The St. Petersburg Paradox

The St. Petersburg Paradox is a seminal problem in probability theory and decision theory that highlights a profound divergence between the calculated expected monetary value (EMV) of a gamble and the actual amount individuals are willing to pay to participate. It demonstrates that simple maximization of expected monetary gain is often an insufficient guide for rational decision-making, particularly when dealing with uncertain outcomes that could theoretically be unbounded.

What is the St. Petersburg Paradox?

The paradox describes a hypothetical game of chance, often called the St. Petersburg Lottery, with the following rules:

1. A fair coin is tossed repeatedly.
2. The game continues until the coin lands on heads for the first time.
3. The payout to the player is determined by the number of tosses required to achieve the first head. Specifically, if the first head appears on the n-th toss, the player wins \(2^n\) dollars.

Let's break down the payouts and their probabilities:

• Heads on the 1^st toss (H): Probability = ½. Payout = \(2^1 = \$2\).
• Tails then Heads (TH): Probability = (½) * (½) = ¼. Payout = \(2^2 = \$4\).
• Tails, Tails, then Heads (TTH): Probability = (½) * (½) * (½) = ⅛. Payout = \(2^3 = \$8\).
• In general, if the first head appears on the n-th toss: Probability = \((1/2)^n\). Payout = \(2^n\).

The Paradoxical Calculation

The paradox emerges when we calculate the expected monetary value (EMV) of this game. The EMV is the sum of the products of each possible outcome's probability and its corresponding payout:

\[ \text{EMV} = \sum_{n=1}^{\infty} P(\text{first head on } n\text{-th toss}) \times (\text{Payout for } n\text{-th toss}) \]

\[ \text{EMV} = \sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n \times 2^n \]

\[ \text{EMV} = \sum_{n=1}^{\infty} \frac{2^n}{2^n} = \sum_{n=1}^{\infty} 1 \]

\[ \text{EMV} = 1 + 1 + 1 + 1 + \dots = \infty \]

Mathematically, the expected monetary value of the St. Petersburg game is infinite. This implies that a purely rational agent, aiming to maximize their wealth, should be willing to pay any finite amount of money to play this game.

However, in practice, when this game is presented to people, they are overwhelmingly unwilling to pay more than a small, finite sum—typically between $10 and $20, and rarely exceeding $50. This vast discrepancy between the theoretically infinite expected payout and the meager amount people are willing to risk is the core of the St. Petersburg Paradox.

Historical Context and Origins

The St. Petersburg Paradox was first described by Nicolaus Bernoulli in a letter to Pierre Raymond de Montmort on September 9, 1713. While Nicolaus posed the problem, it was his cousin, Daniel Bernoulli, who provided the first influential analysis and proposed a resolution in his 1738 paper, "Exposition of a New Theory on the Measurement of Risk," published in the Commentaries of the Imperial Academy of Science of Saint Petersburg. This is how the paradox acquired its name.

Daniel Bernoulli's crucial insight was that individuals do not value money linearly. Instead, he proposed the concept of utility, suggesting that the subjective satisfaction or "usefulness" derived from money diminishes as one's wealth increases. This is known as diminishing marginal utility. For example, an extra $100 might significantly improve the life of someone living in poverty, but it would likely have a negligible impact on the daily life of a billionaire.

Bernoulli suggested that people make decisions based on the expected utility of outcomes, not their expected monetary value. He proposed a logarithmic utility function, \(U(w) = \ln(w)\), where \(w\) represents wealth. When applied to the St. Petersburg game, this function leads to a finite expected utility, thus explaining why people are not willing to pay an infinite amount. Other early thinkers, like Gabriel Cramer, also independently proposed similar ideas about the utility of money.

Resolutions and Explanations

The St. Petersburg Paradox has spurred significant debate and led to various proposed resolutions over centuries:

• Expected Utility Theory: As proposed by Daniel Bernoulli, this is the most widely accepted resolution. It posits that individuals maximize the expected utility of their wealth, not its monetary value. Since utility grows at a decreasing rate, the subjective value of potentially infinite payouts diminishes.
• Finite Bankroll: In reality, no casino or individual has infinite resources. If the game's payout is capped (e.g., by the maximum amount the house can pay), the EMV becomes finite.
• Probability Neglect/Weighting: People may subjectively downweight extremely small probabilities, effectively ignoring the astronomical payouts that occur with very low likelihood. This is related to concepts in Prospect Theory.
• Risk Aversion: Most individuals are risk-averse and prefer a certain, smaller gain over a gamble with a higher expected value but significant risk.
• Ergodicity and Time Averages: Some modern resolutions, like that proposed by Ole Peters, suggest that the paradox arises from conflating ensemble averages (averaging across many players) with time averages (averaging over a single player's lifetime). In a single lifetime, the probability of experiencing the extreme, high-payout outcomes is so low that the average outcome over time remains finite.

Real-World Examples and Applications

While a literal St. Petersburg game is rare, its principles are reflected in many real-world phenomena:

• Lottery Tickets: People buy lottery tickets despite a negative EMV because the minuscule chance of winning a life-altering sum offers immense subjective appeal, a form of seeking extreme outcomes.
• Insurance: Purchasing insurance is an act of paying a premium (a negative EMV, as the expected payout is less than the premiums collected) to avoid the catastrophic financial impact of an unlikely event, demonstrating risk aversion and the utility of security.
• Venture Capital and Startup Investing: Investors might put money into startups with a low probability of success but a potentially massive return, weighing the diminishing utility of potential gains against the risk of total loss.
• Gambling and Casinos: Casinos are designed to profit by offering games with negative EMVs, but they include large jackpots that attract players who are drawn to the possibility of extreme wealth, irrespective of the odds.
• "Black Swan" Events: In finance and risk management, the paradox highlights the importance of considering extremely rare but high-impact events.

Current Relevance and Impact

The St. Petersburg Paradox remains highly relevant today:

• Behavioral Economics: It was a catalyst for the development of behavioral economics, demonstrating that traditional economic models based purely on rational self-interest and expected value were insufficient to explain human behavior.
• Decision Theory: It continues to be a foundational concept for understanding decision-making under uncertainty, risk, and ambiguity.
• Finance and Risk Management: It informs how financial institutions and individuals approach investment, insurance, and risk assessment, emphasizing the need to consider risk preferences and the potential for extreme outcomes.
• Policy Making: Insights from the paradox can help policymakers design interventions and regulations that are more aligned with how people actually make decisions, rather than assuming purely rational, utility-maximizing behavior.

In essence, the St. Petersburg Paradox serves as a powerful reminder that human decision-making is a complex interplay of objective probabilities, subjective values, risk preferences, and psychological biases. It teaches us that the "value" of money is not absolute but relative, and that the pursuit of wealth is tempered by the diminishing utility of each additional dollar and the aversion to ruin.

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References:

• Bernoulli, Daniel. (1738). Exposition of a New Theory on the Measurement of Risk.
• Samuelson, Paul A. (1977). The St. Petersburg Paradox: A Discussion of Some Recent Comments. Journal of Economic Theory, 14(2), 443-445.
• Peters, Ole. (2011). The time resolution of the St Petersburg paradox. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 369(1955), 4913-4931.