Bayes Theorem And Decision Theory In The Media

This is a clip from the show Family Guy. Here’s my transcription:

Salesman: Hold on! You have a choice… you can have the boat, or the mystery box!

Lois: What, are you crazy? We’ll take the boat

Peter: Woa, not so fast Lois. A boat’s a boat, but a mystery box could be anything! It could even be a boat! You know how much we’ve wanted one of those!

Lois: Then let’s just–

Peter: We’ll take the box!

Everyone understands why what Peter Griffin did here was dumb (or maybe you don’t, and only laugh because other people laugh?). Peter’s failure was a failure of probability, deciding on the mystery box when he wanted a boat instead of just going for the boat.

We can put Peter’s dilemma into the format of Bayes Factor and see that the evidence was in favor of him choosing the boat option (if he wanted a boat) instead of the mystery box. So this would be the probability of getting a boat given that he picked the boat option divided by the probability of getting the boat given that he picked the mystery box. Or, let B represent getting a boat and O represent picking the original option, and ~O represent the mystery box.

P(B | O), the probability of getting the boat given that he picked the original option, is 100%. P(B | ~O), the probability of getting the boat given that he picked the mystery box, is some other number. But this number has to include both the probability of getting the boat plus the probability of getting something else. And, since it’s a mystery box, it could have any number of other things to win. Remember that P(B | ~O) + P(~B | ~O) = 100%, so P(B | ~O) = 100% – P(~B | ~O), or 100% minus the probability of not getting the boat given that he picked the mystery box. The big problem is that it’s a mystery box, so it can account for any data.

But this isn’t the whole story behind Peter’s fail of logic.

Remember my post on decision theory? That applies here, as well. A utility function in decision theory is basically where you multiply the probability of the event happening with the amount of utility (an arbitrary number, equivalent to “happiness points”; or “cash” if you’re an economist) you would get from it coming to pass. Combining both Peter’s low probability fail with the mystery box to decision theory, we get one of the classic examples of cognitive biases: the Framing Effect. It goes like this:

Participants were told that 600,000 people were at risk from a deadly disease. They were then presented with the same decision framed differently. In one condition, they chose between a medicine (A) that would definitely save 200,000 lives versus another (B) that had a 33.3 per cent chance of saving 600,000 people and a 66.6 per cent chance of saving no one. In another condition, the participants chose between a medicine (A) that meant 400,000 people will die versus another (B) that had a 33.3 per cent chance that no one will die and 66.6 per cent that 600,000 will die.

For some reason, people choose the saving of 600,000 people at 33.3% probability due to the way the question is framed, even though the two scenarios are equivalent. The brain might be just comparing the sizes and ignoring the probabilities.

Analogously, Peter’s boat option is equivalent to the 100% chance of saving 200,000 people, and his mystery box option is equivalent to the 33.3% chance of saving 600,000 people (though, like I said, a mystery box has some unknown — but necessarily less — probability of being a boat). If you simply replace “people” with “utility” in the Framing Effect example, you realize that the two options are roughly equivalent (33.3 * 600,000 = 200,000) on the positive utility side. We also have to account for the negative utility of having a 66% chance of saving no one. That would be 66.6 * 600,000 compared to 100% * 400,000, which are also equal from a utility perspective.

And this is why Peter shouldn’t have picked the mystery box. We don’t actually know the probability of getting the boat given that he chose the mystery box but, like I said, it’s necessarily lower than 100%. Similarly, if Peter chose the boat outright, that’s a 0% chance of him getting anything else. We also don’t know the amount of utility for him getting anything else, but we do know that his utility of getting the boat seems to be pretty high. This is the crucial difference between Peter’s utility function and the one represented with the Framing Effect: Peter’s utility for getting the boat isn’t the same with either option.

Of course, the thing about logical fallacies is that, due to their non-sequitur nature, they are oftentimes used as jokes. That’s why Peter’s choice is also hilarious.

What’s black and rhymes with Snoop? Dr Dre.

Funny, but also a fallacy of equivocation.

It’s probably just a coincidence, but the creator of Family Guy is an atheist. Peter basically chose the “god” option for his explanation instead of the more precise boat option in the above scenario. God represents the mystery box, not only because theists think that god being mysterious is a good thing, but because god, just like the mystery box, can account for any possible data imaginable (even a boat).