We come from the future
We come from the future

# This Paradox Shows Why Irrational Choices Are Sometimes Better

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

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.)

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.

### DISCUSSION

synthozoic

You know, once I learned about fuzzy logic decades ago, I understood that poor Norman had a way out of that paradox. With an smooth continuum between the binary values of true and false, you can resolve stuff like the excluded middle and the liar paradox simply by stating the conclusion is half-true (Or half false.).

Or, instead of fuzzy logic, Norman could have just done what most humans do when confronted with the excluded middle or the lair paradox, just ignore it as a waste of time and then beat up Kirk and Mudd for having the impudence to fuck with his head.*

I'm reminded of this old Rudy Rucker cartoon:

Where the human mathematician tries to use Godelian thinking to outsmart the robot mathematician he built. Which is doomed from the start if we really dig into it, Godel himself pretty much implied as much. The human can claim that there is con(R) for his robot but simply can't claim there isn't a con(H) for himself. A person can not claim not to have a set of fixed axioms (And algorithm in other words.) governing their mind.

People who use Godel's theorems to disprove strong AI (People like Roger Penrose.) seem to conveniently overlook this.

* Or that episode of the Prisoner, "The General" The computer could have just said, "Nice try, Number Six. To which I reply with a question of my own, why not? You know, you're an imbecile for not having figured out who Number One really is. Think about that for moment, why don't you?"