between the weighted sum-rate $R_{1}+\eta R_{2}$ obtained with the greedy and with the optimal strategy as per Corollary \[cor:ex\] ($R_{b}=0.4$, $\delta=0.6$). []{data-label="fig:greedy-nongreedy"}](plot_greedyvsnongreedy)

We now evaluate the minimum weighted sum-rate $R_{1}+\eta R_{2}$ obtained from (\[eq:ex\_R1+Rb\_ex\_S\])-(\[eq:ex\_R2+Rb\_ex\_S\]) for $R_{b}=0.4$, $\delta=0.6$ and both $\Gamma=0.1$ and $\Gamma=0.9$. Parameter $\eta\geq0$ rules on the relative importance of the two rates. For comparison, we also compute the performance attainable by imposing that the action $A$ be selected independent of $X$, which we refer to as the greedy approach [@Permuter]. Fig. \[fig:greedy-nongreedy\] plots the difference between the two weighted sum-rates $R_{1}+\eta R_{2}$ . It can be seen that, as $\eta$ decreases and thus minimizing rate $R_{1}$ to Node 2 becomes more important, one can achieve larger gains by choosing the action $A$ to be dependent on $X$. Moreover, this gain is more significant when the action cost budget $\Gamma$ allows Node 2 to collect a larger fraction of the side information samples.

Lossy Compression with Common Reconstruction Constraint\[sub:CR\]
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In this section, we turn to the problem of characterizing the rate-distortion-cost region ${\cal R}(D_{1},D_{2}$ $,\Gamma)$ for $D_{1},D_{2}>0$. In order to make the problem tractable [^4], we impose the degradedness condition $X-(A,Y)-Z$ (as in [@Weissman_multi]), which implies the factorization $$\begin{aligned}
