I say this with trepidation, since Peter and Eliezer have both already read this, but...

As an epistemic rationalist, I would say that 1/2 is a better approximation than 0, because the Kullback-Leibler Divergence is (about) 1 bit for the former, and infinity for the latter.

(If the probability distribution peaked at 1/2, it would be not-completely-unreasonable to use a flat distribution, and express a probability as a fixed-point number between 0 and 1. In that case, it would take 60 bits to express 10^-18. With floating point, you'd get a good approximation with 7 bits.)

But you're not really making a fair comparison. You're comparing "probability distribution centered on 1/2" with "0, no probability distribution". If the "centered on 0" choice doesn't get to have a distribution, neither should the "centered on 1/2" choice. Then both give you a divergence of infinity.

The KL-divergence comparison assumes use of a probability distribution. The probability distribution that peaks at zero is going to be able to represent 1E-18 with many fewer bits than the one that peaks at 1/2. So zero wins in both cases, and there is no demonstrated conflict between epistemic and instrumental rationality.