Psychologists and sociologists point out all the ways that human beings are far less rational than we like to consider ourselves. Sometimes, though, their analysis is blind to some of the factors humans consider automatically. And this paradox proves it.

The Saint Petersburg Paradox is also known as the Saint Petersburg Lottery, although no city or state would ever offer it as a lottery. No casino would offer it either. No underground gambling ring would offer it... unless they had a way to rig the game. The paradox takes its name from the academic journal that published it. The 1738 edition of the Commentaries of the Imperial Academy of Science of Saint Petersburg published the work of young Daniel Bernoulli, in which he proposed the following game.

What if you bet on how long it would take for a flipped coin to come up heads? If the coin comes up heads on the first flip, you get two dollars. If it doesn't come up heads until the second flip, you get four dollars. If it comes up heads on the third flip, you get eight dollars. It seems like this is a no-lose scenario for you. You just keep getting paid. The question is, how much would you pay to play the game?

Bernoulli figured out the right price for the game by calculating the expected money that could work as an outcome. The expected value can be calculated by calculating the payoff and the odds, and adding them together.

2 (1/2) + 4(1/4) + 8(1/8) + 16 (1/16) . . . . = Winnings

You'll notice that if multiply the odds and the payoff together, you'll get an infinite series of ones, all added together. Essentially, you could pay an infinite amount of money, and still come out ahead.

#### Bounded Rationality In Saint Petersburg and Out of It

Most people will offer to pay between \$5 and \$10 for the game. This is the "paradox" part of the paradox. The math works out, but there's no way anyone is actually going to pay what the game is worth. (Even if they could pay an infinite amount of money.)