## LESSWRONGLW

So, your suggestion is not just an inconsequential grain of uncertainty, it is an grain of exploration. The agent actually does take 10 with some small probability. If you try to do this with just uncertainty, things would be worse, since that uncertainty would not be justified.

One problem is that you actually do explore a bunch, and since you don't get a reset button, you will sometimes explore into irreversible actions, like shutting yourself off. However, if the agent has a source of randomness, and also the ability to simulate worlds in which that randomness went another way, you can have an agent that with probability does not explore ever, and learns from the other worlds in which it does explore. So, you can either explore forever, and shut yourself off, or you can explore very very rarely and learn from other possible worlds.

The problem with learning from other possible worlds is to get good results out of it, you have to assume that the environment does not also learn from other possible worlds, which is not very embedded.

But you are suggesting actually exploring a bunch, and there is a problem other than just shutting yourself off. You are getting past this problem in this case by only allowing linear functions, but that is not an accurate assumption. Let's say you are playing matching pennies with Omega, who has the ability to predict what probability you will pick but not what action you will pick.

(In matching pennies, you each choose H or T, you win if they match, they win if they don't.)

Omega will pick H if your probability of H is less that 1/2 and T otherwise. Your utility as a function of probability is piecewise linear with two parts. Trying to assume that it will be linear will make things messy.

There is this problem where sometimes the outcome of exploring into taking 10, and the outcome of actually taking 10 because it is good are different. More on this here.