letter (e.g., $X,Y,Z$) and its realization is denoted by a lower case letter (e.g., $x,y,z$). The shorthand notation $X^{n}$ is used to denote the tuple (or the column vector) of random variables $(X_{1},\ldots,X_{n})$, and $x^{n}$ is used to denote a realization. The notation $X^{n}\sim p(x^{n})$ indicates that $p(x^{n})$ is the probability mass function (pmf) of the random vector $X^{n}$. Similarly, $Y^{n}|\{X^{n}=x^{n}\}\sim p(y^{n}|x^{n})$ indicates that $p(y^{n}|x^{n})$ is the conditional pmf of $Y^{n}$ given $\{X^{n}=x^{n}\}$. We say that $X\textrm{---}Y\textrm{---}Z$ form a Markov chain if $p(x,y,z)=p(x)p(y|x)p(z|y)$, that is, $X$ and $Z$ are conditionally independent of each other given $Y$.

Cascade Source Coding with A Side information Vending Machine
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In this section, we first describe the system model for the cascade source coding problem with a side information vending machine of Fig. \[fig:fig2\]. We then present the characterization of the corresponding rate-distortion-cost performance in Sec. \[sub:RD\_cascade\].

![Cascade source coding problem with a side information “vending machine” at Node 2 and Node 3.[]{data-label="fig:fig3"}](fig3)

System Model\[sub:System-Model\_cascade\]
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The problem of cascade source coding of Fig. \[fig:fig2\], is defined by the probability mass functions (pmfs) $p_{XY}(x,y)$ and $p_{Z|AY}(z|a,y)$ and discrete alphabets $\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{A},\mathcal{\hat{X}}_{1},\mathcal{\hat{X}}_{2},$ as follows. The source sequences $X^{n}$ and $Y^{n}$ with $X^{n}\in\mathcal{X}^{n}$ and $Y^{n}\in\mathcal{Y}^{n}$, respectively$,$ are 