# Propensity score matching: bias adjustment

I'm using propensity score matching to match similar individuals. I.e., I first estimate a propensity score (the probability of treatment conditional on some set of variables) and then match on the estimated probabilities. Because this matching uses only one covariate (i.e. the propensity score) I had assumed that there's no need to use bias adjustment?!

Apparently I'm wrong about this, but I've not received a convincing argument why. Can anybody out there give me a compelling reason why I'd need bias adjustment in this case?

It was my understanding that one only needs to use bias adjustment when matching on more than one covariate (for example, nearest neighbour matching uses a bias correction term when matching on more than one covariate to speed up convergence).

Thanks (and sorry for the long question).