# St. Petersburg Paradox

The St. Petersburg Paradox is a strange state of affairs that arises from a game proposed, in 1713, by
Nikolaus (I) Bernoulli. It is named after
the fact that a treatise on the paradox was written by Nikolaus' cousin,
Daniel, and published (1738) in the *Commentaries of the Imperial Academy
of Science of St. Petersburg*. The game goes as follows. You toss a
coin. If it shows heads you get $2. Otherwise, if it shows tails, you toss
again. If the coin now shows heads you get $4, and so on. Whenever you toss
tails the prize is doubled. After *n* tosses you get $2^{n} if heads appear for the first time. The only catch is you have to pay the
play the game. How much should you be willing to pay? Classical decision
theory says that you should be willing to pay any amount up to the expected
prize, the value of which is obtained by multiplying all the possible prizes
by the probability that they are obtained and adding the resulting numbers.
The chance of winning $2 is 1/2 (heads on the first toss); the chance of
winning $4 is 1/4 (tails followed by heads); the chance of winning $8 is
1/8 (tails followed by tails followed by heads); and so on. Since the expected
payoff of each possible consequence is $1 ($2 × 1/2, $4 × 1/4),
etc) and there are an infinite number of them, the total expected payoff
is an infinite sum of money. A rational gambler would enter a game if and
only if the price of entry was less than the expected value. In the St.
Petersburg game, any finite price of entry is smaller than the expected
value of the game. Thus, the rational gambler would play no matter how large
the entry price was! But there's clearly something wrong with this. Most
people would offer between £5 and £20 on the grounds that the chance of
winning more than £4 is only 25% and the odds of winning a fortune are very
small. And therein lies the paradox: If the expected payoff is infinite,
why is no one willing to pay a huge amount to play?

The classical solution to this mystery, provided by Daniel Bernoulli and
another Swiss mathematician, Gabriel Cremer, goes beyond probability theory
to touch areas of psychology and economics. Bernoulli and Cremer pointed
out that a given amount of money isn't always of the same use to its owner.
For example, to a millionaire $1 is nothing, whereas to a beggar it can
mean not going hungry. In a similar way, the utility of $2 million is not
twice the utility of $1 million. Thus, the important quantity in the St.
Petersburg game is the **expected utility** of the game (the
utility of the prize multiplied by its probability) which is far less than
the **expected prize**. This explanation forms the theoretical
basis of the insurance business. The existence of a utility
function means that most people prefer, for example, having $98 in cash
to gambling in a lottery where they could win £70 or £130 each with a chance
of 50%, even though the lottery has the higher expected prize of £100. The
difference of £2 is the premium most of us would be willing to pay for insurance.
That many people pay for insurance to avoid any risk, yet at the same time
spend money on lottery tickets in order to take a risk of a different kind,
is another paradox, which is still waiting to be explained.