This is the first post in the sequence that I fully read since the Introduction. So I’m not going to be able to say anything really useful about the proofs. Still, I was curious about the philosophical aspects of these definitions, so I read this post anyway.
That being said, I still think that I understood some part of the definitions, after checking terms from previous posts. My handwavy understanding of your definitions is
The definitions about subset and conditional policy just rephrase that an observable is something on which the agent can condition it’s own policy. So the agent can observe the partition if it can condition policy on the set of the partition in which it finds itself.
The additive definitions say that that the agent can observe the partition if the cartesian frame can be decomposed into mutually exclusive cartesian frames, one for each set of the partition, in which the agent acts as if it is in a world of this set.
The multiplicative definitions say that the agent can observe the partition if the cartesian frame can be decomposed into a product of cartesian frames, one for each subset of the partition, such that the agent is unable to impact the world if it is outside its subset of the partition. The interpretation of the product is that there’s a supervisor agent that control all agents at the same time, and so here, it controls all of them until the observation, after which he morally only controls the one in the observed set (because the other are powerless).
The internalizing-externalizing definitions say the agent can observe the partition if the cartesian frame can be decomposed into the composition of making the agent able to choose in which set of V it is, but then removing this choice from it, which amount to letting it condition on V, without actually giving it the power to do so.
Is there something really wrong here?
Also, I’m curious if you have an interpretation of the differences between internalizing-externalizing definitions and the others, just like your section on updatelessness compared additive and multiplicative definitions. (Really cool section philosophically, by the way!)
Seems right, except I don’t use the word “product” for the multiplicative definition.
I don’t have much to say about the internalizing-externalizing definition philosophically. One thing to say is that I think the condition that ExternalS(C) observes S is a weaker notion of observability, that might actually agree with philosophical intuition more, and the internalizing-externalizing definition might be easier to interpret if you are thinking in terms of this condition.
This is the first post in the sequence that I fully read since the Introduction. So I’m not going to be able to say anything really useful about the proofs. Still, I was curious about the philosophical aspects of these definitions, so I read this post anyway.
That being said, I still think that I understood some part of the definitions, after checking terms from previous posts. My handwavy understanding of your definitions is
The definitions about subset and conditional policy just rephrase that an observable is something on which the agent can condition it’s own policy. So the agent can observe the partition if it can condition policy on the set of the partition in which it finds itself.
The additive definitions say that that the agent can observe the partition if the cartesian frame can be decomposed into mutually exclusive cartesian frames, one for each set of the partition, in which the agent acts as if it is in a world of this set.
The multiplicative definitions say that the agent can observe the partition if the cartesian frame can be decomposed into a product of cartesian frames, one for each subset of the partition, such that the agent is unable to impact the world if it is outside its subset of the partition. The interpretation of the product is that there’s a supervisor agent that control all agents at the same time, and so here, it controls all of them until the observation, after which he morally only controls the one in the observed set (because the other are powerless).
The internalizing-externalizing definitions say the agent can observe the partition if the cartesian frame can be decomposed into the composition of making the agent able to choose in which set of V it is, but then removing this choice from it, which amount to letting it condition on V, without actually giving it the power to do so.
Is there something really wrong here?
Also, I’m curious if you have an interpretation of the differences between internalizing-externalizing definitions and the others, just like your section on updatelessness compared additive and multiplicative definitions. (Really cool section philosophically, by the way!)
Seems right, except I don’t use the word “product” for the multiplicative definition.
I don’t have much to say about the internalizing-externalizing definition philosophically. One thing to say is that I think the condition that ExternalS(C) observes S is a weaker notion of observability, that might actually agree with philosophical intuition more, and the internalizing-externalizing definition might be easier to interpret if you are thinking in terms of this condition.