The St. Petersburg Paradox is a classic problem in economics that challenges traditional theories of utility and decision-making. It is named after the city of St. Petersburg, Russia, where the paradox was first introduced by the mathematician Daniel Bernoulli in 1738.
The paradox revolves around a hypothetical gambling game. In this game, a player pays an entry fee to participate. A fair coin is then tossed repeatedly until it lands on tails. The player receives $2 raised to the power of the number of tosses required until the first tails appears. For example, if the first tails appears on the third toss, the player would receive $2^3 = $8. The game continues until the first tails appears.
The paradox arises when considering the expected value of the game. The expected value is calculated by multiplying the payoff of each outcome by its respective probability and summing them up. In the St. Petersburg Paradox, the expected value of the game is infinite since the potential winnings keep doubling with each toss of the coin.
However, many individuals would not be willing to pay an arbitrarily high entry fee to play the game, even though the expected value is theoretically infinite. This contradiction challenges the classical assumption that individuals maximize expected utility when making decisions.
The St. Petersburg Paradox highlights the role of diminishing marginal utility. According to this concept, as individuals accumulate wealth, the additional utility derived from each additional unit of wealth diminishes. In the case of the St. Petersburg Paradox, the potential gains are incredibly large, but the marginal utility of those gains decreases significantly as the wealth increases.
This paradox has sparked debates and discussions about alternative theories of utility, such as expected utility theory, which accounts for risk aversion by incorporating a diminishing marginal utility of wealth. It also highlights the limitations of solely relying on expected values in decision-making and the need to consider subjective preferences and attitudes towards risk.