Thank you for your response and raising your score
Thank you for acknowledging our response and for raising your score. Please allow us to clarify our argument "There is no free lunch in causal inference". What we meant to say is, that in the standard CATE setting, CPA corresponds to setting $\Gamma \to \infty$ for our bounds. This is because we obtain our bounds by optimizing over all possible SCMs that are compatible with (i) the causal graph, (ii) the observed data distribution, and (iii) the sensitivity constraints. When setting $\Gamma \to \infty$, we ignore (iii) and only constrain our class of SCMs by (i) and (ii), thus performing CPA. **Hence, any other CPA approach for tighter bounds would provably require stronger assumptions.** For example, there are existing CPA approaches that yield tighter bounds by exploiting valid instrumental variables (see e.g., [3, 4]).
In the following, we characterize our (w.l.o.g. upper) bounds for $\Gamma \to \infty$ in the standard CATE setting: Observed covariates $X$, binary/ continuous treatment $A$, and continuous outcome $Y$. We are interested in the causal query $Q(x, a, \mathcal{M}) = \mathbb{E}\left[Y \mid x, do(A = a)\right]$ (Example 1 from our paper). Our upper bound is $Q^+ = \int_{\ell}^{F^{-1}(c_Y^+)} \frac{y}{s_Y^+} \mathbb{P}(y \mid x, a) dy + \int_{F^{-1}(c_Y^+)}^{u} \frac{y}{s_Y^-} \mathbb{P}(y \mid x, a) dy $, where $c_Y^+ = \frac{\Gamma}{1 + \Gamma}$ and $\ell, u$ are the lower/ upper support points of the distribution $\mathbb{P}(y \mid x, a)$, respectively.
1) For binary treatment $A \in \{0, 1\}$, using the MSM we obtain
\begin{equation}
Q^+ = \int_{\ell}^{F^{-1}(\frac{\Gamma}{1 + \Gamma})} y \left((1 - \Gamma^{-1}) \mathbb{P}(a \mid x) + \Gamma^{-1} \right) \mathbb{P}(y \mid x, a) dy + \int_{F^{-1}(\frac{\Gamma}{1 + \Gamma})}^{u} y \left((1 - \Gamma) \mathbb{P}(a \mid x) + \Gamma \right) \mathbb{P}(y \mid x, a) dy \xrightarrow[\Gamma \to \infty]{} \mathbb{P}(a \mid x) \mathbb{E}[Y \mid x, a] + (1 - \mathbb{P}(a \mid x)) u
\end{equation}
This bound is also known as the "no assumptions bound" or "Manski bound", originally derived in [1]. Of note, with our theory, we can derive similar bounds for distributional effects. Hence, **our paper even makes non-trivial contributions to CPA**. We will add this to our paper.
2) For continuous treatments, using the CMSM we obtain
\begin{equation}
Q^+ = \int_{\ell}^{F^{-1}(\frac{\Gamma}{1 + \Gamma})} \frac{y}{\Gamma} \mathbb{P}(y \mid x, a) dy + \int_{F^{-1}(\frac{\Gamma}{1 + \Gamma})}^{u} y \Gamma \mathbb{P}(y \mid x, a) dy \xrightarrow[\Gamma \to \infty]{} u
\end{equation}
That is, for continuous treatments the corresponding "no assumptions bound" is exactly the right support point of the observed distribution (see also [2]). Hence, informative CPA for continuous treatments and continuous outcomes is **not possible without imposing stronger assumptions** (such as IVs).
**Action**: We will add the derivations above to our discussion on CSA vs CPA.