Infinite Expected Value
A Game Worth
Infinite Dollars
A game has infinite expected value. You should pay any price to play. But everyone goes bankrupt.
Here is a simple game. A fair coin is flipped repeatedly until it lands heads. If heads appears on flip n, you win $2^n.
Heads on flip 1
$2
Heads on flip 2
$4
Heads on flip 5
$32
Heads on flip 20
$1,048,576
Question: How much would you pay to play this game?
Most people say $10-$25. Some adventurous souls might say $100.
But here is the shocking truth: the mathematically "correct" answer is any amount. Pay a million dollars. Pay a billion. The expected value of this game is infinite.
Infinite expected value. Finite willingness to pay.
This is the St. Petersburg Paradox, first described in 1713.
Play the Game
Experience the paradox firsthand. Set an entry fee and play. Watch your balance.
The Rules
- 1.Pay the entry fee to play
- 2.A fair coin is flipped until it lands HEADS
- 3.If heads appears on flip n, you win $2^n
Flip 1 = $2, Flip 2 = $4, Flip 3 = $8, Flip 4 = $16, Flip 5 = $32...
Play a few dozen times. You will notice something: most games pay $2 or $4. Occasionally you hit $8 or $16. Very rarely, you might see $32 or more.
If you set the entry fee at $10 or higher, you are probably losing money. Even though the expected value says you should pay any price.
Why the disconnect?
The infinite EV comes from astronomically large payouts that almost never happen. In any finite number of games, you will almost certainly never see them.
Why EV = Infinity
Expected value is probability times payout, summed over all outcomes.
For each flip number n:
- Probability of heads on exactly flip n: (1/2)^n
- Payout if heads on flip n: $2^n
- Contribution to EV: $1 (they cancel!)
Watch the expected value grow without bound:
Expected Value Formula
Every term adds exactly $1 to the expected value.
The probability halves each time, but the payout doubles. They cancel out, leaving $1 per term.
Infinite terms = infinite expected value
Each term in the sum adds exactly $1. There are infinite terms. Therefore: EV = 1 + 1 + 1 + ... = infinity.
The math is unimpeachable. Yet no rational person would pay $1 million to play. Something is wrong with either the math or our intuition.
The Distribution of Outcomes
Run 1,000 games and see the actual distribution of payouts. This reveals why infinite EV does not translate to infinite value.
The distribution is extremely skewed. About 50% of games pay $2. About 75% pay $4 or less. The "infinite" expected value comes from the right tail - astronomically large payouts that occur with astronomically small probability.
Key insight: In any finite sample, the mean is dominated by the largest single outcome. Run 10,000 games, and one game paying $32,768 can shift the average by $3.
This is why expected value, while mathematically correct, can be a terrible guide for decisions involving unbounded payouts.
What Would You Pay?
Daniel Bernoulli proposed a solution in 1738: people do not value money linearly. The utility of $2 million is not twice the utility of $1 million.
If we use utility = log(wealth) instead of raw dollars, the expected utility converges to a finite value. This value depends on your current wealth.
Probability Analysis
Utility Analysis
For your wealth level, the "fair" entry fee is:
$10
(Using logarithmic utility - Bernoulli's solution from 1738)
Why the fair fee grows with wealth: A billionaire could rationally pay more because losing $100 barely affects them. For you at $1,000, losing $25 represents 2.5% of your wealth.
This explains why billionaires might rationally pay more than you: losing $100 represents a smaller fraction of their wealth, so they can afford to chase the tail.
But Bernoulli's solution raises its own questions. What is the "right" utility function? Why should it be logarithmic? And if we can construct games that break any utility function, have we really solved anything?
Why Every Fix Creates New Problems
Over 300 years, mathematicians have proposed many "solutions" to the St. Petersburg Paradox. Each one fixes the immediate problem but opens new cans of worms.
The St. Petersburg Paradox is a hydra.
Every solution reveals deeper questions about probability, infinity, and rational decision-making.
The paradox persists because it touches fundamental questions:
- What does "expected value" really mean for unbounded distributions?
- How should we make decisions under uncertainty?
- Can probability theory handle infinity?
- What is the relationship between mathematical expectation and practical decision-making?
The St. Petersburg Paradox is not a puzzle to be solved.
It is a boundary marker showing where our intuitions about probability break down. Every solution just moves the boundary somewhere else.
Where This Matters
The St. Petersburg Paradox is not just a mathematical curiosity. Similar structures appear throughout finance, insurance, and risk management.
Fat-Tailed Distributions
Market crashes, pandemics, and natural disasters follow power-law distributions where extreme events dominate the mean. Standard risk models based on expected value fail spectacularly.
Insurance Pricing
How do you price insurance against potentially unlimited losses? The theoretical EV might be infinite, but the policy needs a finite premium.
Pascal's Mugging
A modern variant: someone claims to be a god who will give you infinite utility if you give them $5. If there is any non-zero probability they are telling the truth, the EV is infinite. Should you pay?
Startup Investing
VC returns follow St. Petersburg-like distributions. A few massive winners drive all returns. Most investments lose money. How do you price an entry ticket?
In a world of fat tails and unbounded outcomes,
expected value is not enough.
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Reference: Bernoulli (1738), Menger (1934)