St. Petersburg Paradox Calculator
Explore the paradox of infinite expected value versus real-world decision making
About the St. Petersburg Paradox Calculator
The St. Petersburg Paradox is a theoretical game used in economics and probability theory to demonstrate the limitations of expected value as a decision-making tool. In this game, a fair coin is flipped until it lands on tails. The prize starts at $1 and doubles with every consecutive head. While the math suggests that a player should be willing to pay an infinite amount of money to enter the game due to an infinite expected value, almost no one is willing to pay more than $20 in reality.
This calculator simulates the Paradox by allowing you to test theoretical outcomes and apply real-world constraints. It bridges the gap between pure mathematics and behavioral economics by calculating the 'Effective Expected Value' when the payout is capped or when the player has a logarithmic utility function. Financial analysts, students of game theory, and researchers use this tool to understand risk management and why traditional probability often fails to predict human behavior in high-variance scenarios.
Formula
Expected Value (E) = Σ [(1/2)^n * 2^(n-1)] = 1/2 + 1/2 + 1/2... = ∞The expected value is the sum of all possible outcomes multiplied by their probabilities. In this game, 'n' represents the number of the flip where the first 'Tails' appears. The probability of this happening is (1/2) raised to the power of n, while the payout is 2 raised to the power of (n-1). Each term in the series equals 0.5, and since the game can theoretically go on forever, the sum is infinite.
Worked examples
Example 1: Calculating the fair entry price for a game where the maximum payout is capped at $512 (which occurs at flip 10).
1. Calculate 0.5 for each of the 10 potential flips. \n2. Sum the 10 instances of $0.50 (Flip 1: $0.50, Flip 2: $0.50, ..., Flip 10: $0.50). \n3. Add the residual probability for any flip beyond 10 falling back to the cap ($512 * (1/2^10) = $0.50). \n4. Total = $5.00 + $0.50 = $5.50.
Result: The expected value is $5.50. Even though the payout doubles, the hard ceiling prevents the 'infinite' nature of the paradox from taking effect.
Common use cases
- Teaching university students why the Expected Value theory requires the addition of Utility Theory to be practical.
- Modeling insurance premiums for 'black swan' events that have low probability but extremely high impact.
- Analyzing why gambling houses must set table limits to prevent mathematical exploitation or bankruptcy.
Pitfalls and limitations
- The calculator assumes a perfectly fair coin with exactly 50/50 odds, which rarely exists in physical form.
- Floating-point math in computers can struggle with the extremely high numbers generated after only 1,000 flips.
- Ignoring the 'Gambler's Ruin' aspect where the player might run out of money before hitting a massive payout.
Frequently asked questions
why is the st petersburg paradox a paradox
The paradox occurs because the game has an infinite expected value mathematically, yet a rational person would only pay a few dollars to play it. This highlights the conflict between theoretical probability and utility theory or risk aversion.
how did bernoulli solve the st petersburg paradox
Daniel Bernoulli suggested that the 'utility' of money decreases as your wealth increases (logarithmic utility). Since the utility of winning millions is not significantly higher than winning thousands to most people, the perceived value of the game stays low.
what happens to the paradox if the house has limited money
In the real world, casinos and banks have finite wealth. A calculator that caps the maximum payout will show a very small expected value—usually between $10 and $20—which aligns with what people are actually willing to pay.
how do you calculate the expected value of the st petersburg game
The probability of winning on the nth flip is (1/2)^n. Because the prize grows as 2^n, these two factors cancel each other out in the expected value sum, resulting in an infinite series (1 + 1 + 1...).
why don't people pay infinite money to play the game
The odds of reaching the 30th flip are roughly 1 in 1 billion. Most human lives and financial systems are too short-lived to ever witness the high-payout tail events that drive the paradox's infinite sum.