St. Petersburg Paradox

Saint Petersburg Paradox

The Saint Petersburg Paradox is a logical problem concerning the expected value of a series game. It is named after the Russian city of Saint Petersburg, where it was published in 1738 by Swiss mathematician Daniel Bernoulli. The paradox was first proposed as part of a disagreement between Bernoulli and fellow scientists who believed that the expected value of games of chance should be calculated by multiplying the size of a players reward by the probability of receiving it. Bernoulli argued that it should be calculated by determining the average of all possible outcomes.

The original problem posed by the Saint Petersburg Paradox can be explained as follows: two players participate in a series of coin flips. If the coin lands heads, the player is rewarded with double the amount of money they originally wagered from a pot. If the coin lands tails, they receive nothing. The expected value of the game is calculated by multiplying the probability of heads (50%) by the size of the reward (2x the pot). Therefore, the expected value of the game is one.

However, when calculating the expected value of the game using Bernoulli’s method of finding the average of all outcomes, the paradox becomes apparent. If the pot contains one unit of currency, the expected value of the game is infinite. Because the probability of heads is 50%, the expected value of the game should be one. However, as the size of the pot increases, the expected value of the game increases exponentially. For example, if the pot consists of two units of currency, the expected value of the game is four. Conversely, if the pot contains one million units of currency, the expected value of the game is one million.

This apparent contradiction can be seen in chart form. As the size of the pot increases, the expected value of the game also increases exponentially.

Size of Pot 1 2 4 8 16 32 64 128 256 512

Expected Value 1 4 16 64 256 1024 4096 16384 65536 262144

One of the most widely accepted theories as to how this paradox occurs is the law of diminishing marginal utility. The principle of diminishing marginal utility holds that additional units of a good become less valuable in utility as more units are consumed. This concept can be seen in the Saint Petersburg Paradox — as the size of the pot increases, the utility of additional units of currency decreases, thus resulting in an expected value that is higher than the expected value when calculated using the traditional method.

It is important to note that the Saint Petersburg Paradox does not always result in an infinite expected value — in some cases, the expected value is finite. For example, if the size of the pot is infinitely large, then the expected value of the game is also finite. This is because, at a certain point, the expected value of the game will stabilize and reach a point of diminishing returns.

The Saint Petersburg Paradox is an interesting paradox that helps to illustrate the concept of expected value and marginal utility. Despite its counter-intuitive nature, the paradox can be explained using basic mathematical principles. Moreover, it has led to important discoveries in the fields of mathematics, economics, and game theory.

The St. Petersburg Paradox, also known as the St. Petersburg Lottery, was first proposed by Swiss mathematician Daniel Bernoulli in 1738. Bernoulli postulated that by repeatedly flipping a fair coin, one could expect to win $2^n once in a while, and that the expected value of the lottery would be infinite, as the probability of winning increases with each coin flip.

The paradox lies in the fact that the expected value of the lottery is infinite, yet no rational person would be willing to pay an infinite amount of money to purchase it. This paradox has puzzled philosophers, mathematicians and economists since it was published, and has at times been used to criticize the rational expectations theory of neoclassical economics.

In the St. Petersburg Paradox, the probability of winning varies exponentially with each coin toss. This means that each additional flip adds a finite amount to the expected value of winning the lottery, but the amount of money that one must pay to enter the lottery increases (perhaps exponentially in some cases) with each flip.

This is despite the fact that the expected value of the lottery is still considered to be infinite. Thus, the paradox is essentially one of expected utility—we should expect to pay an infinite amount of money for an infinite expected gain, but in practice the amount of money necessary to purchase the lottery ticket is simply too large for most people to be willing to pay.

In order to resolve this paradox, some economists have suggested that people simply don’t possess an infinite amount of money, or that the risk of failure is too great to be willing to pay an infinite amount. Others have suggested that while the expected value is infinite, the expected utility (i.e., the amount of satisfaction one gets from entering the lottery) is actually finite and limited, due to the possibility of losing all the money.

Whatever the resolution, the St. Petersburg Paradox remains one of the most fascinating and perplexing puzzles in economics.