How to understand random assignment eliminates selection bias in the potential outcomes framework

Question

In Angrist & Pischke's book mostly harmless econometrics, they explain that if the treatment in an RCT $D_i$ is randomly assigned, then $D_i$ is independent of potential outcomes and the following holds:

$$ \begin{align} E[Y_i|D_i = 1] - E[Y_i|D_i = 0] &= E[Y_{1i}|D_i = 1] \color{red}{- E[Y_{0i}|D_i = 0]} \\\\ &= E[Y_{1i}|D_i = 1] \color{red}{- E[Y_{0i}|D_i = 1]} \end{align} $$.

The main thing here, as I understand it, is that we can switch the terms in red $E[Y_{0i}|D_i = 0]$ $E[Y_{0i}|D_i = 1]$. But How do I interpret this? I get why we want to randomly assign treatment (because we want the treatment group and control group to be identical), but I don't understand what the above equation says. For me it looks like the expected outcome for the control group is the same even if they get treatment, i.e, $D_i = 1$. I think I am misinterpreting something here.

For completeness. After this, they go on to say that we can simplify further. Starting from the last line above:

$$ \begin{align} E[Y_{1i}|D_i = 1] - E[Y_{0i}|D_i = 1] &= E[Y_{1i} - Y_{0i} | D_i = 1] \\\\ &= E[Y_{1i} - Y_{0i}] \end{align} $$, noting that "The main thing, however, is that random assignment of $D_i$ eliminates selection bias".

Let me know if I need to add some context!

Answer 1

an intutive explanation is that given independence of treatment from potential outcomes the average of $Y_0$ in the control group (roughly E[Y_{i0}|D=0]) is as good an estimate of E[$Y_0$] as the average of $Y_0$ in the experimental group. Thus, given the aforementioned independence you can 'exchange' participants in the experimental and the control group; they are just two random samples from the larger population.

Hope that helps. Best Stefan