at hand, it can be advantageous to provide more side information to one of the two encoders depending on the desired trade-off between the rates $R_{1}$ and $R_{2}$ in the achievable rate-cost region.

**S*-Channel between $X$ and $W$*: \[S-Channel\]We now consider the special case of Corollary \[cor:ex\] in which $(X,W)$ are jointly distributed so that $p(x)=1/2$ and $p(w|x)$ is the S-channel characterized by $p(0|0)=1-\delta$ and $p(1|1)=1$ (see Fig. \[fig:S-Channel\]). Moreover, we let $\lambda_{1}=1$, $\lambda_{2}=0$, $\lambda_{0}=\lambda_{3}=\infty$ as above, while the cost constraint is set to $\Gamma\leq1$. As discussed in [@Weissman_multi Sec. III] for this example with $R_{1}=R_{2}=0$, providing side information to Node 2 is more costly and thus should be done efficiently. In particular, given Fig. \[fig:S-Channel\], it is expected that biasing the choice $A=2$ when $X=1$ (i.e., providing side information to Node 2) may lead to some gain (see [@Weissman_multi]). Here we show that in the cascade model, this gain depends on the relative importance of rates $R_{1}$ and $R_{2}$.

To this end, we set $p(a|x)$ as $p(1|0)=\alpha$ and $p(1|1)=\beta$ for $\alpha,\beta\in[0,1]$. We now evaluate the inequality (\[eq:Ra\_lossless\_ex\]) as $R_{b}\geq0$; inequality (\[eq:R1+Rb\_lossless\_ex\]) as $$\begin{aligned}
R_{1}+R_{b} & \geq & 1-\Bigl(\frac{\alpha+\beta}{2}\Bigr)\Bigl(H\Bigl(\frac{(1-\delta)\alpha}{\alpha+\beta}\Bigr)-H(1-\delta)\frac{\alpha}{\alpha+\beta}\Bigr);\label{eq:ex_R1+Rb_ex_S}\end{aligned}$$ and inequality (\[eq:R2+Rb\_lossless\_ex\]) as $$\begin{aligned}
R_{2}+R_{b} & \geq & 1-\Bigl(\frac{2-\alpha-\beta}{2}\Bigr)\Bigl(H\Bigl(\frac{(1-\delta)(1-\alpha)}{2-\alpha-\beta}\Bigr)-H(1-\delta)\frac{1-\alpha}{2-\alpha-\beta}\Bigr),\label{eq:ex_R2+Rb_ex_S}\end{aligned}$$

![Difference 