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.
I was talking about a discrete probability distribution over two possible states: {meteorite, no meteorite}. You seem to be talking about something else.
What is the probability that my apartment will be struck by a meteorite tomorrow? Based on the information I have, I might say something like 10-18. Now suppose I wanted to approximate that probability with a different number. Which is a better approximation: 0 or 1/2?
The answer depends on what we mean by "better," and this is a situation where epistemic (truthseeking) and instrumental (useful) rationality will disagree.
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. This means that my expected Bayes Score drops by one bit if I use 1/2 instead of 10-18, but it drops to minus infinity if I use 0, and any probability conditional on a meteorite striking my apartment would be undefined; if a meteorite did indeed strike, I would instantly fall to the lowest layer of Bayesian hell. This is too horrible a fate to imagine, so I would have to go with a probability of 1/2.
As an instrumental rationalist, I would say that 0 is a better approximation than 1/2. Even if a meteorite does strike my apartment, I will suffer only a finite amount of harm. If I'm still alive, I won't lose all of my powers as a predictor, even if I assigned a probability of 0; I will simply rationalize some other explanation for the destruction of my apartment. Assigning a probability of 1/2 would force me to actually plan for the meteorite strike, perhaps by moving all of my stuff out of the apartment. This is a totally unreasonable price to pay, so I would have to go with a probability of 0.
I hope this can be a simple and uncontroversial example of the difference between epistemic and instrumental rationality. While the normative theory of probabilities is the same for any rationalist, the sorts of approximations a bounded rationalist would prefer can differ very much.