$A^{n}\in\mathcal{A}^{n}.$ Given $A^{n}$ and $X^{n}$, the sequences $Y^{n}$ and $Z^{n}$ are distributed as $p(y^{n},z^{n}|a^{n},x^{n})=\prod_{i=1}^{n}p_{YZ|A,X}(y_{i},z_{i}|a_{i},x_{i})$. The cost of the action sequence is defined as in previous section. A formal description of the operations at encoder and decoder follows.

\[def\_BC\_lossless\]An $(n,R_{1},R_{2},R_{b},D_{1},D_{2},\Gamma,\epsilon)$ code for the set-up of Fig. \[fig:fig4\] consists of two source encoders, namely $$\begin{aligned}
\mathrm{g}_{1}\text{: \ensuremath{\mathcal{X}^{n}\rightarrow[1,2^{nR_{1}}]\times[1,2^{nR_{b}}],}} &  & \text{ }\label{eq:encoder1}\end{aligned}$$ which maps the sequence $X^{n}$ into messages $M_{1}$ and $M_{b}$, respectively; $$\mathrm{g}_{2}\text{:}\text{ }[1,2^{nR_{1}}]\times[1,2^{nR_{b}}]\times\mathcal{Y}^{n}\rightarrow[1,2^{nR_{2}}]\label{encoder2_BC}$$ which maps the sequence $Y^{n}$ and messages $(M_{1},M_{b})$ into a message $M_{2};$ an “action function $$\mathrm{\ell}\text{: }[1,2^{nR_{b}}]\rightarrow\mathcal{A}^{n},\label{action_fun_BC}$$ which maps the message $M_{b}$ into an action sequence $A^{n};$ two decoders, namely $$\mathrm{h}_{1}\text{: }\text{ }[1,2^{nR_{1}}]\times[1,2^{nR_{b}}]\times\mathcal{Y}^{n}\rightarrow\mathcal{\hat{X}}_{1}^{n},\label{decoder1_BC}$$ which maps messages $M_{1}$ and $M_{b}$ and the measured sequence $Y^{n}$ into the estimated sequence $\hat{X}_{1}^{n};$ and $$\mathrm{h}_{2}\text{: }[1,2^{nR_{2}}]\times[1,2^{nR_{b}}]\times\mathcal{Z}^{n}\rightarrow\mathcal{\hat{X}}_{2}^{n},\label{decoder2_BC}$$ which maps the messages $M_{2}$ and $M_{b}$ into the the estimated sequence $\hat{X}_{2}^{n};$ such that the action cost constraint (\[action cost\]) and distortion constraint (\[dist const\]) are satisfied.

Achievable rates $(R_{1},R_{2},R_{b})$ and rate-distortion-cost region are defined analogously to Definitions \[def\_ach\] and \[def\_reg\].

The rate–distortion–cost region for the system model described above is open even for the case without VM at Node 2 and Node 3 (see [@Vasudevan]). Hence, in the following subsections, we characterize the rate 