I made a mistake again. As described above, complete only pseudodominates incomplete.
But this is easily patched with the trick described in the OP. So we need the choice complete to make two changes to the downstream decisions. First, change decision 1 to always choose up (as before), second, change the distribution of Decision 2 to {1−q(1−p), q(1−p)}, because this keeps the probability of B constant. Fixed diagram:
Now the lottery for complete is {B: q(1−p), A+: 1−q(1−p), A:0}, and the lottery for incomplete is {B: q(1−p), A+: pq, A:(1−q)}. So overall, there is a pure shift of probability from A to A+. [Edit 23/7: hilariously, I still had the probabilities wrong, so fixed them, again].
I think the above money pump works, if the agent sometimes chooses the A path, but I was incorrect in thinking that the caprice rule sometimes chooses the A path.
I misinterpreted one of EJT’s comments as saying it might choose the A path. The last couple of days I’ve been reading through some of the sources he linked to in the original “there are no coherence theorems” post and one of them (Gustafsson) made me realize I was interpreting him incorrectly, by simplifying the decision tree in a way that doesn’t make sense. I only realized this yesterday.
Now I think that the caprice rule is essentially equivalent to updatelessness. If I understand correctly, it would be equivalent to 1. choosing the best policy by ranking them in the partial order of outcomes (randomizing over multiple maxima), then 2. implementing that policy without further consideration. And this makes it immune to money pumps and renders any self-modification pointless. It also makes it behaviorally indistinguishable from an agent with complete preferences, as far as I can tell. The same updatelessness trick seems to apply to all money pump arguments. It’s what scott uses in this post to avoid the independence money pump.
So currently I’m thinking updatelessness removes most of the justification for the VNM axioms (including transitivity!). But I’m confused because updateless policies still must satisfy local properties like “doesn’t waste resources unless it helps achieve the goal”, which is intuitively what the money pump arguments represent. So there must be some way to recover properties like this. Maybe via John’s approach here. But I’m only maybe 80% sure of my new understanding, I’m still trying to work through it all.
I made a mistake again. As described above, complete only pseudodominates incomplete.
But this is easily patched with the trick described in the OP. So we need the choice complete to make two changes to the downstream decisions. First, change decision 1 to always choose up (as before), second, change the distribution of Decision 2 to {1−q(1−p), q(1−p)}, because this keeps the probability of B constant. Fixed diagram:
Now the lottery for complete is {B: q(1−p), A+: 1−q(1−p), A:0}, and the lottery for incomplete is {B: q(1−p), A+: pq, A:(1−q)}. So overall, there is a pure shift of probability from A to A+.
[Edit 23/7: hilariously, I still had the probabilities wrong, so fixed them, again].
I think the above money pump works, if the agent sometimes chooses the A path, but I was incorrect in thinking that the caprice rule sometimes chooses the A path.
I misinterpreted one of EJT’s comments as saying it might choose the A path. The last couple of days I’ve been reading through some of the sources he linked to in the original “there are no coherence theorems” post and one of them (Gustafsson) made me realize I was interpreting him incorrectly, by simplifying the decision tree in a way that doesn’t make sense. I only realized this yesterday.
Now I think that the caprice rule is essentially equivalent to updatelessness. If I understand correctly, it would be equivalent to 1. choosing the best policy by ranking them in the partial order of outcomes (randomizing over multiple maxima), then 2. implementing that policy without further consideration. And this makes it immune to money pumps and renders any self-modification pointless. It also makes it behaviorally indistinguishable from an agent with complete preferences, as far as I can tell.
The same updatelessness trick seems to apply to all money pump arguments. It’s what scott uses in this post to avoid the independence money pump.
So currently I’m thinking updatelessness removes most of the justification for the VNM axioms (including transitivity!). But I’m confused because updateless policies still must satisfy local properties like “doesn’t waste resources unless it helps achieve the goal”, which is intuitively what the money pump arguments represent. So there must be some way to recover properties like this. Maybe via John’s approach here.
But I’m only maybe 80% sure of my new understanding, I’m still trying to work through it all.