On reflection, I didn’t quite understand this exploration business, but I think I can save a lot of it.
>You can do exploration, but the problem is that (unless you explore into non-fixed-point regions, violating epistemic constraints) your exploration can never confirm the existence of a fixed point which you didn’t previously believe in.
I think the key here is in the word “confirm”. Its true that unless you believe p is a fixed point, you can’t just try out p and see the result. However, you can change your beliefs about p based on your results from exploring things other than p. (This is why I call the thing I’m objecting to humean trolling.) And there is good reason to think that the available fixed points are usually pretty dense in the space. For example, outside of the rule that binarizes our actions, there should usually be at least one fixed point for every possible action. Plus, as you explore, your beliefs change, creating new believed-fixed-points for you to explore.
>I think your idea for how to find repulsive fixed-points could work if there’s a trader who can guess the location of the repulsive point exactly rather than approximately
I don’t think thats needed. If my net beliefs have a closed surface in propability space on which they push outward, then necessarily those beliefs have a repulsive fixed point somewhere in that surface. I can then explore that believed fixed point. Then if its not a true fixed point, and I still believe in the closed surface, theres a new fixed point in that surface that I can again explore (generally more in the direction I just got pushed away from). This should converge on a true fixed point. The only thing that can go wrong is that I stop believing in the closed surface, and it seems like I should leave open that possibility—and even then, I might believe in it again after I do some checking along the outside.
>However, the wealth of that trader will act like a martingale; there’s no reliable profit to be made (even on average) by enforcing this fixed point.
This I don’t understand at all. If you’re in a certain fixed point, shouldn’t the traders that believe in it profit from the ones that don’t?
So I’ve reread your section on this, and I think I follow that, but its arguing a different claim. In the post, you argue that a trader that correctly identifies a fixed point, but doesn’t have enough weight to get it played, might not profit from this knowledge. That I agree with.
But now you’re saying that even if you do play the new fixed point, that trader still won’t gain?
I’m not really calling this a proof because it’s so basic that something else must have gone wrong, but:
btrue has a fixed point at p, and bfalse doesn’t. Then bfalse(p)=p′≠p. So if you decide to play p, then bfalse predicts p′, which is wrong, and gets punished. By continuity, this is also true in some neighborhood around p. So if you’ve explored your way close enough, you win.
On reflection, I didn’t quite understand this exploration business, but I think I can save a lot of it.
>You can do exploration, but the problem is that (unless you explore into non-fixed-point regions, violating epistemic constraints) your exploration can never confirm the existence of a fixed point which you didn’t previously believe in.
I think the key here is in the word “confirm”. Its true that unless you believe p is a fixed point, you can’t just try out p and see the result. However, you can change your beliefs about p based on your results from exploring things other than p. (This is why I call the thing I’m objecting to humean trolling.) And there is good reason to think that the available fixed points are usually pretty dense in the space. For example, outside of the rule that binarizes our actions, there should usually be at least one fixed point for every possible action. Plus, as you explore, your beliefs change, creating new believed-fixed-points for you to explore.
>I think your idea for how to find repulsive fixed-points could work if there’s a trader who can guess the location of the repulsive point exactly rather than approximately
I don’t think thats needed. If my net beliefs have a closed surface in propability space on which they push outward, then necessarily those beliefs have a repulsive fixed point somewhere in that surface. I can then explore that believed fixed point. Then if its not a true fixed point, and I still believe in the closed surface, theres a new fixed point in that surface that I can again explore (generally more in the direction I just got pushed away from). This should converge on a true fixed point. The only thing that can go wrong is that I stop believing in the closed surface, and it seems like I should leave open that possibility—and even then, I might believe in it again after I do some checking along the outside.
>However, the wealth of that trader will act like a martingale; there’s no reliable profit to be made (even on average) by enforcing this fixed point.
This I don’t understand at all. If you’re in a certain fixed point, shouldn’t the traders that believe in it profit from the ones that don’t?
This is the fundamental obstacle according to me, so, unfortunate that I haven’t successfully communicated this yet.
Perhaps I could suggest that you try to prove your intuition here?
So I’ve reread your section on this, and I think I follow that, but its arguing a different claim. In the post, you argue that a trader that correctly identifies a fixed point, but doesn’t have enough weight to get it played, might not profit from this knowledge. That I agree with.
But now you’re saying that even if you do play the new fixed point, that trader still won’t gain?
I’m not really calling this a proof because it’s so basic that something else must have gone wrong, but:
btrue has a fixed point at p, and bfalse doesn’t. Then bfalse(p)=p′≠p. So if you decide to play p, then bfalse predicts p′, which is wrong, and gets punished. By continuity, this is also true in some neighborhood around p. So if you’ve explored your way close enough, you win.