Suppose we have random variables $\theta$ and $c$. We want to maximize the expected value of the following function with respect to $q$: $$E_{\theta, c}[V(q(\theta,c),\theta)-C(q(\theta,c),c)]$$ where $E_{\theta, c}$ is taking the expected value with respect to both $\theta$ and $c$. The above expression was copied directly from the text I'm working with, although I'm not quite sure if it's accurate in that $q$ is a choice variable who's optimized value will depend on $\theta$ and $c$, but the variable itself is not a function of $\theta$ and $c$ (sorry if this is confusing). Thus, I suppose we're trying to solve the following: $$q^* = \underset{q}{\arg\max}\ E_{\theta, c}[V(q,\theta)-C(q,c)]$$ The text then claims that this would be the as solving: $$q^*(\theta,c) = \underset{q}{\arg\max} \{V(q,\theta)-C(q,c)\}$$ I was wondering if this is generally true and what is the reasoning behind this. Because intuitively it seems that the text implies $$\underset{q}{\arg\max} \{V(q,\theta)-C(q,c)\} = \underset{q}{\arg\max}\ E_{\theta, c}[V(q,\theta)-C(q,c)]$$
Thanks!