School Choice: A Mechanism Design Approach

Transcription

1 School Choice: A Mechanism Design Approach Atila Abdulkadiroglu, Tayfun Sonmez Presented by : Saba Ahmadi

2 School Choice Problem -N Students, -M Schools, with a maximum capacity Each student should be assigned a seat in one of the schools. Find a Matching!

3 School Choice Problem Each student has strict preferences over all schools. Each school has a priority ordering over students, different with preferences, based on: A sibling already attending a school Living in the walking zone

4 Boston Student Assignment Mechanism Each student submits a preference ranking of schools. Each school has a priority ordering according to: Sibling and walking zone Sibling Walking zone Students in the same priority group are ordered based on a previously announced lottery.

5 Boston Student Assignment Mechanism Step k: Only the k th choices of the students are considered. For each school consider the students who have listed it as their k th choice and assign seats following their priority order.

6 Boston Student Assignment Mechanism Strategy-proof? Not strategy-proof. Even if a student has very high priority at school s unless she lists it as her top choice she loses her priority to students who have listed it as their top choices. Students are incentivized to misrepresent their preferences by improving ranks of those schools for which they have high priority.

7 Difference Between School Choice and College Admissions Problems Schools are objects to be consumed by students Colleges are agents who have preferences over students

8 Two New Mechanisms Two Competing Mechanisms Gale-Shapley Student Optimal Stable Mechanism Top Trading Cycles Mechanism

9 Gale-Shapley Student Optimal Stable Mechanism Interpret priorities of schools over students as their preferences, we ll get College Admission Problem. Each student proposes to her first choice. Each school s_i keeps a list of c_i most-preferred students. Rejects other proposers. Each student who is rejected proposes to her next choice. Algorithm terminates when no student proposal is rejected.

10 Pareto Optimal Matching A matching is Pareto Optimal if there is no other matching which assigns each student a weakly better school and at least one student a strictly better school.

11 Pareto Optimal? s 1 : i 1, i 3, i 2 s 2 : i 2, i 1, i 3 Stable Matching: i 1 i 2 i 3 s 1 s 2 s 3 s 3 : i 2, i 1, i 3 i 1 : s 2, s 1, s 3 i 2 : s 1, s 2, s 3 Pareto Dominated by: i 1 i 2 i 3 s 2 s 1 s 3 i 3 : s 2, s 1, s 3

12 Gale-Shapley Student Optimal Stable Mechanism Gale-Shapley results in a stable matching: There is no unmatched pair (i,s) where student i prefers school s to her assignment and i has higher priority than one of the students assigned to s. Not Pareto Optimal.

13 Two Competing Mechanisms Gale-Shapley Student Optimal Stable Mechanism Top Trading Cycles Mechanism

14 Top Trading Cycles Mechanism Step i: Each remaining student points to her favourite school among the remaining schools with nonzero capacity and each remaining school points to a remaining student with highest priority. There is at least one cycle. Every student in a cycle is assigned a seat at the school that she points to and is removed. Capacity of each school in the cycle is decreased by one. s 1 i 1 s 2 i 2 s 3 i 3.. s k i k

15 Top Trading Cycles Mechanism The Mechanism is Pareto Optimal. The Mechanism is Strategy-proof. Does not guarantee Stability.

16 Two Competing Mechanisms Stability? Gale-Shapley Student Optimal Stable Mechanism Pareto Optimality? Top Trading Cycles Mechanism

17 Controlled Choice Maintain the racial and ethnic balance at schools. Example: In a school 75 seats should be assigned to majority students and 55 should be assigned to minority students.

18 Gale-Shapley with Rigid Controlled Choice Constraints For each type of students, separately implement the mechanism. Example: Assign exactly 50 seats to type A and 60 seats to type B.

19 Gale-Shapley with Flexible Controlled Choice Constraints Step k: Each student who was rejected proposes to her next choice. Each school tentatively assigns its seats to its proposers following their priority. If the quota of a type fills the remaining proposers of that type are rejected and the tentative assignment proceeds with the students of the other types.

20 Gale-Shapley with Flexible Controlled Choice Constraints Stability: j i 1.Students i and j have different types 2. Assigned seats for type of student i are all full at school s. s r

21 Two Competing Mechanisms - Controlled Choice Gale-Shapley Student Optimal Stable Mechanism Top Trading Cycles Mechanism

22 Top Trading Cycles Mechanism with Rigid Controlled Choice Constraints For each type of students, separately implement the mechanism.

23 Two Competing Mechanisms - Controlled Choice Gale-Shapley Student Optimal Stable Mechanism Top Trading Cycles Mechanism

24 Top Trading Cycles Mechanism with Flexible Controlled Choice Constraints Keep a counter for pair (t i, s j ) which shows the remaining capacity in school s j for type t i. In step k each remaining student points to her favorite remaining school among those which has room for her type. Each school points to the student with highest priority. Detect a cycle, remove assigned students and update counters.

25 Top Trading Cycles Mechanism with Flexible Controlled Choice Constraints The result will be constrained Pareto Optimal. Which means there is no other matching that satisfies the controlled choice constraints and assigns all students a weakly better school and at least one student a strictly better school. Does not guarantee stability.

26 Two Competing Mechanisms - Controlled Choice Stability? Gale-Shapley Student Optimal Stable Mechanism Pareto Optimality? Top Trading Cycles Mechanism