pairs $(R_{1},R_{2})$ that satisfy the conditions

\[eqn: R\_action\_cascade\_lossless\] $$\begin{aligned}
R_{1} & \geq & I(X;A|Y)+H(X|A,Y)\label{eq:R1_lossless}\\
\ce{and}\mbox{ }R_{2} & \geq & I(X,Y;A)+H(X|A,Z),\label{eq:R2_lossless}\end{aligned}$$

where the mutual information terms are evaluated with respect to the joint pmf $$\begin{aligned}
p(x,y,z,a)=p(x,y)p(a|x,y)p(z|y,a) & ,\label{eq:joint_lossless}\end{aligned}$$ for some pmf $p(a|x,y)$ such that $\ce{E}[\Lambda(A)]\leq\Gamma$.

Cascade-Broadcast Source Coding with A Side Information Vending Machine
=======================================================================

In this section, the cascade-broadcast source coding problem with a side information vending machine illustrated in Fig. \[fig:fig3\] is studied. At first, the rate-cost performance is characterized for the special case in which the reproductions at Node 2 and Node 3 are constrained to be lossless. Then, the lossy version of the problem is considered in Sec. \[sub:CR\], with an additional common reconstruction requirement in the sense of [@Steinberg] and assuming degradedness of the side information sequences.

System Model\[sub:lossless\]
----------------------------

In this section, we describe the general system model for the cascade-broadcast source coding problem with a side information vending machine. We emphasize that, unlike the setup of Fig. \[fig:fig2\], here, the vending machine is at both Node 2 and Node 3. Moreover, we assume that an additional broadcast link of rate $R_{b}$ is available that is received by Node 2 and 3 to enable both 