The St. Petersburg Paradox: When Infinite Value Isn’t Worth Much

The game

A fair coin is flipped until it lands heads. If heads comes on the first flip, you win $2. If it comes on the second, $4. On the third, $8 — the prize doubles with every tails. How much should a rational person pay to play once?

Nicolas Bernoulli posed the puzzle in 1713. His cousin Daniel Bernoulli published an influential answer in 1738 in the journal of the Imperial Academy in St. Petersburg, which gave the paradox its name.

Why the expected value is infinite

Expected value multiplies each prize by its probability and adds the results. The first heads has a 1/2 chance and pays $2 — worth $1. The second has a 1/4 chance and pays $4 — also $1. Every possible outcome contributes exactly $1, and there is no limit to how many there are.

Expected value = ½ × $2 + ¼ × $4 + ⅛ × $8 + … = $1 + $1 + $1 + … = ∞

Yet most people asked say they would pay only a few dollars, and they are right to hesitate: half of all games pay just $2, and three-quarters pay $4 or less.

Resolution one: utility

Daniel Bernoulli argued that people do not value money linearly. An extra $1,000 means much more to someone with $1,000 than to someone with $1 million. He proposed measuring the value of wealth on a logarithmic scale — an early version of what economists now call diminishing marginal utility. Measured that way, the game is worth only a modest amount, depending on how much wealth the player starts with.

Resolution two: nobody can pay infinity

A second answer is practical. Any real banker has a limited bankroll. If the most the house could ever pay were about $1.07 billion (2³⁰ dollars), the expected value would collapse to about $31. Infinite value depends on payouts that could never actually be honored.

Why it still matters

The paradox is a reminder that expected value alone is not a complete guide to decisions. Insurance, savings buffers, and diversification all make sense because people reasonably care about the spread of possible outcomes, not just the average. When an option has a tiny chance of a huge payoff, ask what happens in the typical case — and whether you could afford the most likely outcome.