School Choice Activity

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451


email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk

Large cities have many public schools while suburbs and small towns will often have many fewer. Public schools typically reflect three levels of education, which cover the grades K-5, 6-8, and 9-12. These levels are often described as elementary school, middle (intermediate) school, and high school. Historically, the pattern of attendance for schools was that if there were several elementary schools in a town, each school served the students who were closest to that school. This was often described as the neighborhood school system. However, sometimes different schools had very different reputations for quality, and because of this, as well as for other reasons there arose a competitive (market) approach to placing students in schools. Simply described, the idea was that schools would all get better if they competed with each other for students, that is, students could attend the school they wanted to without regard to whether that school was the closest to where they lived or not. In either system there was the issue of whether there were enough "seats" at a given school to accommodate the students who wanted to come there. Sometimes communities have special schools which require competitive examinations to gain admission. Sometimes schools have "preferences" for students.

The problem of how to assign students to schools where the students (or more precisely their parents) have preference about the schools their children attend and the schools have preferences about the students who come is usually called the school choice problem. A "simple" version of this is where the "rankings" that the students have for the schools include all of the schools without ties allowed. The schools may have slots for more than one student and where the schools may also have more slots than there are students who are applying. The school choice problem falls within the class of problems which are sometimes known as two-sided markets. There are different "algorithms" or procedures which allow for the matching of schools and students. Here is an example to which you can apply your own ideas about matching students to schools.

Example:

The names of the schools are S1, S2, S3, S4.

S1 2 slots; S2 2 slots; S3 3 slots; S4 3 slots

The students are named k1, k2, ..., k8.

The schools rank the students as shown in the table below. The students further to the left are more preferred. School S3 likes student K2 5th best.

S1	K1	K2	K3	K4	K5	K6	K7	K8
S2	K3	K5	K4	K8	K7	K2	K1	K6
S3	K5	K3	K1	K7	K2	K8	K6	K4
S4	K6	K8	K7	K4	K2	K3	K5	K1

The students rank the schools as indicated in the columns of the array below. The student K2 likes school S3 third best.

Activity: Using some system that could be used on other similar problems, assign students to schools and explain why you think the system you used results in assignments that are desirable, i.e. maximize the "satisfaction" for the schools, the students, or both.