In a previous post, I argued that Kolmogorov complexity/simplicity priors do not help when learning human values - that some extreme versions of the reward or planners were of roughly equal complexity.

Here I'll demonstrate that it's even worse than that: the extreme versions are likely simpler than a "reasonable" one would be.

Of course, as with any statement about Kolmogorov complexity, this is dependent on the computer language used. But I'll aim to show that for a "reasonable" language, the result holds.

So let (p,R) be a reasonable pair that encodes what we want to encode in human rationality and reward. It is compatible with the human policy πH, in that p(R)=πH.

Let (pr,Rr) be the compatible pair where pr is the rational Bayesian expected reward maximiser, with Rr the corresponding reward so that pr(Rr)=πH.

Let (pi,0) be the indifferent planner (indifferent to the choice of reward), chosen so that pi(R′)=πH for all R′. The reward 0 is the trivial reward.

Information content present in each pair

The planer pi is simply a map to πH, so the only information in pi (and (pi,0)) is the definition of πH.

The policy πH and the brief definition of an expected reward maximiser pr are the only information content in (pr,Rr).

On the other hand, (p,R) defines not only πH, but, at every action, it defines the bias or inefficiency of πH, as the difference between the value of πH and the ideal R-maximising policy πR. This is a large amount of information, including, for instance, every single human bias and example of bounded rationality.

None of the other pairs have this information (there's no such thing as bias for the flat reward 0, nor for the expected reward maximiser pr), so (p,R) contains a lot more information than the other pairs, so we expect it to have higher Kolmogorov complexity.

A putative new idea for AI control; index here.In a previous post, I argued that Kolmogorov complexity/simplicity priors do not help when learning human values - that some extreme versions of the reward or planners were of roughly equal complexity.

Here I'll demonstrate that it's even worse than that: the extreme versions are likely simpler than a "reasonable" one would be.

Of course, as with any statement about Kolmogorov complexity, this is dependent on the computer language used. But I'll aim to show that for a "reasonable" language, the result holds.

So let (p,R) be a reasonable pair that encodes what we want to encode in human rationality and reward. It is compatible with the human policy πH, in that p(R)=πH.

Let (pr,Rr) be the compatible pair where pr is the rational Bayesian expected reward maximiser, with Rr the corresponding reward so that pr(Rr)=πH.

Let (pi,0) be the indifferent planner (indifferent to the choice of reward), chosen so that pi(R′)=πH for all R′. The reward 0 is the trivial reward.

## Information content present in each pair

The planer pi is simply a map to πH, so the only information in pi (and (pi,0)) is the definition of πH.

The policy πH and the brief definition of an expected reward maximiser pr are the only information content in (pr,Rr).

On the other hand, (p,R) defines not only πH, but, at every action, it defines the bias or inefficiency of πH, as the difference between the value of πH and the ideal R-maximising policy πR. This is a large amount of information, including, for instance, every single human bias and example of bounded rationality.

None of the other pairs have this information (there's no such thing as bias for the flat reward 0, nor for the expected reward maximiser pr), so (p,R) contains a lot more information than the other pairs, so we expect it to have higher Kolmogorov complexity.