In two-sided many-to-one matching markets some mechanisms induce worse allocations for one side of the market following a capacity reduction on the other side. This prediction, however, is not true for all matching mechanisms. Assuming preferences are strict and responsive, we are able to provide comparative statistics involving subsets of agents on both sides of the market for all stable mechanisms. Within the larger domain of substitutable preferences capacity reductions may have ambiguous consequences. Nevertheless, if preferences satisfy the law of aggregate demand a similar result does hold. These results are an extension of the one-to-one results on entry of Roth and Sotomayor (Econometric Society Monographs No. 18, 1990) to many-to-one environments. Finally, we consider truncation strategies, and describe how agents may manipulate the matching process to their advantage, without knowing which stable mechanism is being used either by reporting a truncated preference or by shading capacity.
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