I believe that the argument in my previous comment applies to any case satisfying the following.
Assume again that intelligence is measured by a quantity between 0 and 1. Assume there are two worlds, both with a prior distribution p for intelligence applying to the entire population. Furthermore, in the second world, the total population is divided into two equal sub-populations, f-people and g-people, with respective posterior distributions f and g for their intelligence. Assume the following about these distributions:
The support of each distribution p, f, and g is the entire interval [0,1]. That is, these distributions are all nonzero over the entire interval.
The prior distribution p is symmetric about 0.5. That is, p(x) = p(1 − x) for 0 ≤ x ≤ 1.
The distributions f and g are mirror images of each other about x=0.5. That is, f(x) = g(1 − x) for 0 ≤ x ≤ 1.
The expected intelligence for the f-people is below 0.5.
I believe that these assumptions suffice for the conclusions in my previous comment to follow. That is, in both worlds, exactly half the people are paid below their intelligence. But, in World 2 alone, some smart people are treated as dumb.
(Here I use the definitions from my previous comment, which I repeat here for convenience: Each employer computes an expected intelligence E for an employee and then pays that employee at a rate of E utilons-per-hour. “Dumb” means “intelligence less than 0.5“. “Smart” means “intelligence greater than or equal to 0.5”. Finally, “treating a smart person as dumb” means “paying an employee at a rate less than 0.5 when that employee’s intelligence is greater than or equal to 0.5″.)
I believe that the argument in my previous comment applies to any case satisfying the following.
Assume again that intelligence is measured by a quantity between 0 and 1. Assume there are two worlds, both with a prior distribution p for intelligence applying to the entire population. Furthermore, in the second world, the total population is divided into two equal sub-populations, f-people and g-people, with respective posterior distributions f and g for their intelligence. Assume the following about these distributions:
The support of each distribution p, f, and g is the entire interval [0,1]. That is, these distributions are all nonzero over the entire interval.
The prior distribution p is symmetric about 0.5. That is, p(x) = p(1 − x) for 0 ≤ x ≤ 1.
The distributions f and g are mirror images of each other about x=0.5. That is, f(x) = g(1 − x) for 0 ≤ x ≤ 1.
The expected intelligence for the f-people is below 0.5.
I believe that these assumptions suffice for the conclusions in my previous comment to follow. That is, in both worlds, exactly half the people are paid below their intelligence. But, in World 2 alone, some smart people are treated as dumb.
(Here I use the definitions from my previous comment, which I repeat here for convenience: Each employer computes an expected intelligence E for an employee and then pays that employee at a rate of E utilons-per-hour. “Dumb” means “intelligence less than 0.5“. “Smart” means “intelligence greater than or equal to 0.5”. Finally, “treating a smart person as dumb” means “paying an employee at a rate less than 0.5 when that employee’s intelligence is greater than or equal to 0.5″.)