Propensity score matching is a well known method of estimating treatments effects with observational data where the treatment is binary and it is assumed there is no selection on unobservables. To recall: one has data on individuals who have been treated. One would like to form a control who are otherwise identical (on average) but who were not treated.
One could match on characteristics but if the dimension of the X's is high that gets very difficult. It turns out that, due to a famous result of Rosenbaum & Rubin, given a key assumption, matching on the probability of being treated (the propensity) is equivalent.
So the norm is to model this probability with say a logit or probit, estimate the predicted probability and form your control group. There are several ways of doing this.
So what if you used a linear probability model instead? Well it turns out that, like speaking prose, you may have been doing this all along without realizing it. In a recent paper P. Kline shows that such a procedure is equivalent to our old friend the Oaxaca-Blinder estimator well known to de-composers. Aside from being easy to do it has several other nice features like being unbiased in finite samples and ensuring exact covariate balance between the two groups in circumstances where it is not guaranteed by other estimators.
For a nice introduction to matching methods see Conniffe et al.