The St. Petersburg Paradox is a classic problem in probability theory and economics that was first introduced by Daniel Bernoulli in 1738. The paradox challenges the theory of expected value by presenting a scenario in which a player can potentially win an infinite amount of money, yet the expected value of the game is finite. In this article, we will discuss Bernoulli's solution to the St. Petersburg Paradox.
The Setup of the St. Petersburg Paradox
In the St. Petersburg Paradox, a player is offered a game of chance with the following rules:
- The player pays an entry fee of a certain amount, let's say $10.
- A fair coin is flipped repeatedly until it lands on tails.
- The number of coin tosses is recorded, let's say the coin lands on heads 4 times.
- The player wins 2 to the power of the number of coin tosses in dollars, so in this case, the player would win $16 (2 to the power of 4).
The Paradox
The paradox arises from the fact that the potential payout of the game increases rapidly as the number of coin tosses increases. For example, if the coin tosses resulted in 20 heads, the player would win $1,048,576, which is a vast amount of money. However, the probability of winning such a large amount is very low, as it requires 20 consecutive heads, which has a probability of only 1 in 1,048,576.
The expected value of the game can be calculated as the sum of the products of the payout and the probability of winning that payout, which is:
EV = (1/2) x $2 + (1/4) x $4 + (1/8) x $8 + ... + (1/2 to the power of n) x $2 to the power of n
Using this formula, we can see that the expected value of the game is infinite, as each payout is multiplied by a probability that decreases exponentially with each additional coin toss. However, the actual payout of the game is limited by the initial entry fee of $10.
Bernoulli's Solution
Bernoulli's solution to the St. Petersburg Paradox was to introduce the concept of utility, which is a measure of the satisfaction or happiness that a player derives from a certain amount of money. Bernoulli argued that the value of money to an individual is not simply determined by the amount of money itself but also by the individual's utility function.
Bernoulli posited that the marginal utility of money decreases as the amount of money increases. This means that the satisfaction derived from winning an additional $1 decreases as the individual's wealth increases. For example, the marginal utility of an extra $1 is likely to be higher for a person with very little money than for a person who is already very wealthy.
Using the concept of utility, Bernoulli argued that the expected utility of the game can be calculated as the sum of the products of the utility of the payout and the probability of winning that payout. The expected utility of the game would then be:
EU = (1/2) x U($2) + (1/4) x U($4) + (1/8) x U($8) + ... + (1/2 to the power of n) x U($2 to the power of n)
Bernoulli argued that the player's utility function is likely to be concave, meaning that the marginal utility of money decreases as the amount of money increases. Therefore, as the potential payout of the game increases, the marginal utility of each additional dollar decreases. This means that even though the expected utility of the game may be high, the player may not be willing to pay a high entry fee to play the game.
Bernoulli's solution to the paradox is essentially that the utility gained from playing the game is not infinite, despite the infinite expected value of the game. He argued that people do not value money solely on the basis of its nominal value but also on the basis of the subjective value that it holds for them. Therefore, in practice, people are unlikely to be willing to pay very large sums of money to play a game, even if the potential payout is infinite.
Critiques of Bernoulli's Solution
While Bernoulli's solution to the St. Petersburg Paradox has been widely accepted, it has also been subject to criticism. One criticism is that Bernoulli's utility function assumes that the marginal utility of money is always decreasing, which may not be the case for all individuals. For example, some individuals may experience increasing marginal utility of money due to a psychological need for security or a desire for power and status.
Another criticism of Bernoulli's solution is that it assumes that individuals have a fixed utility function. In reality, people's utility functions are likely to be influenced by various contextual factors, such as their current level of wealth, their personal values, and the social norms of their community.
The St. Petersburg Paradox is a classic problem in probability theory and economics that challenges the theory of expected value. Bernoulli's solution to the paradox introduced the concept of utility and argued that people do not value money solely on the basis of its nominal value but also on the basis of the subjective value that it holds for them. While Bernoulli's solution has been widely accepted, it has also been subject to criticism and further research is needed to fully understand the complex nature of human decision-making in the face of uncertainty.
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