St. Petersburg Paradox

The St. Petersburg Paradox is a problem related to probability and decision theory in economics. It’s based on a lottery game that results in a random variable with infinite expected value, but still seems to be worth only a small amount to the players.

The paradox is as follows:

“A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The pot starts at 2 dollars and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if a tail appears on the first toss, 4 dollars if a head appears on the first toss and a tail on the second, and so on. Expressing this mathematically,  the player wins 2k dollars, where k equals number of tosses and is a positive integer. What would be a fair price to pay the casino for entering the game?”

In order to answer this we must consider the average payout. As the probability of getting 2 dollars is 1/2,the probability of getting 4 dollars is 1/4 and so on, the expected value is:

{\displaystyle {\begin{aligned}E&={\frac {1}{2}}\times 2+{\frac {1}{4}}\times 4+{\frac {1}{8}}\times 8+{\frac {1}{16}}\times 16+\cdots \\&=1+1+1+1+\cdots \\&=\infty \,.\end{aligned}}}

Note that we must assume that the game can continue infinitely, i.e. the casino has unlimited resources, hence the sum grows without bound. Thus, as the expected value is infinite, one should play the game at any price if offered the opportunity. However, as described by Ian Hacking “few of us would pay even $25 to enter”. The paradox arises from the difference in what people are willing to pay in real life, versus the infinite expected value.

Expected Utility?

To Daniel Bernouilli, “the determination of the value of an item must not be based on the price, but rather on the utility it yields”, and hence suggested that the answer relied on the expected utility, rather than expected value. Bernoulli proposed a utility model which involves the logarithmic function U(w) = ln(w), where w is the gambler’s total wealth. Note that each possible event in weighted by the probability of that event happening. If c is the cost of entering the game, the expected incremental utility converges to a finite value:

{\displaystyle \Delta E(U)=\sum _{k=1}^{\infty }{\frac {1}{2^{k}}}\left[\ln(w+2^{k}-c)-\ln(w)\right]<\infty \,.}

This formula gives a relationship between the gambler’s wealth and how much he should pay. For example, a millionaire should be willing to pay up to $20.88 and a person with $1,000 should pay up to $10.95.

However, this solution was criticised by other economists. Alfred Marshall replaced ‘wealth’ in Bernoulli’s solution by ‘income’, which changed the main conclusions of the paradox. Marshall explains how, when assuming decreasing marginal utility, no rational individual would play the game since losses would be greater than gains.

St Petersburg paradox-marshall

Consider the probability of loss being equal to the probability of winning = ½. If a player loses, they will lose the money paid to play the game (x-1) but if they win, the player will win exactly the amount paid (x+1). Since area A is greater than B no rational player would play.

Alfred Marshall puts it in such way:

“The clerk with £100 a-year will walk to business in a much heavier rain than the clerk with £300 a-year; for the cost of a ride by tram or omnibus measures a greater benefit to the poorer man than to the richer. If the poorer man spends the money, he will suffer more from the want of it afterwards than the richer would. The benefit that is measured in the poorer man’s mind by the cost is greater than that measured by it in the richer man’s mind.”

Hope you enjoyed today’s post! M x

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