Let’s tweak it, make it better. Suppose we run a national campaign to raise awareness about estimating Mesd-su-Re’s nose. TV ads, radio spots, pundits, community organizers, teachers, even bureaucrats all getting the word out about this important subject. That’ll really bring ’em in!
Before we begin, I should tell you that only vile racists guess lengths under four inches. Only those sympathetic with the war on women go short. The right side of history is with the long nose! Elites stand as one. Stars and starlets agree: length matters.
Okay, everybody. Put down your numbers. I’ll wait here.
Argumentum ad populum.
He said what?
Ignore the blatant political prompting and suppose only that I had asked for the length of Mesd-su-Re’s nose. If people had no information, other than the usual olfactory arcana we all possess—e.g., none of us has seen a human nose longer than one meter and noses can’t have negative length—there is no reason to suppose just guessing-and-averaging is helpful. How could it be? Ignorance plus ignorance divided by two is still ignorance. Ignorance-averaging is a fallacy which is actually well known; goes by the name The Chinese Emperor’s Nose. I changed it to an Egyptian Prince here for variety and for a second reason to be revealed below.
The proof the Wisdom of Crowds is the Chinese Emperor’s Nose fallacy is somewhat involved, but here’s a rough sketch. People’s guesses about Mesd-su-Re’s nose will have a minimum, maximum, and some arithmetic mean which lies between (or at one of) these two. If people have no idea about the length (except possibly rough bounds; this is the key) then the mean of guesses is probably near the midpoint, the center of maximum minus minimum. Then regardless of where the real answer lies, the error (distance from guess to real answer) averaged across people will be the same as the error using the mean. In other words, crowds have no wisdom of subjects in which they are ignorant.
Wait! That the Wisdom of Crowds can sometimes provide reasonable predictions is obviously true. It could work in the sense as when an economist throws an equation at a list of stocks which sticks. But that’s (credentialed) luck. Crowd wisdom is also successful when people have some idea, some unbiased idea, of the answer. If individuals in a group had opinions like, “I don’t know exactly what the answer is, but I know or can see it’s X plus-or-minus” then averaging might provide superior guesses to the average individual.
Yet when a crowd is fed biased information the game is off.
Example. You might look at that jar of pennies (long-time reader and contributer DAV reminded me of this example) and know that it can’t contain a million pennies; no, nor a hundred thousand. But we all know pennies and many of us have jars of change, so we could all form a crude but not insane idea of the number. The average of many in this case is likely to be a good guess.
Then imagine a moustachioed slickster standing by the jar whispering, “Psst, buddy. There’s a solid cone of cork in the middle. Only looks like there’s a lot of pennies. Word to the wise.” Finger on the nose and everything. Hey, he might be in on it: could be a hot tip—and if many think so there goes the accuracy of the average (supposing he’s fibbing).
Recapitulation. Wisdom of the crowds isn’t worth squat when individuals are ignorant of the subject matter they’re guessing. Averaging is okay, but only when folks are using unbiased information. The bad news is already well known: spreading misinformation works. People, even groups of them, will come to wrong conclusions conditioning on flawed premises.
So what does this have to do with voting? Well, everything. But that’s for another time.
Meanwhile, here’s the answer. Zero, you racist. Inches or centimeters. The length of Mesd-su-Re’s nose at the end of his life. Ramses had it sliced off for daring to corrupt his (Ramses’s) harem. Ouch. The lesson is: don’t guess unless you have to, and when you do, be less confident.
See also Voting (And Wisdom of the Crowds).