As an instrumental rationalist, I would say that 0 is a better approximation than 1/2 ....
How is this train of thought "instrumental"? You aren't making any choices or decisions outside of your own brain.
To make it a real instrumental example, consider whether or not you should go buy a meteorite shield. Lets say the shield costs S and if the meteorite hits you it costs M, and the true probability of the strike is p. So buying the shield is best if pM > S.
Now if you go with 0, you'll never buy the shield, so if pM > S you have an expected loss of (pM - S) due to your approxamation.
If you go with 1/2 then you'll buy the shield if M/2 > S. If M/2 > S and pM <= S then you bought the shield when you shouldn't have, and you lose and expected (S - pM).
So you see, it all depends on how big M is compared to S
M < S/p : 0 is the better instrumental approximation
M > S/p : 1/2 is better
In other words, if the risks (or payoffs) are small compared to the probabilities involved and the costs of shields, round to 0. Otherwise round to 1/2.
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.