Problem 88 - An Introduction to Mathematical Statistics and Its Applications

Question

How much would you have to ante to make the St. Petersburg game "fair" (recall Example 3.5.5) if the most you could win was \(\$ 1000 ?\) That is, the payoffs are \(\$ 2^{k}\) for \(1 \leq k \leq 9\), and \(\$ 1000\) for \(k \geq 10\).

Step-by-Step Solution

Answer

To make the St. Petersburg game 'fair', you should bet approximately \$ 11.00.

Step 1: Calculate payoffs in terms of probability

You will start by calculating the payoffs in terms of their probabilities for k = 1 to 9. The odds of winning go down by a half with each round, so for round \(k\), the probability is \(1/2^k\), and the payoff is obtained by multiplying the probability with \(2^k\). Thus, the expected payoff for each such round is 1.

Step 2: Calculating payoff when k ≥ 10

Given that the probability halves every round, the probability for any round k≥10 is \(1/2^{k}\). The payoff for these rounds is constant at $1000. Therefore, the expected payoff for any round k≥10 can be calculated as \(1000 \times 1/2^{k}\).

Step 3: Adding up all payoffs

Add up all the expected payoffs calculated in previous steps to get the total expected payoff. This means addition of 9 from rounds 1 to 9 and the resulting series (which is a geometric series) for all the rounds k≥10. The sum of geometric series \(a/(1-r)\), where a is the first term and r is the common ratio, can be used.

Step 4: Calculating the 'fair' bet

The 'fair' bet value corresponds to the total expected payoff from all the rounds. This can be verified by comparing your value with the payoff table.

Key Concepts

Probability Theory

Probability theory is a mathematical framework that helps us understand and quantify uncertainty. In the context of the St. Petersburg Paradox, it involves calculating the likelihood of different outcomes in a game. The St. Petersburg game is a seemingly simple gambling game where a fair coin is tossed until it lands heads. Each round doubles the previous payout.

Key principles of probability that apply here include:

• Probability of an event: This is a measure of the likelihood that the event will occur, represented by a value between 0 and 1. In this game, the probability is halved with each subsequent win (i.e., for round 1 it is \(\frac{1}{2}\), round 2 is \(\frac{1}{4}\), and so on).
• Cumulative probability: This refers to the sum of probabilities of all possible outcomes. It is crucial for determining the odds of reaching a certain round in the game.

Understanding these elements allows us to calculate the expected value of the game, which reflects an average outcome if the game is played an infinite number of times.

Expected Value

Expected value is a central concept in calculating the fairness or attractiveness of various probabilistic scenarios, such as betting games. In the St. Petersburg game, expected value tells us what a player can anticipate winning on average per game, over many iterations.

To find the expected value, we multiply the payoff of each outcome by its probability and sum all these up. For rounds 1 to 9, this is straightforward since the reward is \(2^k\) and the odds are \(\frac{1}{2^k}\), leading to an expected payoff of 1 for each round. Hence, the sum for these rounds is 9.

For rounds where \(k \geq 10\), the payoff stabilizes at \$1000. Each of these outcomes still has a probability of \(\frac{1}{2^k}\), but because the reward doesn’t increase, the calculation changes a bit. Summing these expected values gives an infinite series, requiring methods from geometric series for evaluation.

In essence, understanding expected value helps determine how much someone should logically pay to enter this game, seeking a balance between potential gains and the risks involved.

Geometric Series

A geometric series is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the St. Petersburg game, this concept comes into play when calculating expected value for rounds where \(k \geq 10\).

For these rounds, each payout of \$1000 has diminishing probabilities \(\frac{1}{2^{10}}, \frac{1}{2^{11}}, \ldots\). This forms a geometric series with the first term \(a = \frac{1000}{2^{10}}\) and common ratio \(r = \frac{1}{2}\). The sum of such a series can be calculated using the formula:\[S = \frac{a}{1 - r}\]where \(a\) is the first term, and \(r\) is the common ratio (\(|r| < 1\)).

Understanding geometric series is crucial to determine the total expected payoff from the game lifetime when higher payouts are considered. This, in turn, helps in figuring out the 'fair' ante value for the game.