St. Petersburg Paradox

Definition

The St. Petersburg Paradox is a famous problem in probability and decision theory that illustrates the conflict between expected value and actual decision-making behavior. It presents a gambling game where a fair coin is flipped until it comes up heads, with the payoff doubling with each flip, leading to an infinite expected value. This paradox challenges our understanding of rationality and utility in games of chance.

5 Must Know Facts For Your Next Test

1. The game involves flipping a coin until it lands on heads, with the player winning $2^n dollars, where n is the number of flips before the first heads.
2. Despite the theoretically infinite expected payout, most people would only be willing to pay a small amount to play the game, demonstrating a discrepancy between mathematical expectation and human behavior.
3. The paradox highlights the limitations of expected value as a decision-making tool in real-life situations, especially under uncertainty.
4. This problem sparked significant discussions in probability theory and economics, influencing later developments in utility theory and behavioral economics.
5. The St. Petersburg Paradox serves as a foundational example for exploring concepts like diminishing marginal utility and risk management in gambling contexts.

Review Questions

• How does the St. Petersburg Paradox illustrate the difference between expected value and actual decision-making in gambling?
  • The St. Petersburg Paradox shows that while the expected value of the game is theoretically infinite due to the increasing payouts with each coin flip, most people would only bet a small amount to play. This reveals a gap between mathematical expectation and real-world decision-making, where individuals often weigh potential outcomes differently than what calculations suggest. It highlights how actual choices can be influenced by psychological factors rather than just numerical analysis.
• Discuss how the St. Petersburg Paradox has influenced concepts in utility theory and our understanding of rational behavior in economics.
  • The St. Petersburg Paradox led to significant developments in utility theory by demonstrating that individuals do not always act according to expected value calculations. Instead, the concept of diminishing marginal utility emerged, suggesting that as potential gains increase, individuals perceive their value differently based on personal risk tolerance and subjective preferences. This shift has been crucial for understanding rational behavior in economic contexts, emphasizing that decisions are often made based on utility rather than sheer mathematical expectation.
• Evaluate how the St. Petersburg Paradox contributes to our understanding of risk aversion and its implications for economic decisions.
  • The St. Petersburg Paradox contributes significantly to our understanding of risk aversion by illustrating how people tend to prioritize certainty over potentially infinite rewards. In practice, many individuals choose not to engage in high-risk gambling scenarios despite high expected values, reflecting their preference for guaranteed outcomes. This insight into human behavior impacts economic decisions across various fields, influencing everything from investment strategies to insurance models, where risk assessment plays a critical role in shaping choices.

Related terms

Expected Value: A calculated average outcome of a random variable, factoring in all possible outcomes and their probabilities.

Utility Theory:

A theory in economics and psychology that explains how individuals make decisions based on their preferences and the satisfaction they expect to gain.

Risk Aversion: The tendency of individuals to prefer certainty over uncertainty, leading them to make choices that minimize potential losses.