Toss a coin until it lands heads. Heads on the first toss pays 1 ducat, on the second 2, then 4, 8, doubling every toss. Half of all rounds pay just 1 ducat; one in 64 pays 64 or more. Nicolaus Bernoulli posed a version of this game in a letter in 1713 (with a die), and his cousin Daniel wrote it up in St Petersburg, which is how it got its name.

The average prize: half a chance of 1, a quarter chance of 2, an eighth of 4. Each row adds another half ducat, forever. On average the game is worth more than all the money in the world. Yet when students bid real money to play, the middle bid was $1.75.

Daniel's answer: a ducat is worth less the more you already have. Ten ducats to someone with a hundred are like twenty to someone with two hundred. Counted that way, the game is worth a few ducats. In 2011 Daniel Kahneman added what Bernoulli missed: we feel changes, not totals.

The game is a thought experiment about valuing an uncertain future, not a betting tip; the curve and the bars are drawings.

So, what would you pay?

Sources: Daniel Bernoulli, "Exposition of a New Theory on the Measurement of Risk", 1738, translated in Econometrica 22, 1954; Benjamin Hayden and Michael Platt, 2009; Daniel Kahneman, Thinking, Fast and Slow, 2011, chapter 25.
Sounds: our own, made in code (earshot, CC0).