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 $2n 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.