St. Petersburg Expected Value
A game with infinite expected value yet little real worth.
Formula
Variables
Example
The sum diverges to infinity, yet nobody would pay much to play.
How it works
This formula computes the expected payoff of the classic St. Petersburg game, where the coin is flipped until the first heads appears. If the first heads occurs on round k, that happens with probability 1/2^k, and the payoff is 2^(k-1), so each term is probability × payoff. Every term equals 1/2, so the infinite sum grows without bound even though the game feels small in practice; this is the famous gap between mathematical expectation and what people are actually willing to pay.
When to use it
Use this when analysing paradoxes involving infinite expected value, especially in decision theory, utility, and expected-utility critiques. It is also useful whenever you need to compare a theoretical average payoff with realistic willingness-to-pay under risk, budget limits, or utility constraints.
Worked example, step by step
Suppose a game pays 1 unit if heads appears on the 1st flip, 2 units on the 2nd flip, 4 units on the 3rd flip, and so on. Round 1 contribution: (1/2^1) × 2^(1-1) = (1/2) × 1 = 1/2 Round 2 contribution: (1/2^2) × 2^(2-1) = (1/4) × 2 = 1/2 Round 3 contribution: (1/2^3) × 2^(3-1) = (1/8) × 4 = 1/2 In general, each round contributes 1/2, so EV = 1/2 + 1/2 + 1/2 + ... Because the series adds 1/2 infinitely many times, EV diverges to infinity. Final answer: EV = ∞
Common mistakes to avoid
- Forgetting that the first heads on round k has probability 1/2^k, not 1/2
- Using the payoff 2^k instead of 2^(k-1)
- Assuming infinite expected value means a rational person should pay an unlimited amount to play
Did You Know?
The St. Petersburg paradox (1738) forced mathematicians to invent utility — value isn't just expected cash.
Share this formula
More in Game Theory & Decision
View allExpected Value
BasicProbability-weighted average outcome of a decision.
Kelly Criterion
AdvancedOptimal fraction of bankroll to bet for long-run growth.
Expected Utility
AdvancedWeighted average of the utility of outcomes, not just their cash value.
Payoff Comparison
IntermediateOne strategy dominates another if its payoff is always higher.
Minimax Value
AdvancedBest worst-case payoff a player can guarantee.
Mixed Strategy (Indifference)
AdvancedProbability that makes an opponent indifferent between their options.