Stability-Preserving, Incentive-Compatible, Time-Efficient Mechanisms for Increasing School Capacity
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We address the following dynamic version of the school choice question: a city, named City, admits students in two temporally-separated rounds, denoted R_1 and R_2. In round R_1, the capacity of each school is fixed but in round R_2, the City is happy to allocate extra seats to specific schools per the recommendation of the mechanism; in turn, the latter has to meet specified requirements. We study three natural settings of this model, with the requirements getting increasingly stringent. For Settings I and II, we give pairs of polynomial time mechanisms (M_1, M_2) which, besides addressing the specific requirements, find stable matchings, are dominant strategy incentive compatible (DSIC) w.r.t. reporting preference lists of students, and never break, in round R_2, a match created in round R_1. In Setting III, the mechanism needs to deal with residents of the City who try to game the system by not appearing in round R_1 and only showing up in round R_2, in addition to gaming by misreporting preference lists. We note that the mechanisms described above were all oblivious in that they needed to know only the preference lists of students being considered for admission and not those who were not participating. After proving that no oblivious mechanism can satisfy the rather stringent requirements of Situation III in a stability-preserving, DISC manner, we turn to non-oblivious mechanisms. Moreover, since we were unable to achieve DSIC, we relax this notion to weak incentive compatible (WIC) and give such a pair of mechanisms. Finally, we also give a procedure that outputs all possible stability-preserving extensions of a given stable matching (which may be exponentially many) with polynomial delay.
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