Note that, given this definition of $\Lambda(a)$, at each time, Node 1 can choose whether to provide the side information $W$ to Node 2 *or* to Node 3 with no further constraints. By symmetry, it can be seen that we can set the pmf $p(a|x)$ with $x\in\{0,1\}$ and $a\in\{1,2\}$ to be a BSC with transition probability $q$. This implies that $p_{1}=\textrm{Pr}[A=1]=q$ and $p_{2}=\textrm{Pr}[A=2]=1-q$. We now evaluate the inequality (\[eq:Ra\_lossless\_ex\]) as $R_{b}\geq0$; inequality (\[eq:R1+Rb\_lossless\_ex\]) as $R_{1}+R_{b}\geq1-p_{1}I(X;W|A=1)=1-qH(\delta)$; and similarly inequality (\[eq:R2+Rb\_lossless\]) as $R_{2}+R_{b}\geq1-(1-q)H(\delta).$ From these inequalities, it can be seen that, in order to trace the boundary of the rate-cost region, in general, one needs to consider all values of $q$ in the interval $[0,1]$. This

![The side information S-channel $p(w|x)$ used in the example of Sec. \[sub:lossless\_ex\].[]{data-label="fig:S-Channel"}](S-Channel)

corresponds to appropriate time-sharing between providing side information to Node 2 (for a fraction of time $q$) and Node 3 (for the remaining fraction of time). Note that, as shown in [@Weissman_multi Sec. III], if $R_{1}=R_{2}=0$, it is optimal to set $q=\frac{1}{2}$, and thus equally share the side information between Node 2 and Node 3, in order to minimize the rate $R_{b}$. This difference is due to the fact that in the cascade model 