Investment Decisions in Coin Toss Game

Solution & Explanation

The St. Petersburg paradox presents an intriguing scenario where the expected value of a coin-tossing game is infinite, yet practical decision-making often diverges from this theoretical prediction. Let's delve into the nuances of this paradox and how it impacts investment decisions in the game.

Understanding the Game

• Game Structure:
  • A fair coin is tossed repeatedly until it lands on tails.
  • The player wins $2^k$ dollars, where $k$ is the number of tosses needed to obtain the first tails.
• Expected Value Calculation:
  • The probability of getting the first tails on the $k^{th}$ toss is $\frac{1}{2^k}$.
  • The expected payout for each possible outcome is calculated as $\frac{1}{2^k} \times 2^k = 1$.
  • Summing over all possible outcomes yields an infinite expected value: $\sum_{k=1}^{\infty} 1 = \infty$.

Theoretical vs. Practical Decision-Making

• Theoretical Perspective:
  • According to expected value theory, a rational player should be willing to pay any finite amount to play the game due to its infinite expected value.
• Practical Considerations:
  • Risk Aversion: Most individuals are risk-averse, meaning they prefer certain outcomes over uncertain ones, even if the uncertain outcome has a higher expected value.
  • Diminishing Marginal Utility: The utility or satisfaction gained from each additional dollar decreases as wealth increases. Thus, the potential high payouts may not justify the risk for many players.
  • Wealth Constraints: A player's financial situation significantly influences their willingness to pay. A wealthier individual might be more inclined to pay a higher entry fee than someone with limited financial resources.

Alternative Theories

• Bernoulli's Utility Theory:
  • Bernoulli proposed that individuals make decisions based on expected utility rather than expected monetary value.
  • Utility is a subjective measure of satisfaction or happiness derived from wealth.
• Prospect Theory:
  • Developed by Kahneman and Tversky, this theory suggests that people evaluate potential losses and gains relative to a reference point, often leading to risk-averse behavior for gains and risk-seeking behavior for losses.

Conclusion

The St. Petersburg paradox highlights the limitations of expected value theory in predicting real-world decision-making. While the game theoretically offers an infinite expected value, practical considerations such as risk aversion, diminishing marginal utility, and personal wealth constraints play a crucial role in determining how much one would be willing to invest. Understanding these factors is essential for making informed investment decisions in uncertain scenarios.