Include constant to estimate propensity scores? Should we include the constant when we estimate propensity scores?
We model the probability that $D_i=1$ using a set of controls $x$
$$ p (D_i=1 | X=x) = \alpha + x \beta + \varepsilon $$
to created predicted values $\hat{p}$. Should we include the $\alpha$?

Why wouldn't you? The goal is to get good predicted probabilities, and surely you will get better predicted probabilities with an intercept in the model. But you don't have to wonder; you can try your way, assess balance after matching or weighting on the resulting propensity scores, try the other, and compare performance. I really can't imagine too many scenarios in which you wouldn't want an intercept. It removes 1 extra degree of freedom but will likely dramatically improve the performance of the model.
