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.
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.