Friday, 31 January 2014

The St Petersburg Paradox

In Lecture 7 I mentioned that $EX$ may not exist, and that for a random variable $X$ such that $X\geq 0$ it is possible for $EX=\infty$. Here's a famous example arising from the case $EX=\infty$. See also, Examples sheet 3, #19.

The St Petersburg Paradox
How much would you be willing to pay to play the following game? We will toss a fair coin repeatedly. We start you with £1 and then double this every time the coin shows a head. We stop when the first tail occurs. You keep your winnings, denoted by the random variable $X$. Now

$EX = (1/2)1+(1/4)2+(1/8)4+(1/16)8 +\cdots = \infty$.

Since the expected value of your winnings is infinite, it seems like you should be willing to pay an arbitrarily large amount to play this game! But would you really be willing to pay £1,000,000? (if you could!) Surely not. If you paid £1,000,000 for the right to play this game, the probability is 0.9749 that you will win less than you paid.

This is the St. Petersburg paradox  posed by Nicolas Bernoulli 1713.The paradox concerns what the mathematics predicts versus what a rational person might really do in practice. One resolution is that a person's utility for money is not linear. We should really be looking at $Eu(X)$, where perhaps $u(X)=\log X$. In that case $Eu(X) < \infty$.