If this were anything like my high school math class, everyone else in the class would decide to copy my answer. In some cases, I have darn good reasons to believe I am significantly better than the average of the group I find myself in. For example, I give one of my freshman chemistry midterms. The test was multiple choice, with five possible answers for each question. My score was an 85 out of 100, among the highest in the class. The average was something like 42. On the final exam in that class, I had such confidence in my own answer that I declared that, for one of the questions, the correct answer was not among the responses offered - and I was right; one of the values in the problem was not what the professor intended it to be. I was also the only one in the class who had enough confidence to raise an objection to the question.
On the other hand, there are situations in which I would reasonably expect my estimate to be worse than average. If I wandered into the wrong classroom and had no idea what the professor was talking about, I'd definitely defer to the other students. If you ask me to predict the final score of a game between two well-known sports teams, I probably wouldn't have heard of either of them and just choose something at random. (The average American can name the two teams playing in the Super Bowl when it occurs. I rarely can, and I don't know whether to be proud or ashamed of this.) I also suspect that I routinely overestimate my chances of winning any given game of Magic. ;)
I'm not a random member of any group; I'm me, and I have a reasonable (if probably biased, given the current state of knowledge in psychology) grasp of my own relative standing within many groups.
Also, when you're told that there is a hidden gotcha, sometimes you can find it if you start looking; this is also new information. Of course, you can often can pick apart any given hypothetical situation used to illustrate a point, but I don't know if that matters.
I've always been annoyed at the notion that the bias-variance decomposition tells us something about modesty or Philosophical Majoritarianism. For example, Scott Page rearranges the equation to get what he calls the Diversity Prediction Theorem:
I think I've finally come up with a nice, mathematical way to drive a stake through the heart of that concept and bury it beneath a crossroads at midnight, though I fully expect that it shall someday rise again and shamble forth to eat the brains of the living.
Why should the bias-variance decomposition be relevant to modesty? Because, it seems to show, the error of averaging all the estimates together, is lower than the typical error of an individual estimate. Prediction Diversity (the variance) is positive when any disagreement exists at all, so Collective Error < Average Individual Error. But then how can you justify keeping your own estimate, unless you know that you did better than average? And how can you legitimately trust that belief, when studies show that everyone believes themselves to be above-average? You should be more modest, and compromise a little.
So what's wrong with this picture?
To begin with, the bias-variance decomposition is a mathematical tautology. It applies when we ask a group of experts to estimate the 2007 close of the NASDAQ index. It would also apply if you weighed the experts on a pound scale and treated the results as estimates of the dollar cost of oil in 2020.
As Einstein put it, "Insofar as the expressions of mathematics refer to reality they are not certain, and insofar as they are certain they do not refer to reality." The real modesty argument, Aumann's Agreement Theorem, has preconditions; AAT depends on agents computing their beliefs in a particular way. AAT's conclusions can be false in any particular case, if the agents don't reason as Bayesians.
The bias-variance decomposition applies to the luminosity of fireflies treated as estimates, just as much as a group of expert opinions. This tells you that you are not dealing with a causal description of how the world works - there are not necessarily any causal quantities, things-in-the-world, that correspond to "collective error" or "prediction diversity". The bias-variance decomposition is not about modesty, communication, sharing of evidence, tolerating different opinions, humbling yourself, overconfidence, or group compromise. It's an algebraic tautology that holds whenever its quantities are defined consistently, even if they refer to the silicon content of pebbles.
More importantly, the tautology depends on a particular definition of "error": error must go as the squared difference between the estimate and the true value. By picking a different error function, just as plausible as the squared difference, you can conjure a diametrically opposed recommendation:
Here we are taking the square root of the difference between the true value and the estimate, and calling this the error function, or loss function. (It goes without saying that a student's utility is linear in their grade.)
And now, your expected utility is higher if you pick a random student's estimate than if you pick the average of the class! The students would do worse, on average, by averaging their estimates together! And this again is tautologously true, by Jensen's Inequality.
A brief explanation of Jensen's Inequality:
(I strongly recommend looking at this graph while reading the following.)
Jensen's Inequality says that if X is a probabilistic variable, F(X) is a function of X, and E[expr] stands for the probabilistic expectation of expr, then:
Why? Well, think of two values, x1 and x2. Suppose F is convex - the second derivative is positive, "the cup holds water". Now imagine that we draw a line between x=x1, y=F(x1) and x=x2, y=F(x2). Pick a point halfway along this line. At the halfway point, x will equal (x1 + x2)/2, and y will equal (F(x1)+F(x2))/2. Now draw a vertical line from this halfway point to the curve - the intersection will be at x=(x1 + x2)/2, y=F((x1 + x2)/2). Since the cup holds water, the chord between two points on the curve is above the curve, and we draw the vertical line downward to intersect the curve. Thus F((x1 + x2)/2) < (F(x1) + F(x2))/2. In other words, the F of the average is less than the average of the Fs.
So:
If you define the error as the squared difference, F(x) = x^2 is a convex function, with positive second derivative, and by Jensen's Inequality, the error of the average - F(E[X]) - is less than the average of the errors - E[F(X)]. So, amazingly enough, if you square the differences, the students can do better on average by averaging their estimates. What a surprise.
But in the example above, I defined the error as the square root of the difference, which is a concave function with a negative second derivative. Poof, by Jensen's Inequality, the average error became less than the error of the average. (Actually, I also needed the professor to tell the students that they all erred in the same direction - otherwise, there would be a cusp at zero, and the curve would hold water. The real-world equivalent of this condition is that you think the directional or collective bias is a larger component of the error than individual variance.)
If, in the above dilemma, you think the students would still be wise to share their thoughts with each other, and talk over the math puzzle - I certainly think so - then your belief in the usefulness of conversation has nothing to do with a tautology defined over an error function that happens, in the case of squared error, to be convex. And it follows that you must think the process of sharing thoughts, of arguing differences, is not like averaging your opinions together; or that sticking to your opinion is not like being a random member of the group. Otherwise, you would stuff your fingers in your ears and hum when the problem had a concave error function.
When a line of reasoning starts assigning negative expected utilities to knowledge - offers to pay to avoid true information - I usually consider that a reductio.