$\mathcal{R}(D_{1},D_{2},\Gamma)$ is defined as the closure of all rate tuples $(R_{1},R_{2})$ that are achievable given the distortion-cost tuple $(D_{1},D_{2},\Gamma)$.

For side information $Z$ available causally at Node 3, i.e., with decoding function (\[decoder2\]) at Node 3 modified so that $\hat{X}_{i}$ is a function of $M_{2}$ and $Z^{i}$ only, the rate-distortion region $\mathcal{R}(D_{1},D_{2},\Gamma)$ has been derived in [@Ahmadi_DSC].

In the rest of this section, for simplicity of notation, we drop the subscripts from the definition of the pmfs, thus identifying a pmf by its argument.

Rate-Distortion-Cost Region \[sub:RD\_cascade\]
-----------------------------------------------

In this section, a single-letter characterization of the rate-distortion-cost region is derived.

\[prop:RD\_action\_cascade\]The rate-distortion-cost region $\mathcal{R\mbox{\ensuremath{(D_{1},D_{2},\Gamma)}}}$ for the cascade source coding problem illustrated in Fig. \[fig:fig2\] is given by the union of all rate pairs $(R_{1},R_{2})$ that satisfy the conditions

\[eqn: RD\_action\_cascade\] $$\begin{aligned}
R_{1} & \geq & I(X;\hat{X}_{1},A,U|Y)\label{eq:R1}\\
\ce{and}\mbox{ }R_{2} & \geq & I(X,Y;A)+I(X,Y;U|A,Z),\label{eq:R2}\end{aligned}$$

where the mutual information terms are evaluated with respect to the joint pmf $$\begin{aligned}
p(x,y,z,a,\hat{x}_{1},u)=p(x,y)p(\hat{x}_{1},a,u|x,y)p(z|y,a) & ,\label{eq:joint}\end{aligned}$$ for some pmf $p(\hat{x}_{1},a,u|x,y)$ such that the inequalities

\[eqn: action\_cascade\_const\] $$\begin{aligned}
\ce{E}[d_{1}(X,\hat{X}_{1})] & \leq & D_{1},\label{eq:dist1}\\
\ce{E}[d_{2}(X,\ce{f}(U,Z))] & \leq & D_{2},\label{eq:dist2}\\
\ce{and}\textrm{ }\ce{E}[\Lambda(A)] & \leq & \Gamma,\label{eq:action_bound}\end{aligned}$$

are satisfied for some function $\ce{f}\textrm{: }\mathcal{U}\times\mathcal{Z}\rightarrow\hat{\mathcal{X}}_{2}$. Finally, $U$ is an auxiliary random variable 