If there is a linear relationship between test score (X, ability) and test score self-assessment (Y, self-perception), then the random variables are modeled as:

$$ Y \sim aX+b+N $$

Where N is some statistically independent noise, mean zero.

This means the covariance between them is

$$ Cov(Y-X,X) = E[ ((a-1)X+b+N -(a-1)E[X]-b) (X - E[X]) ] $$

Which is

$$ Cov(Y-X,X) = E[(a-1)(X-E[X])(X-E[X])] + E[N(X-E[X])]= (a-1) Var[X] $$

To get a "DK effect" we need (a-1) < 0, or a < 1. If a=0, in the case of the blog post, then this is absolutely true. If a=1 (which, along with b=0, is the ideal scenario), then this is barely not true. If a > 1, then we'd have a whole new effect about arrogant experts.

So the only thing that matters from this "auto-correlation perspective" is the rate at which an individual's self-assessment increases with their ability. As long as they underestimate the increase, a "DK effect" will occur.

However, in the above analysis, we ignored the variable b. If a = 0.8 and b=0, we'd never have the so-called "DK effect" even though it matches the "auto-correlation perspective" because everyone would underestimate their ability.

This tells me that the value of b matters. It is sort of like the prior ability everyone assumes they have. What the DK papers shows is that b > .5, which I think is in line with the spirit of the popular interpretation of the "DK effect". People should not be assuming they have, at a minimum, a capacity higher than the average.

At the same time, the value b isn't insanely higher than .5, which also makes me want to cut those unskilled and unaware some slack. It "seems reasonable" to assume your baseline is average. That can't be the case, but it feels intuitive.