Wisdom of the Crowd: not always so wise

That's kind of Eliezer's point when he talks about how astounding it is that human beings are unbiased estimators of beans in a jar. I'd agree that it's astounding, but there are plenty of other statistical phenomena that astound me equally, so I've learned to not treat my level of astonishment as a precision tool for judging incredibility.

To some extent, I suspect the mechanism of estimation plays a significant role. I doubt very much that human beings have built-in heuristics for appraising large numbers of objects. Arithmetic is a fairly novel concept, evolutionarily speaking, and some cultures don't even have the natural numbers.

So when we try and guess the number of beans in a jar, there's presumably no single go-to mechanism we're using to come up with that value. It will be some sort of aggregate of sources, such as our past experience of beans in jars, visualisations of what 200 or 400 or 600 beans all in one place might look like, or rough guesses of volume and packing density. It isn't even necessarily a transparent process. If you try and make a rough estimate of something, aren't you using some sort of basis for that? It's not like the number just pops into your head. You wrestle with it for a little while.

Individual components of that estimation may be subject to bias in a given direction, but over enough sources, over enough people with many different estimation criteria, I wouldn't trust there to necessarily be a demonstrable bias over repeated experiments without deliberate intervention on the part of the experimenter, such as using a container of an unusual shape that would result in a known overestimation of its volume.

Edit: I should also add an expectation of bias idiosyncratic to specific questions. For example, I think it was Yvain's most recent LW membership poll that asked for the date Newton published his Philosophiæ Naturalis Principia Mathematica. If there was a widely-believed false date for this event, that would be an obvious source of noise that wouldn't be cancelled out by corresponding noise on the other side of the true value.

Let's say you're asking a thousand people to guess the date of the Battle of Bosworth Field. If I asked this right now in Less Wrong, I imagine it would receive some wildly different answers.

If you're me, and you remember it because its anniversary is on your birthday, (or if you were paying attention in a specific history class) you'll know the exact year (1485). These people are probably not very numerous, but their answers will all coincide and converge. This subgroup would also have a variance of zero.

All the people who were paying only a little bit of attention in that history class, or watched the first series of Blackadder, will not know the exact date, but they'll probably guess to within a few decades. This subgroup has a wider variance, but it's still pretty tight, and they're answering a convergent question. There's a correct answer, and the answer these people give is informed by it, even if it's not correct. In the absence of systematic bias, we would expect roughly the same number of people to answer 1480 as 1490, and so the mean of this group should converge.

We now look at a wider variance subgroup, which includes all the people who only have a sketchy idea of when this battle was and what it was about. Some people will recall it's got something to do with the Tudor dynasty, and Henry VIII was early 16th century. Some will recall that there was a King Richard involved, and dig up a late 14th century connection. They are all contributing some information to proceedings, (14th-16th Century), but in the absence of systematic bias, we'd expect people to be as wrong on one side as they are on the other. Even greater variance subgroups, who aren't sure whether this battle was fought by Romans or Crusaders or Confederates, are still contributing some small quantity of information by giving answers in the range of human history. No-one's going to say 3991 AD, or 6,000,000 BC.

As the variance gets wider, the population of any given subgroup gets larger, but the coherence of their answers gets smaller. If you take a hundred people who have absolutely no knowledge of human history and ask them when the Battle of Bosworth Field occurred, you're basically asking them to pick a number. Their answers aren't going to converge on anything, so they won't systematically interfere with the overall distribution, while the answers that are more informed will converge on the correct answer.

But systematic bias does occur. American education on non-American history is notoriously sketchy. If our participants included a large number of Americans, they're more likely to guess a date in American history through the availability heuristic. All of a sudden, the uninformed answers will start converging at some point in the late 19th Century, which will skew the overall distribution and pull the mean forward in time. The least wise parts of the crowd suddenly found a way to be a whole lot louder.

That's what I meant by your noise converging on the same answer. In giving people an incorrect choice, you're giving all the people who have no knowledge an opportunity to pick the same incorrect answer. If they didn't have that answer to converge on, the mean of their answer wouldn't be able to exert as much influence on the overall distribution.

Does that make sense?

(This also does point to an obvious source of systematic bias when dealing with dates: we have better records [and hence more available knowledge] of events closer to the present. History is lumpy, and forward-weighted, so any uninformed guess on the date of an event in the past is going to be distorted around points of greater historical interest, many of which occurred over the last century).

Could you summarize any differences in performance between the two question formats?

Closer to the truth than the other answer. That is, if the truth is 1000 million and the available options are 1300 million and 200 million, then the first answer is 300 million away from the truth, while the second is 800 million away from the truth. One of those two answers is more approximately correct than the other.

Actually, looking back at the question asked, approximate truth isn't even an issue. The question asked which of the two numbers was closer to the (true) population of Africa. One is closer, the other is not. So, when one answers 1300 million in that forced choice, one is answering the question correctly.

Does that make sense?

Closer to the truth than what?

Closer to the truth than coin flipping?