St Petersburg paradox

3 key takeaways

• The paradox demonstrates that people do not always act according to expected value calculations in the face of infinite outcomes.
• It highlights the limitations of expected value theory when applied to real-world decision-making.
• The paradox has led to the development of concepts like utility and risk aversion in economics.

What is the St. Petersburg Paradox?

The St. Petersburg Paradox is named after the city where Daniel Bernoulli first presented it in 1738. The paradox is based on a hypothetical lottery game with the following rules:

1. A fair coin is tossed until it comes up heads.
2. The payoff for the game is $2 raised to the power of the number of tosses required to get heads (i.e., $2 for one toss, $4 for two tosses, $8 for three tosses, etc.).

The expected value (EV) of the game can be calculated by summing the infinite series of possible outcomes, each weighted by its probability:

EV = sum from n=1 to ∞ ( (1 / 2^n) * 2^n ) = sum from n=1 to ∞ 1 = ∞

According to the expected value calculation, a rational person should be willing to pay any finite amount of money to play this game, since the expected value is infinite. However, in practice, people are not willing to pay large amounts to participate, revealing a discrepancy between theoretical predictions and actual behavior.

Why is it paradoxical?

The paradox lies in the fact that the expected value suggests an infinite payoff, yet individuals exhibit reluctance to engage in the game for substantial amounts of money. This inconsistency challenges the notion that people make decisions purely based on expected value.

The St. Petersburg Paradox highlights the limitations of expected value theory in capturing human preferences and behaviors, particularly when dealing with highly improbable but theoretically infinite outcomes. It underscores the need for alternative frameworks to explain decision-making under uncertainty.

Resolving the paradox

Economists and mathematicians have proposed several resolutions to the paradox:

• Risk aversion: Many individuals are risk-averse, meaning they prefer a certain outcome over a gamble with a higher expected value but more uncertainty. Risk aversion helps explain why people would not pay a large amount to play the St. Petersburg game despite its high expected value.

• Practical limitations: In reality, no one has infinite resources, and the potential payoffs in the St. Petersburg game would be limited by the finite wealth of the participants and the economy. This practical constraint makes the infinite expected value less relevant in real-world scenarios.

• Expected utility theory: Daniel Bernoulli himself proposed a solution by introducing the concept of utility. Instead of focusing on monetary value, he suggested that people derive utility (satisfaction) from money, and the utility function is concave, reflecting diminishing marginal utility. This means that the additional satisfaction gained from an extra dollar decreases as wealth increases. By using a logarithmic utility function, the expected utility of the game becomes finite, resolving the paradox.

  U(x) = log(x)

  Using this utility function, the expected utility is:

  EU = sum from n=1 to ∞ ( (1 / 2^n) * log(2^n) ) = sum from n=1 to ∞ ( (n * log 2) / 2^n )

This series converges to a finite value, making the game more realistic in terms of how much someone would be willing to pay to play. 

Impact of the St. Petersburg Paradox

The St. Petersburg Paradox has had a significant impact on the fields of economics and decision theory. It has led to the development of more sophisticated models of human behavior, such as expected utility theory, which accounts for the subjective value individuals place on different outcomes. The paradox also highlights the importance of considering psychological and practical factors when analyzing economic decisions.

Understanding the St. Petersburg Paradox provides valuable insights into the complexities of human decision-making under uncertainty. For further exploration, you might look into related topics such as utility theory, risk aversion, and behavioral economics.