extending [@Chiru]. Note that the effect of adaptive actions on rate–distortion–cost region was open even for simple point-to-point communication channel with decoder side non-causal side information VM until recently, when [@Chiru] has shown that adaptive action does not decrease the rate–distortion–cost region of point-to-point system. In this paper we have extended this result to the multi-terminal framework and we conclude that, in all of the considered examples, where applicable, adaptive

    ![Cascade source coding problem with a side information “vending machine” at Node 3.[]{data-label="fig:fig2"}](fig2)

    selection of the actions does not improve the achievable rate-distortion-cost trade-offs.

Our results extends to multi-hop scenarios the conclusion in [@Permuter] that a joint representation of data and control messages enables an efficient use of the available communication links. In particular, layered coding strategies prove to be optimal for all the considered models, in which, the base layer fulfills two objectives: determining the actions of downstream nodes and simultaneously providing a coarse description of the source. Moreover, the examples provided in the paper demonstrate the dependence of the optimal coding design on network topology action costs.

Throughout the paper, we closely follow the notation in [@Ahmadi_CR]. In particular, a random variable is denoted by an upper case 