A Lottery With Infinite Expected Payoff That Almost Nobody Will Pay $20 For

Here is the gamble. A casino flips a fair coin until it lands heads. If heads comes up on the first flip you win $2. If on the second flip you win $4. On the third, $8. On the n-th, $2^n$. The casino sets the entry fee. How much would you pay to play?

The expected payoff is straightforward arithmetic. The probability of stopping on flip n is 1/2^n. The payoff in that case is $2^n$. Multiply: $2^n × 1/2^n = $1$. Sum that contribution across all n and you get $1 + $1 + $1 + … indefinitely. The expected value of the gamble is infinite. By the standard rule of decision under risk — pay any finite price for an infinite expected return — you should be willing to wager your house, your retirement, and your liver to play.

Nobody actually does. When the puzzle has been put to subjects in classroom studies, real bidders typically offer somewhere between two and twenty units of currency. The discrepancy was first stated as a problem by the Swiss probabilist Nicolas Bernoulli in a 1713 letter to Pierre Raymond de Montmort. It was made famous in 1738 by Nicolas's cousin Daniel Bernoulli, who published his analysis in Commentarii Academiae Scientiarum Imperialis Petropolitanae — the Imperial Academy of Sciences journal in St Petersburg, where Daniel was working at the time. The puzzle has had the city's name attached to it ever since.

Daniel's resolution introduced what would become a foundational idea in economics: utility is not the same as money. If utility is the logarithm of wealth — Bernoulli's working assumption — the same gamble yields a finite expected utility, worth a few dollars to a starting bidder. The entire mathematical apparatus of risk-averse decision-making descends from that paragraph.