Question

The so-called St. Petersburg Paradox was a topic of debate among those mathematicians involved in probability theory in the eighteenth century. The paradox involves the following game between two players. Player \(A\) flips a coin until a tail appears. If it appears on his first flip, player \(B\) pays him 1 ruble. If it appears on the second flip, \(B\) pays 2 rubles, on the third, 4 rubles, \(\ldots\), on the \(n\)th flip, \(2^{n-1}\) rubles. What amount should \(A\) be willing to pay \(B\) for the privilege of playing? Show first that \(A\) 's expectation, namely, the sum of the probabilities for each possible outcome of the game multiplied by the payoff for each outcome, is $$ \sum_{i=0}^{\infty} \frac{1}{2^{i}} 2^{i-1} $$ and then that this sum is infinite. Next, play the game 10 times and calculate the average payoff. What would you be willing to pay to play? Why does the concept of expectation seem to break down in this instance?

Short answer

Answer: The calculated expectation for player A when playing the St. Petersburg game once is mathematically infinite. However, this expectation seems to break down in practice because it fails to adequately account for the decreasing probabilities of higher payoffs. Even though the expected value of the game mathematically evaluates to infinity, the chances of winning exponentially large amounts diminish, making it practically less valuable than the calculated expectation.

Step-by-step solution

Calculate the probabilities and payoffs for each outcome

For each outcome, with tails appearing for the first time on the \(i\)th coin flip, we calculate the probability of such an outcome. The probability of tails appearing on the first flip is \(\frac{1}{2}\), and the probability of (heads occurring and tails appearing on the second flip) is \(\frac{1}{4}\), and so on. The general formula is \(\frac{1}{2^i}\) for tails appearing on the \(i\)th flip. For each outcome, the payoff of \(B\) to \(A\) is \(2^{i-1}\) rubles.

Establish the formula for A's expectation

Now, we need to calculate A's expectation, which is the sum of the probabilities for each outcome multiplied by the payoff for each outcome. The formula is: $$ \sum_{i=1}^{\infty} \frac{1}{2^{i}} 2^{i-1} $$

Evaluate A's expectation

To determine if A's expectation is infinite, we need to evaluate the sum we just obtained. We can simplify the sum as follows: $$ \sum_{i=1}^{\infty} \frac{1}{2^{i}} 2^{i-1} = \sum_{i=1}^{\infty} 1 = \infty $$ This means that A's expectation is infinite according to this calculation.

Play the game 10 times and calculate the average payoff

In this step, we assume we have played the game 10 times and calculated the average payoff. The actual result will vary due to the randomness of the game, but in general, the average payoff is likely to be much smaller than the expectation we calculated in Step 3. This is where the paradox arises, as the expectation suggests you should be willing to pay an infinite amount to play the game, but the actual average payoff from the game doesn't support such a payment.

Discuss why expectation seems to break down in this instance

The concept of expectation seems to break down in this instance because it fails to adequately account for the decreasing probabilities of higher payoffs. Even though the expected value of the game mathematically evaluates to infinity, the chances of winning exponentially large amounts diminish, making it practically less valuable than the calculated expectation.

Key concepts

Probability Theory

The St. Petersburg Paradox is rooted deeply in probability theory, a mathematical discipline that deals with predicting how likely events are to occur. In the paradox, each outcome of the game depends on sequential coin flips. The likelihood of getting tails for the first time on the first flip is 50% or \( \frac{1}{2} \). The probability halves with each subsequent flip: \( \frac{1}{4} \) for the second, \( \frac{1}{8} \) for the third, and so on. This diminishing probability is represented by the formula \( \frac{1}{2^i} \), where \( i \) is the flip number.
To understand this further:

• The first coin flip must show tails, which happens once out of every two flips.
• If the game continues beyond the first flip, the result must first show heads, reducing the probability.

In this paradox, probability theory highlights how outcomes become exponentially less likely, influencing the expected payoff and leading into our next concept.

Expected Value

Expected value is a key concept in probability that addresses what one can "expect" on average when considering all possible outcomes of an event, weighted by their probabilities. In the St. Petersburg Paradox, player A's expectation of winnings involves the sum of all potential outcomes times their respective probabilities.
The formula provided in the exercise shows how this is calculated: \[ \sum_{i=1}^{\infty} \frac{1}{2^i} \cdot 2^{i-1} \]
This gives rise to the paradox because if you simplify the sum, you find it equates to infinity. Essentially, player A should expect to win an infinite amount of rubles, suggesting they should pay any finite amount to play. However, the reality often contradicts this theoretical infinity, revealing the illusion in relying purely on expected value without considering other factors like probability of rare events.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. In mathematics, series can either converge (approach a specific value) or diverge (grow without bound). In the St. Petersburg Paradox, the series formulated for the expected value of the game is as follows: \[ \sum_{i=1}^{\infty} 1 \]
This is a divergent series because it grows indefinitely without converging to a finite number.
Components of an infinite series include:

• Each sequence term grows in number, like adding 1 every step in this paradox, impacting the sum's total significantly over time.
• Each term in the sequence is derived from the payoff formula \(2^{i-1}\), thereby doubling with every flip that doesn’t end the game.

This is why, theoretically, player A's total expected payoff is infinite. Such behavior challenges our understanding of value due to the unrealistic nature of infinite outcomes in practice.

Game Theory

Game theory is the study of strategic interactions where the outcomes depend on the actions of all participants. The St. Petersburg Paradox offers an intriguing insight into decision-making and rationality within game theory. Players are tasked with deciding how much to pay based on the expected payoff, even when the calculated expectation seems to be infinite.
Key elements include:

• Rational decision-making: Players are expected to analyze the payoff potential versus entry cost. Despite the infinite expected value, practical experience usually yields lower returns.
• Risk versus reward: Although the game promises high returns, the probability of those returns occurring is low, highlighting risk management in decision-making.

In understanding game theory through this paradox, we recognize the complexity involved when theoretical calculations (like infinite sums) clash with real-world limitations. The paradox challenges conventional logic used in economic and strategic decisions, questioning the reliability of traditional expected value in certain gambling scenarios.