let $k_{s^{\prime }}:P_j\to R^{U_{\top }}$be a helper function that maps each $a\in V\left(P_i\right)$ to $e_{\top }(s^{\prime },a)$.

This function is ill-defined outside the vertices.

or $s:=(s^{\prime },s^{\prime \prime })\in S_{\wedge }$, we want $P_{\wedge }\left(s\right):=P_{{\sigma }_{\top }\left(s^{\prime }\right)}\left(s^{\prime \prime }\right)$ and $A_{\wedge }\left(s\right):=A_{{\sigma }_{\top }\left(s^{\prime }\right)}\left(s^{\prime \prime }\right)$, so that the actions available are just those of the state in the sub-environment. To achieve this we define ${\sigma }_{\wedge }\left(s\right):={\sigma }_{{\sigma }_{\top }\left(s^{\prime }\right)}\left(s^{\prime \prime }\right)$

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.

such that $f_i\left(\right(1-{\gamma }_{\perp }\left)Q_i\right)=A\left(E_i\right)$

Is A(Ei) supposed to be just Ai?

μ×:=μ1×⋯×μk×δ

Unclear what delta is here. Is it supposed to be p?

An atomic environment is constructed by directly providing

The transition kernel is missing from this list.

• a vector space $W$ and linear maps $g:R^U\to W$ and $R^Q:W\to R$ such that for any $q\in Q$, $R^Q\left(q\right)={max}_{\nu \in M\text{s.t.}g\left(\nu \right)=q}{R}^{\nu }\left(\nu \right)$.
• a H-polytope $Q:=g\left(M\right)\subseteq W$ that we call the occupancy polytope

Confusing: you're using Q before you defined it. Also, instead of writing "s.t." in the subscript, you can write ":"

Let's view each accessible action space $A\left(s\right)$ as the set of randomized policies over $V\left(A\right(s\left)\right)$.

Seems worth to clarify that this representation is non-unique: multiple distribution over V(A) can correspond to the same point in A.

where each $A_i$ and $P_i$ is an HV-polytope

Too restrictive. P can be an H-polytope, doesn't need to be an HV-polytope.

efficiently^[1]

The footnote is missing