The Research On The Matching Mechanism Of School Choose Problem

The school choice movement is one of the most influential educational reforms in the United States since the 1980s.Its core is to introduce the market mechanism into the education field so that students and their parents have opportunity to choose the school by their own.Since the beginning of the 21st century,the school choice movement has been widely supported in the United States.The number of participants has shown a strong growth trend,and educational service providers and ways are increasingly diversified.However,the reform of school choice based on the market is controversial,among which the issue of racial segregation caused by the marketization of school choice has been a hot issue.Because minority students lack competitive advantages,they always occupy a weak position in the school choice market.With the rise of affirmative action,the support policies for minority students in the school choice market have been implemented.The most widely studied are the quota-based affirmative action and the reserve-based affirmative action.The quota-based affirmative action policy gives minority students higher chances to attend their desired schools by limiting the number of admitted majority students at some schools.But quota-based affirmative action can lead to unnecessary efficiency losses.The reserve-based affirmative action changes the way of quota-based affirmative action,provides a certain number of seats to support minority students and these seats can be provided for majority students under certain conditions.The reserve-based affirmative action can avoid the unnecessary efficiency loss of the quota-based affirmative action.Also,the matching of reserve-based affirmative action is Pareto dominant the matching of quota-based affirmative action.A central issue in school choice is the design of student assignment mechanism.The research on school choice mainly focuses on the deferred acceptance algorithm(DA)and the top trading cycles algorithm(TTC).The outcome of DA algorithm is stable but not Pareto efficient and the outcome of TTC algorithm is Pareto efficient but not stable.It’s important to get a stable matching in school choice,so our research is based on DA algorithm.With reserve-based affirmative action,Hafalir(2013)proposed the DAr algorithm.DAr algorithm produce stable and strategy-proof matching.However,with the DAr algorithm,there are some minority students with high priority will occupy the support seats,resulting in the limited role of affirmative action.We propose a new matching rule(DAa algorithm)to solve the above problem.This algorithm can produce stable and strategy-proof matching.According to the results of the simulation,DAa algorithm can improve the welfare of minority students compared with the DAr algorithm.The purpose of affirmative action is to improve the welfare of minority students,so affirmative action should be minimally responsive.In this paper,we prove that DAa is not minimally responsive and a stronger affirmative action may be hurt the welfare of minority students.The reason is that a minority student who has lower priority than a majority student at a school,is temporarily accepted by that school while the majority student is rejected.However,the majority student being rejected initiates a sequence of further rejections that may end with the minority student being rejected by the same school.Dogan call such minority student "interferers".Providing support to"disruptors" can’t improve their welfare,but will hurt the welfare of others.Based on the research of Dogan(2016),we treat the "interferers" as majority students and propose the MDAa algorithm.MDAa algorithm can get stable matching and have the property of minimally responsive.The impossibility theorem shows that no rule is stable,minimally responsive,and strategy-proof.We prove that MDAa algorithm is not strategy-proof.But we find that in the limiting information setting,at each problem,each student reporting her true preferences is an ordinal Bayesian Nash equilibrium of the preference revelation game induced by the MDAa algorithm.