to the case in which the rate $R_{1}$ and/or $R_{2}$ are non-zero;

-   *Cascade-broadcast source coding problem with VM at Node 2 and Node 3, lossy compression with CR constraint* (Fig. \[fig:fig3\]): In Sec. \[sub:CR\], we tackle the problem in Fig. \[fig:fig3\] but under the more general requirement of lossy reconstruction. Conclusive results are obtained under the additional constraints that the side information at Node 3 is degraded and that the source reconstructions at Node 2 and Node 3 can be recovered with arbitrarily small error probability at Node 1. This is referred to as the CR constraint following [@Steinberg], and is of relevance in applications in which the data being sent is of sensitive nature and unknown distortions in the receivers’ reconstructions are not acceptable (see [@Steinberg] for further discussion). This characterization extends the result of [@Ahmadi_CR] mentioned above to the set-up with a side information VM, and also in that both rates $R_{1}$ and $R_{b}$ are allowed to be non-zero;

-   *Adaptive actions*: Finally, we revisit the results above by allowing the decoders to select their actions in an adaptive way, based not only on the received messages but also on the previous samples of the side information 