whose alphabet cardinality can be constrained as $|\mathcal{U}|\leq|\mathcal{X}||\mathcal{Y}||\mathcal{A}|+3$, without loss of optimality.

For side information $Z$ independent of the action $A$ given $Y$$,$ i.e., for $p(z|a,y)=p(z|y),$ the rate-distortion region $\mathcal{R}(D_{1},D_{2},\Gamma)$ in Proposition \[prop:RD\_action\_cascade\] reduces to that derived in [@Chia].

The proof of the converse is provided in Appendix A for a more general case of adaptive action to be defined in Sec \[sec:Adaptive-Actions\]. The achievability follows as a combination of the techniques proposed in [@Permuter] and [@Chia Theorem 1]. Here we briefly outline the main ideas, since the technical details follow from standard arguments. For the scheme at hand, Node 1 first maps sequences $X^{n}$ and $Y^{n}$ into the action sequence $A^{n}$ using the standard joint typicality criterion. This mapping requires a codebook of rate $I(X,Y;A)$ (see, e.g., [@Elgammal pp. 62-63])$.$ Given the sequence $A^{n}$, the sequences $X^{n}$ and $Y^{n}$ are further mapped into a sequence $U^{n}$. This requires a codebook of size $I(X,Y;U|A)$ for each action sequence $A^{n}$ from standard rate-distortion considerations [@Elgammal pp. 62-63]. Similarly, given the sequences $A^{n}$ and $U^{n},$ the sequences $X^{n}$ and $Y^{n}$ are further mapped into the estimate $\hat{X}_{1}^{n}$ for Node 2 using a codebook of rate $I(X,Y;\hat{X}_{1}|U,A)$ for each codeword pair $(U^{n},A^{n})$. The 