Tâtonnement in matching markets

I study tâtonnement processes in a matching market without transfers. In each period, schools set cutoffs, i.e. the preference ranks of the least preferred students they are willing to admit, and students accept their most preferred offers. Cutoffs are adjusted on the basis of demand-supply imbalances. A school's adjustment from one period to the next is moderate if it is bounded by the most recently observed imbalance at that school. I show that for any period in which all schools adjust moderately, the sum of demand-supply imbalances across all schools weakly decreases. Moreover, if all schools always adjust moderately and there is a unique stable matching, then adjustments converge to a market clearing cutoff vector. If there is more than one stable matching, moderate adjustments may cycle indefinitely but the supremum and the infimum of all cutoff vectors observed along a cycle are both market clearing.