optimization loss due to misperceived probability Suppose $a$ is chosen to maximize the expected value of $u(a,x)$ under a probability measure of $x$. Image the true distribution is $P(x)$, but the optimization may be conducted under a misperceived distribution $Q(x)$. We denote the optimal action under $P$ and $Q$ as $a^*(P),a^*(Q)$ respectively,
\begin{align}
a^*(P)&=\text{argmax}_a \int u(a,x)d P(x),\\
a^*(Q)&=\text{argmax}_a \int u(a,x)d Q(x).
\end{align}
Now we investigate the loss of the misoptimization
\begin{align}
\Delta(P,Q) = \int \left[u\left(a^*(P),x\right) - u\left(a^*(Q),x\right) \right]d P(x).
\end{align}
My question: Is there a good bound or approximation for $\Delta(P,Q)$ in general, in terms of some function (or value function) of $u$ and some metric to measure the distance between two distributions $P$ and $Q$? It's fine to assume good properties for $u$, say smooth and bounded and assume $P$ and $Q$ are close in some intuitive sense.
It would be nice to also allow the possibility that $P$ and $Q$ are indeed Dirac measure placed at $p$ and $q$ with $p,q$ being close to each other.
It seems like a well-motivated question, but I failed to find any literature on this. Thanks in advance for discussions or pointing me to some extant results.
For an ideal solution, it's better not to use KL-divergence, because it's infinite between Dirac $P$ and Dirac $Q$. Wasserstein metric appears more likely to be related. I guess some form of envelope theorem might be useful, since an optimization is involved.
 A: One simple bound arises from the Cauchy–Schwarz inequality.
For notational convenience, I will write $u_a(x)$ to indicate the $u(a,x)$. Assume that $P$ and $Q$ have well-defined pdfs, indicated by $p(x)$ and $q(x)$. Also assume that $p(x)$, $q(x)$, and $u_a(x)$ for all $a$ are in $L^2$ (which would eliminate delta function, unfortunately).
First, we rewrite 
\begin{align}
\Delta (P,Q)&=\int [u_{a^*(P)}(x)−u_{a^*(Q)}(x)] p(x) dx \\
 & =\int [u_{a^*(P)}(x)−u_{a^*(Q)}(x)] (p(x) - q(x)) dx + \int [u_{a^*(P)}(x)−u_{a^*(Q)}(x)] q(x) dx \\
& = \int [u_{a^*(P)}(x)−u_{a^*(Q)}(x)] (p(x) - q(x)) dx - \Delta(Q,P)
\end{align}
Clearly $\Delta(Q,P) \ge 0$, so we can bound
$$
\Delta(P,Q) \le \int [u(a^*(P),x)−u_{a^*(P)}(x)] (p(x) - q(x)) dx
$$
We then apply the Cauchy–Schwarz inequality to the RHS to get
$$
\Delta(P,Q) \le \left\Vert u_{a^*(P)}-u_{a^*(Q)} \right\Vert_2 \left\Vert p-q \right\Vert_2
$$
Define  $u^* := \max_{a,a'} \left\Vert u_{a}-u_{a'} \right\Vert_2$, that is the maximal difference between utility functions for different actions. This gives 
$$
\Delta(P,Q) \le u^* \left\Vert p-q \right\Vert_2
$$
