  The model of a side information "vending machine" (VM) accounts for scenarios
in which the measurement of side information sequences can be controlled via
the selection of cost-constrained actions. In this paper, the three-node
cascade source coding problem is studied under the assumption that a side
information VM is available and the intermediate and/or at the end node of the
cascade. A single-letter characterization of the achievable trade-off among the
transmission rates, the distortions in the reconstructions at the intermediate
and at the end node, and the cost for acquiring the side information is derived
for a number of relevant special cases. It is shown that a joint design of the
description of the source and of the control signals used to guide the
selection of the actions at downstream nodes is generally necessary for an
efficient use of the available communication links. In particular, for all the
considered models, layered coding strategies prove to be optimal, whereby the
base layer fulfills two network objectives: determining the actions of
downstream nodes and simultaneously providing a coarse description of the
source. Design of the optimal coding strategy is shown via examples to depend
on both the network topology and the action costs. Examples also illustrate the
involved performance trade-offs across the network.
