What Is St. Petersburg Paradox?

August 31, 2024 5 min read Probability Decision Theory Economics Mathematics Paradox Expected Value Utility Theory

The St. Petersburg Paradox highlights the discrepancy between the theoretical expected value of a game and the amount individuals are willing to pay to play, despite an infinite expected payoff.

St. Petersburg Paradox: A Paradox in Probability and Decision Theory

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The St. Petersburg Paradox is a famous problem in probability and decision theory that raises questions about human decision-making, utility, and economic behavior. This paradox highlights the discrepancy between the theoretical expected value of a particular game and the comparatively small amount of money individuals are willing to pay to play it.

Historical Context

The St. Petersburg Paradox was first introduced by the Swiss mathematician Daniel Bernoulli in 1738. He presented this paradox in the context of decision-making and economic behavior, challenging the prevailing notions of expected value and rational choice. Bernoulli’s work laid the groundwork for the development of utility theory, which aims to explain the real-world behavior of individuals when confronted with uncertain outcomes.

The Game and Expected Value

Consider a game where a fair coin is tossed repeatedly until it lands on tails for the first time. The payoff structure is as follows:

If the first tail appears on the 1st toss, the player wins 2^1 = 2 units of currency.
If the first tail appears on the 2nd toss, the player wins 2^2 = 4 units of currency.
If the first tail appears on the 3rd toss, the player wins 2^3 = 8 units of currency.
And so on…

The payoff if the first tail appears on the \( t^{th} \) toss is \( 2^t \).

The expected payoff \( E \) of the game can be calculated as follows:

E = \sum_{t=1}^{\infty} \left( \frac{1}{2^t} \times 2^t \right)

Since \( \frac{1}{2^t} \) is the probability of getting the first tail on the \( t^{th} \) toss:

E = \sum_{t=1}^{\infty} 1

Given this series diverges, the expected value of the game is theoretically infinite.

Paradox Explanation

Despite the infinite expected value, empirical observations show that individuals are generally willing to pay only a small amount to participate in this game, typically no more than a few dollars. This paradoxical behavior challenges traditional economic theory, which suggests that rational individuals should be willing to pay a price close to the game’s expected value.

Utility Theory and Risk Aversion

Daniel Bernoulli proposed that the paradox could be resolved by considering the concept of utility rather than monetary value. According to utility theory, individuals derive utility from wealth, and the marginal utility of wealth decreases as wealth increases. Bernoulli suggested a logarithmic utility function \( U(x) = \log(x) \) to capture this diminishing marginal utility.

Using this utility function, the expected utility \( EU \) of the game can be calculated:

EU = \sum_{t=1}^{\infty} \left( \frac{1}{2^t} \times \log(2^t) \right)

Simplifying, we get:

EU = \sum_{t=1}^{\infty} \left( \frac{t \log(2)}{2^t} \right)

This series converges to a finite value, explaining why individuals would only pay a finite amount to play the game.

Charts and Diagrams

To visualize the utility function and its implications, consider the following Mermaid diagram illustrating the utility curve:

    graph TD
	    A[Utility] --> B[Wealth]
	    B --> C{Diminishing Marginal Utility}
	    C --> D[Logarithmic Function]

Importance and Applicability

The St. Petersburg Paradox is crucial in various fields, including economics, finance, and behavioral psychology. It has led to the development of expected utility theory, risk aversion models, and behavioral economics, influencing how economists and financial analysts understand decision-making under uncertainty.

Expected Value: The weighted average of all possible outcomes of a random variable.
Utility: A measure of preferences over a set of goods and services.
Risk Aversion: A preference for certainty over uncertainty, even when the uncertain option has a higher expected payoff.
Diminishing Marginal Utility: The principle that as the consumption of a good increases, the marginal utility of each additional unit decreases.
Behavioral Economics: A field of economics that examines psychological factors influencing economic decisions.

Comparison with Other Paradoxes

Allais Paradox: Demonstrates how people’s choices violate expected utility theory by showing preferences that depend on the potential outcomes, rather than probabilities.
Ellsberg Paradox: Illustrates ambiguity aversion, where people prefer known risks over unknown risks, even when the known risk has a lower expected value.

Interesting Facts

The paradox is named after the St. Petersburg Academy of Sciences, where Daniel Bernoulli published his original paper.
The paradox remains a topic of debate and research in contemporary economics and decision theory.

Inspirational Story

Consider an investor faced with the St. Petersburg Paradox who opts to invest wisely in diversified portfolios rather than chasing the theoretically infinite expected value of risky ventures. This decision mirrors the prudent approach advocated by Bernoulli, emphasizing utility over mere monetary gain.

Famous Quotes

“Risk comes from not knowing what you’re doing.” – Warren Buffett
“Not everything that counts can be counted, and not everything that can be counted counts.” – William Bruce Cameron

Proverbs and Clichés

“A bird in the hand is worth two in the bush.” – This proverb reflects the preference for certain gains over uncertain, potentially larger gains.

Jargon and Slang

EV (Expected Value): The anticipated value for an investment or decision.
Risk Premium: The extra return expected by an investor for taking on additional risk.
Utility Function: A mathematical representation of how much satisfaction or value an individual derives from different outcomes.

FAQs

Why is the expected value of the St. Petersburg Paradox infinite?

Because the sum of the series that calculates the expected value diverges, leading to an infinite result.

How does the St. Petersburg Paradox challenge traditional economic theory?

It demonstrates a discrepancy between theoretical expected value and real-world behavior, where individuals are unwilling to pay a price close to the infinite expected value.

What is diminishing marginal utility?

It is the principle that as wealth increases, the additional satisfaction or utility gained from an additional unit of wealth decreases.

References

Bernoulli, D. (1738). Exposition of a New Theory on the Measurement of Risk. St. Petersburg Academy.
Savage, L. J. (1954). The Foundations of Statistics. Wiley.

Summary

The St. Petersburg Paradox is a seminal problem in probability and decision theory, highlighting the gap between theoretical and practical decision-making. By considering utility and risk aversion, the paradox provides crucial insights into human behavior, guiding economic theories and applications. It remains a cornerstone in understanding rational choice and utility theory.