A game with infinite expected value that no one wants to play
Click to flip!
1. A fair coin is flipped until it lands HEADS.
2. If heads appears on flip n, you win $2ⁿ.
3. Heads on flip 1 = $2, flip 2 = $4, flip 3 = $8, etc.
How much would you pay to play this game?
| Flip # | Result | Probability | Prize | Expected Contribution |
|---|---|---|---|---|
| 1 | H | 1/2 | $2 | $1 |
| 2 | TH | 1/4 | $4 | $1 |
| 3 | TTH | 1/8 | $8 | $1 |
| 4 | TTTH | 1/16 | $16 | $1 |
| n | T...TH | 1/2ⁿ | $2ⁿ | $1 |
| Total Expected Value | $1 + $1 + $1 + ... = ∞ | |||
Every flip contributes exactly $1 to the expected value. Since there are infinitely many possible flips, the sum is INFINITE!
Mathematically, you should pay any finite amount to play this game—
even $1 million! But psychological studies show most people won't pay more than $10-25.
Why the huge gap between mathematical expectation and human intuition?
Daniel Bernoulli (1738): The utility of money isn't linear. Winning $1 billion doesn't make you 1000× happier than $1 million. If utility = log(wealth), expected utility becomes finite!
No casino has infinite money. If the house has $1 trillion max, the expected value becomes finite (about $40). Real-world constraints matter!
Getting 40 tails in a row (needed for a trillion-dollar win) would take impossibly long. Physical time constraints cap the realistic expectation.
Kahneman & Tversky: Humans overweight small probabilities but also ignore extremely tiny ones. Very unlikely huge payoffs get mentally discounted.
The expected value is infinite, but the median outcome is just $2! (50% chance of heads on first flip). Perhaps median matters more than mean.
Even with fair odds, people prefer certainty. The game is extremely high-variance: usually you win $2-4, very rarely you hit jackpots.
The St. Petersburg Paradox reveals that expected value isn't everything. Human decision-making involves utility, risk tolerance, probability perception, and real-world constraints. This insight founded modern behavioral economics and decision theory!