for $A=2$ (only Node 3 observes the side information $W$); and $Y=Z=W$ for $A=3$ (both nodes observe the side information $W$)[^3]. We also select the cost function such that $\Lambda(j)=\lambda_{j}$ for $j\in{\cal A}$. When $R_{1}=R_{2}=0$, this model reduces to the ones studied in [@Weissman_multi Sec. III]. The following is a consequence of Proposition 2.

\[cor:ex\]For the setting of switching-dependent side information described above, the rate-cost region (\[eqn: RD\_BC\_lossless\]) is given by

\[eqn: RD\_BC\_lossless\_ex\] $$\begin{aligned}
R_{b} & \geq & I(X;A)\label{eq:Ra_lossless_ex}\\
R_{1}+R_{b} & \geq & H(X)-p_{1}I(X;W|A=1)-p_{3}I(X;W|A=3)\label{eq:R1+Rb_lossless_ex}\\
\ce{and}\mbox{ }R_{2}+R_{b} & \geq & H(X)-p_{2}I(X;W|A=2)-p_{3}I(X;W|A=3)\label{eq:R2+Rb_lossless_ex}\end{aligned}$$

where the mutual information terms are evaluated with respect to the joint pmf $$\begin{aligned}
p(x,y,z,a)=p(x,a)p(y,z|a,x),\label{eq:joint-BC_lossless-1}\end{aligned}$$ for some pmf $p(a|x)$ such that $\sum_{j=0}^{3}p_{j}\lambda_{j}\leq\Gamma$, where we have denoted $p_{j}=\ce{Pr}[A=j]$ for $j\in{\cal A}$.

The region (\[eqn: RD\_BC\_lossless\_ex\]) is obtained from the rate-cost region (\[eqn: RD\_BC\_lossless\]) by noting that in (\[eq:R1+Rb\_lossless\]) we have $I(X;A)+H(X|A,Y)=H(X)-I(X;Y|A)$ and similarly for (\[eq:R2+Rb\_lossless\]).

In the following, we will elaborate upon two specific instances of the switching-dependent side information example.

*Binary Symmetric Channel (BSC) between $X$ and $W$*: Let $(X,W)$ be binary and symmetric so that $p(x)=p(w)=1/2$ for $x,w\in\{0,1\}$ and $\Pr[X\neq W]=\delta$ for $\delta\in[0,1]$. Moreover, let $\lambda_{j}=\infty$ for $j=0,3$ and $\lambda_{j}=1$ otherwise. We set the action cost constraint to $\Gamma=1$. 