The St. Petersburg Paradox: Why Expected Value Can Diverge…

What the St. Petersburg Paradox Is

The St. Petersburg paradox is a thought experiment involving a game where a player flips a fair coin until it comes up tails, and then wins an amount of money equal to $2 raised to the power of the number of heads flipped (i.e., if 3 heads are flipped before the first tail, the player wins $8). The paradox arises when calculating the expected value of this game.

The expected value is calculated by summing the products of each possible outcome and its probability. For the St. Petersburg game, as the number of flips increases, the potential winnings grow exponentially while the probabilities decrease rapidly, leading to an infinite expected value.

Why It Happens

The paradox occurs because the expected value calculation assumes that all outcomes are equally likely and weighted by their probability. However, in practice, such a game would not be played indefinitely due to practical constraints like finite resources or time.

In reality, players would stop playing once they reach a certain threshold of winnings, making the infinite expected value irrelevant.

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Why It Matters

The St. Petersburg paradox highlights the limitations of using expected values to make decisions in situations with extreme outcomes or large variances.

It also demonstrates the importance of considering practical constraints and risk aversion when making financial decisions.

Real-World Examples

The paradox has applications in economics, where it can explain why people might be willing to pay a premium for insurance or other financial products that offer protection against rare but catastrophic events.

It also serves as a cautionary tale in the field of finance, reminding us that expected values alone do not always provide a complete picture of risk and reward.

Frequently asked questions

What is the flip-cap slider for?

The flip-cap slider truncates the number of coin flips in the simulation to demonstrate how practical constraints affect the expected value, making it finite and more realistic.

How does the entry-fee slider work?

Adjusting the entry-fee slider allows you to compare your net winnings against the running average after each game, showing how initial costs can impact profitability in this type of game.

Why is the expected value infinite for the St. Petersburg game?

The expected value is calculated as an infinite sum because the potential payouts grow exponentially with each additional coin flip, and there's no upper limit to the number of flips that could occur in theory.

Does this paradox apply only to the St. Petersburg game?

No, similar paradoxes can arise in other games or situations where outcomes are highly variable and extreme values have a non-negligible probability.

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