  We study a school choice problem under affirmative action policies where
authorities reserve a certain fraction of the slots at each school for specific
student groups, and where students have preferences not only over the schools
they are matched to but also the type of slots they receive. Such reservation
policies might cause waste in instances of low demand from some student groups.
To propose a solution to this issue, we construct a family of choice functions,
dynamic reserves choice functions, for schools that respect within-group
fairness and allow the transfer of otherwise vacant slots from low-demand
groups to high-demand groups. We propose the cumulative offer mechanism (COM)
as an allocation rule where each school uses a dynamic reserves choice function
and show that it is stable with respect to schools' choice functions, is
strategy-proof, and respects improvements. Furthermore, we show that
transferring more of the otherwise vacant slots leads to strategy-proof Pareto
improvement under the COM.
