Imagine a game beginning with a stake of $2. The stake is the amount the player will be paid at the end. The player flips a coin, and if it lands on tails, the stake doubles. Otherwise, the game ends and the player collects the stake.
Calculating the Expected Value
We can calculate the expected value of the player’s earnings. The expected value is the average value that you would predict from a random process.
There is a 1/2 chance that the player lands on heads immediately, earning $2. Given the 1/2 chance of landing on tails to start with, there is then a 1/2 chance of landing on heads on the next flip and earning $4, giving an overall chance of 1/4 to earn $4. This logic continues for all remaining flips.
We can calculate the expected value by multiplying each payout by its probability and adding the results together:
(1/2 × $2) + (1/4 × $4) + (1/8 × $8) + …
= 1 + 1 + 1 + …
= ∞
The Paradox
What would be a fair cost to play this game? Hypothetically, no matter the cost, even if it were $1 billion, the casino always loses. However, according to Ian Hacking, few people would pay even $25 to play this game.
Why Don’t People Pay More?
This disconnect between infinite expected value and low willingness to pay is what makes the St. Petersburg paradox so fascinating. Several explanations have been proposed over the centuries. Daniel Bernoulli, who first analyzed this problem in 1738, suggested that people do not value money linearly. The jump from $0 to $1,000 feels far more significant than the jump from $1,000,000 to $1,001,000, even though the dollar amount is the same. This idea, called diminishing marginal utility, means that the massive but extremely unlikely payouts contribute far less psychological value than their dollar amounts suggest. Others have pointed out that any real casino has finite resources, so the game could never truly pay out unlimited sums. In practice, the expected value is bounded by the casino’s wealth, which dramatically reduces the fair price of the game.
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