and that in [@Ahmadi_CR], which we briefly review here for their relevance to the present work. In [@Chia], the cascade model in Fig. \[fig:intro\](a) was considered for the special case in which the side information $Y$ measured at Node 2 is also available at Node 1 (i.e., $X=(X,Y)$) and we have the Markov chain $X-Y-Z$ so that the side information at Node 3 is degraded with respect to that of Node 2. Instead, in [@Ahmadi_CR], the cascade-broadcast model in Fig. \[fig:intro\](b) was considered for the special case in which either rate $R_{b}$ or $R_{1}$ is zero, and the reconstructions at Node 1 and Node 2 are constrained to be retrievable also at the encoder in the sense of the Common Reconstruction (CR) introduced in [@Steinberg] (see below for a rigorous definition).

Contributions
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In this paper, we investigate the source coding models of Fig. \[fig:intro\] by assuming that some of the side information sequences can be affected by the actions taken by the corresponding nodes via VMs. The main contributions are as follows.

-   *Cascade source coding problem with VM at Node 3* (Fig. \[fig:fig2\]): In Sec. \[sub:RD\_cascade\], we derive the achievable rate-distortion-cost trade-offs for the set-up in Fig. \[fig:fig2\], 