Is this derivation in Manski (1990) correct? Consider the following setting. There are two treatments, $A,B$. Individuals in the population are described by a tuple $(y_A,y_B,z)$ where $z \in \{A,B\}$ denotes the treatment received. Only $y_A$ is observed when $z = A$ and only $y_B$ is observed when $z = B$.
In a classic paper on non-parametric bounds on treatment effects, Manski proposes bounds based on the following economic assumption: individuals always choose the "more effective" treatment, which is formalized by his Equation (8):
$$\{z = B\} = \{y_B \geq y_A\}$$
From this, he derives in Equation (9)
$$E[y_A|z=B] = E[y_A | y_A \leq y_B] \leq E[y_A | y_A > y_B] = E[y_A|z = A]$$
I am pretty sure the inequality is simply incorrect. For example, suppose the pair $(y_A,y_B)$ only takes on two possible values. Either $y_A = 2, y_B = 1$ or $y_A = 8, y_B = 9$. In this case, $E[y_A | y_A \leq y_B] = 8$ while $E[y_A | y_A > y_B] = 2$. In this example, we have $E[y_A | y_A \leq y_B] = 8 > 2 = E[y_A | y_A > y_B]$, which provides a counterexample to the claimed inequality.
I wanted to ask this question to double check my understanding, since I know this is a fairly well-known paper, so I wanted to make sure I am not missing something.

From my read, I agree with you.
I'm almost positive that what Manski meant to write is that under the considered assumption, $E[Y_A|Z=B] \leq E[Y_B|Z=B]$ and $E[Y_B|Z=A] \leq E[Y_A|Z=A]$, and since the RHSs are identified by the data, you can upper bound the LHSs. You can then construct bounds on the average treatment effect by combining this information with the bounded outcome assumption.
