respectively. A formal description of the operations at the encoder and the decoder follows.

![Cascade-broadcast source coding problem with a side information “vending machine” at Node 2.[]{data-label="fig:fig4"}](fig4_modified)

\[def\_cascade\]An $(n,R_{1},R_{2},D_{1},D_{2},\Gamma,\epsilon)$ code for the set-up of Fig. \[fig:fig2\] consists of two source encoders, namely $$\mathrm{g}_{1}\text{: }\mathcal{X}^{n}\times\mathcal{Y}^{n}\rightarrow[1,2^{nR_{1}}],\label{encoder1}$$ which maps the sequences $X^{n}$ and $Y^{n}$ into a message $M_{1};$ $$\mathrm{g}_{2}\text{:}\text{ }\mathcal{Y}^{n}\times[1,2^{nR_{1}}]\rightarrow[1,2^{nR_{2}}],\label{encoder2}$$ which maps the sequence $Y^{n}$ and message $M_{1}$ into a message $M_{2};$ an “action function $$\mathrm{\ell}\text{: }[1,2^{nR_{2}}]\rightarrow\mathcal{A}^{n},\label{action_fun}$$ which maps the message $M_{2}$ into an action sequence $A^{n};$ two decoders, namely $$\mathrm{h}_{1}\text{: }[1,2^{nR_{1}}]\times\mathcal{Y}^{n}\rightarrow\mathcal{\hat{X}}_{1}^{n},\label{decoder1}$$ which maps the message $M_{1}$ and the measured sequence $Y^{n}$ into the estimated sequence $\hat{X}_{1}^{n};$ $$\mathrm{h}_{2}\text{: }[1,2^{nR_{2}}]\times\mathcal{Z}^{n}\rightarrow\mathcal{\hat{X}}_{2}^{n},\label{decoder2}$$ which maps the message $M_{2}$ and the measured sequence $Z^{n}$ into the the estimated sequence $\hat{X}_{2}^{n};$ such that the action cost constraint $\Gamma$ and distortion constraints $D_{j}$ for $j=1,2$ are satisfied, i.e., $$\begin{aligned}
\frac{1}{n}\underset{i=1}{\overset{n}{\sum}}\mathrm{E}\left[\Lambda(A_{i})\right] & \leq\Gamma\label{action cost}\\
\text{ and }\frac{1}{n}\underset{i=1}{\overset{n}{\sum}}\mathrm{E}\left[d_{j}(X_{ji},\textrm{h}_{ji})\right] & \leq D_{j}\text{ for }j=1,2,\label{dist const}\end{aligned}$$ where we have defined as $\textrm{h}_{1i}$ and $\textrm{h}_{2i}$ the $i$th symbol of the function $\textrm{h}_{1}(M_{1},Y^{n})$ and $\textrm{h}_{2}(M_{2},Z^{n})$, respectively.

\[def\_ach\]Given a distortion-cost tuple $(D_{1},D_{2},\Gamma)$, a rate tuple $(R_{1},R_{2})$ is said to be achievable if, for any $\epsilon>0$, and sufficiently large $n$, there exists a $(n,R_{1},R_{2},D_{1}+\epsilon,D_{2}+\epsilon,\Gamma+\epsilon)$ code.

\[def\_reg\]The *rate-distortion-cost region* 