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.
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.)
Bernoulli recognized that people weren't being stupid to refuse to pay billions of dollars for a game, no matter what the payoff works out to be mathematically. He believed that, when looking at how people make decisions, we need to consider the practical aspects of a game, not just the mathmatical outcome. Practicality constrains rationality in human decision making. It took until 1950 for academics to come up with a formal term for it. Psychologist and sociologist Herbert Simon called it "bounded rationality."
Simon thought it was important to look at decision-making not just as a statistical problem, but as a practical one in human lives. Sometimes, he acknowledged, people were stupid or stubborn when making choices. For example, studies show that sometimes when people who have researched a certain product walk into a store, they will reject obviously better deals for a lesser deal that they've already decided on. They've narrowed down their decision and won't consider new options.
On the other hand, sometimes the only way to make a decision is to cut down on options. Simon pointed out that researching different options was a cost in and of itself. People are rational, therefore, if they do a quick search, they realize that all the decent options for a new phone are within $200 of each other, and decide they don't need the extra money if it means slogging through 50 different data plans. So cutting down on time and effort might be well worth not playing around with coin-flip variations.