The St. Petersburg Paradox
The St. Petersburg Paradox is a famous problem in economics and game theory that has fascinated numerous economists, philosophers and mathematicians since it was first presented in the late 18th century. The paradox is based on a simple game, in which a coin is flipped a certain amount of times, and the player receives a certain amount of money for each heads that comes up. The more heads the player gets, the more money they receive, and the payouts increase exponentially. According to the traditional formulation of the game, if a head comes up on the infinity-flip, the player will receive an infinite amount of money.
Despite the seemingly obvious conclusion that the player should be willing to pay an infinite amount of money to play this game, empirical studies have shown that most people are unwilling to pay more than a few hundred dollars to play. This is the heart of the paradox - why would people, when faced with the possibility of gaining an infinite amount of money, not be willing to pay the initial cost?
In order to answer this question, we must first take a look at the utility (or satisfaction) of money, and how it changes as wealth increases. Studies have found that as wealth increases, the utility of money decreases - put simply, while having some money is better than having none, having more money does not provide proportional amounts of additional satisfaction. This concept, known as the diminishing marginal utility of wealth, is the key to understanding the St. Petersburg Paradox.
Given this principle, it is easy to see why people would be unwilling to pay an infinite amount of money to play the game - in most cases, the satisfaction gained from winning even a large sum of money would not be enough to justify such a large expenditure. If, for example, a player stands to win one million dollars from a single coin flip, then the utility of money will be much less than if the same person were to win an infinite amount - the difference in utility is far greater than the difference in monetary value.
This can also be used to explain why people are willing to pay a few hundred dollars to play the game - while the utility of the expected gain is still lower than the amount paid to play, the difference is small enough to make it a rational decision. After all, the possibility of winning an infinite amount of money is still present, even if it is extremely unlikely.
Ultimately, the St. Petersburg Paradox highlights the fact that humans make decisions based on utility rather than monetary value. While money can certainly be an important factor in making decisions, it is not the only one, and it is important to understand the limitations of monetary value in order to make rational decisions.
