or s:=(s′,s′′)∈S∧, we want P∧(s):=Pσ⊤(s′)(s′′) and A∧(s):=Aσ⊤(s′)(s′′), so that the actions available are just those of the state in the sub-environment. To achieve this we define σ∧(s):=σσ⊤(s′)(s′′)
It seems that you’re using Ai and Pi to denote both the action spaces of the top environments and the action space assignment functions of the bottom environments. In addition, there is an implicit assumption that the bottom environments share the same list of action spaces. This is pretty confusing.
I’m not. I guess this is the part that makes it confusing
for readability we define A(s):=Aσ(s) and Pσ(s) to be the accessible and outer action spaces of s respectively
Do you have a suggestion for alternate notation? I use this because we often need to refer to the action space corresponding to a state. I think this would be needed even with the language framing.
(I also assigned j:=σ(s′) to make it more readable)
Yes, my point is that currently subscripts refer to both subenvironments and entries in the action space list. I suggest changing one of these two into superscripts.
This function is ill-defined outside the vertices.
It seems that you’re using Ai and Pi to denote both the action spaces of the top environments and the action space assignment functions of the bottom environments. In addition, there is an implicit assumption that the bottom environments share the same list of action spaces. This is pretty confusing.
I’m not. I guess this is the part that makes it confusing
Do you have a suggestion for alternate notation? I use this because we often need to refer to the action space corresponding to a state. I think this would be needed even with the language framing.
(I also assigned j:=σ(s′) to make it more readable)
You can use e.g. subscripts to refer to indices of the action space list and superscripts to refer to indices of the subenvironment list.
I don’t think this will work because we are already using subscripts to denote which environment’s list we are referring to
Yes, my point is that currently subscripts refer to both subenvironments and entries in the action space list. I suggest changing one of these two into superscripts.
Is A(Ei) supposed to be just Ai?
Unclear what delta is here. Is it supposed to be p?
The transition kernel is missing from this list.
Confusing: you’re using Q before you defined it. Also, instead of writing “s.t.” in the subscript, you can write “:”
Seems worth to clarify that this representation is non-unique: multiple distribution over V(A) can correspond to the same point in A.
Too restrictive. P can be an H-polytope, doesn’t need to be an HV-polytope.
The footnote is missing