Bernoulli who wrestled it into fame in a paper published in 1738, while he was working in St. Petersburg, which is how the paradox got its name.
Daniel’s answer changed economics. He argued that people do not value money in a flat, straight-line way. A second million matters far less to you than your first, so what we really weigh is not the raw cash but its usefulness, its utility, and utility does not run off to infinity the way the coin payouts do. That single idea, born from a strange coin game, grew into the modern theory of how everyone from gamblers to insurance companies decides what a risky bet is truly worth.
