= Solution
Under the displayed <causal directed acyclic graphs>, with nonzero effects along the relevant arrows and no exact cancellation of the effects on $B$, \b[the valid <instrumental variables> are scenarios (i), (ii), and (iv)]. The path arguments are:
* In (i), $G\to B$ supplies <instrument relevance>, and every directed path from $G$ to $Y$ goes through $B$. The unmeasured $U$ does not cause $G$, so <instrumental-variable independence> holds.
* In (ii), $G\to A\to B$ supplies <instrument relevance>. Although $A$ is unmeasured, every directed path from $G$ to $Y$ still passes through $B$. The path $G\to A\leftarrow U\to Y$ is blocked at its collider $A$ without conditioning, so it does not violate <instrumental-variable independence>.
* In (iii), $G\to C\to Y$ bypasses $B$, violating the <exclusion restriction>. There is also no directed effect of $G$ on $B$, so <instrument relevance> is absent in this graph.
* In (iv), both $G\to B$ and $G\to A\to B$ affect $B$, and all directed paths to $Y$ still pass through $B$. The unconditioned collider $G\to A\leftarrow U$ blocks the route to $U$, so both <instrumental-variable independence> and the <exclusion restriction> hold. Conditioning on $A$ could introduce <collider bias>, but $A$ is not being conditioned on here.
* In (v), $G\to C\to Y$ bypasses $B$, so the <exclusion restriction> fails despite the additional $G\to B$ arrow.
* In (vi), $G\to U\to Y$ bypasses $B$, and $G$ affects the unmeasured common cause $U$. Thus the <exclusion restriction> and the stated <instrumental-variable independence> requirement fail.
An upstream intermediate variable need not invalidate an <instrumental variable> if its routes to the outcome all pass through the target <exposure>; an outcome intermediate reached without that <exposure> does invalidate it.
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