Two recent papers model some of the timing issues facing families thinking about school (or daycare) placement, when there are preferences for incumbent students, and the siblings of incumbent students. Needless to say, mechanisms that are strategy proof for static choice problems are no longer strategy proof when the strategy sets are expanded to take account of the dynamic problem. Both papers will be presented at the July 11 - 15, 2011: International Conference on Game Theory at Stony Brook.
The papers are:
The Daycare Assignment Problem, by John Kennes, Daniel Monte and, Norovsambuu Tumennasan
Dynamic School Choice by Umut Dur, a student at UT Austin, who I heard in Montreal...
The incentive problems in the daycare problem are akin to the incentive problems of the Boston mechanism. In the Boston mechanism changing your report changes your priority (the Boston mechanism is equivalent to DA where priorities are changed so students who ranked a school as their k-th choice get higher priority than student who rank the school > k). In the daycare problem your report changes your priority since priorities are history dependent (for example, kids are guaranteed to be allowed to stay where they first got admitted). Misreporting preferences to get a more advantageous priority can be a profitable manipulation, and these manipulations will probably still persist even when the market becomes large."