Thomas Kwa comments on Fixing The Good Regulator Theorem

[Thomas Kwa]

Vivek points out the following variant. If the effects of the regulator and system are both modulated by some intermediate functions $I_1,I_2$ which are deterministic.

$X\to R\to I_1\to Z$

$X\to S\to I_2\to Z$

Then the “consequential parts” of every good regulator R (defined as the parts of R’s state whose information is preserved by the intermediate function) are a deterministic function of the consequential parts of every system S. This is also technically implied by the lemma, but framing it this way makes it more clear.

Intuitive argument: The lemma implies that $Z$ is a deterministic function of $I_2$. The “simplest regulator” assumption on $I_1$ says that $I_1$ is a deterministic function of $I_2$. We can propagate this back to the “consequential parts” of R and S.

This is meaningful because the real world often throws away information from the perspective of agents. It might be useful in a selection theorem that starts with optimization as a premise.

[johnswentworth]

Note for others: this is a variant of the original Good Regulator theorem. In particular, it assumes that the regulator R has perfect information about system state S, which is why $I_1$ is able to be a deterministic function of $I_2$.