How can we justify using "with replacement" in propensity score matching I am trying to understand how it is justified to use "with replacement" methods in propensity score matching. In the literature there are many statements like this:

However, situations arise when there are not enough controls in the
  overlapping region to fully provide one match per treated unit. In
  this case it can help to use some control observations as matches for
  more than one treated unit. This approach is often called matching
  with replacement, a term which commonly refers to with one-to-one
  matching but could generalize to multiple control matches for each
  control. Such strategies can create better balance, which should yield
  estimates that are closer to the truth on average. Once such data are
  incorporated into a regression, however, the multiple matches reduce
  to single data points, which suggests that matching with replacement
  has limitations as a general strategy.



*

*How is the bold sentence above proven? I thought we were trying to find the truth, not cook the books that give us what we want. 

*How, even philosophically, can this possibly be justified? If this is real data, can you go around making clones and claiming the new set means anything? With replacement methods in Monte Carlo world make sense; we are taking limits and we want to preserve uniformity in our sample space - I can't see how that works here. 


Any insight would be appreciated. 
 A: Matching without replacement can yield very bad matches if the number of comparison observations comparable to the treated observations is small. It keeps variance low at the cost of potential bias.
Matching with replacement keeps bias low at the cost of a larger variance since you are using the same info over and over. So there's no free lunch (as long as you do actually adjust the variance). It is also good practice when matching with replacement to report the distribution of the number of times that untreated units are used in the matching. This is a second guardrail. 
Matching without replacement is also order dependent, which doesn't seem like a great feature. Suppose the treated observations have PS = (10, 20, 30) and the untreated units have PS = (5, 14, 27, 28). If we match the treated units in order the selected comparison units are (14, 27 and 28). Now change the order of the treated units to (20, 10, 30). The matched units are then (14, 5, and 28).
Philosophically, imagine doing an experiment where you test two types of fertilizer on two separate plots of land, and you have a single control plot that you leave alone. Is that control plot a reasonable counterfactual for what would happen if you did not apply fertilizer? Seems so, so you can "recycle" it again for each treatment-control comparison. But if you want to say something about fertilizer in general, you would have smaller standard errors if you had a second control plot. 
A: You're not creating clones in matching with replacement. You're matching one control to several treated units. You don't artificially double the size of your control group if you match each control to two treated units. The method of estimating the standard error for your treatment effect takes the reuse of control units into account.
Imagine the parallel scenario of propensity score weighting. When you up-weight individuals, you don't artificially increase the size of your sample; the standard error for your estimate accommodates the weights so you don't create something out of nothing. In the matching scenario, you can think of it as giving each control unit a weight of 2 (if they are matched to two treated units), but again, the standard error accounts for this weighting and does not allow for your sample size to be artificially inflated.
