1. Stable Matchings for Assigning Students to Dormitory-Groups
Nitsan Perach

2. About myself
PhD. Student at TAU under supervision of Prof. Shoshana Anily
Senior SAP consultant
The Stable Matching Model with an Entrance Criterion

3. Bibliography N. Perach , J. Polak and U. G. Rothblum,
A stable matching model with an entrance criterion applied to the assignment of students to dormitories at the Technion,
International Journal of Game Theory 36: , 2007
N. Perach and U. G. Rothblum,
Incentive compatibility for the stable matching model with an entrance criterion,
International Journal of Game Theory 39: , 2010
Stable Matchings for Assigning Students to Dormitory-Groups

4. Summary Case study of dormitory assignment at the Technion
Variant of the (GS) stable matching model which includes an “entrance criterion”
Modification of the (MV-W) Deferred Acceptance Algorithm and some of its properties
Unresolved issues and current research topics
Note: A new method for assigning students to dormitories at the Technion was implemented in the fall of 2004
Stable Matchings for Assigning Students to Dormitory-Groups

5. The case study: dormitory assignment at the technion
~5000 applications each year
~3500 beds
8 dormitory-groups
The groups are different:
Location
Setup
Age
Convenience
Price
Stable Matchings for Assigning Students to Dormitory-Groups

6. Assignment Principles
Student-eligibility for housing: Should depend on “personal characteristics.” (merit score)
Assigning students found eligible to a dormitory-group: Should depend on “academic seniority.”
(credit score)
All beds should be occupied.
Stable Matchings for Assigning Students to Dormitory-Groups

7. Old process Preprocessing Applying:
Personal data
Specification of the preferred dorm-group
“Merit-score” and “credit-score” determination (freshmen are handled differently).
Global-ranking of dormitory-groups
Capacity determination
Spare beds are held in each dorm-group (up to 20%).
Stable Matchings for Assigning Students to Dormitory-Groups

8. Old process ctd. Assignment Appeals and declines I – “crying”
Students with highest “merit-score” are determined “eligible.”
Filling dorm-groups from highest to lowest:
D1  q1 highest “credit score” students that selected it
The “selection” of those who listed D1 and did not get it is changed to D2.
D2  q2 highest “credit score” students that selected it
etc.
Appeals and declines I – “crying”
Assignment of rooms and Appeals II
Stable Matchings for Assigning Students to Dormitory-Groups

9. Disadvantages of previous assignment method
Students stated only one desired dormitory-group
A joint ladder of preferences
Students couldn’t state they prefer to live off- campus over getting some dormitory-group
Stable Matchings for Assigning Students to Dormitory-Groups

10. The (GS) classic model: 2-sided matching markets
One-to-one model:
Two groups
Each person having a ranking over a subset of members of the other group
Goal: finding a stable matching
Individually rational
No blocking pairs
(blocking pair: a pair of individuals that are not matched to each other but both individuals prefer the other over their match)
Stable Matchings for Assigning Students to Dormitory-Groups

11. Importance Amusing story – boys and girls
Results with interesting interpretation
Gale-Shapley and McVitie-Wilson Algorithms
Captures many concepts and ideas
Important applications
Interns assigned to residence (14,000/year)
Other junior-level job markets
Assigning students to schools in NYC & Boston
Kidney transplants by live donors
Rich mathematical analysis
Stable Matchings for Assigning Students to Dormitory-Groups

12. The stable matching model with an entrance criterion
2 finite disjoint sets:
S – students
T – dormitory-groups
For each student sS :
preferences over dormitory-groups, allowing to find some dormitory-groups unacceptable
ms - merit score
For each dormitory-group t:
qt - The number of beds in it
Preferences over students, allowing to find some students unacceptable
(extension of using a common and complete preferences over students determined by a credit score)
Stable Matchings for Assigning Students to Dormitory-Groups

13. Stability in our model Outcome (µ, W, R):
µ - An assignment of students to dorm-groups
W - Waiting list
R - Refugees
(W, R) partitions the set of unassigned students
Outcome (µ, W, R) is stable if:
All pairs in µ are mutually acceptable
No blocking pairs (s,t) with s in S \ W
ms < ms’ for each s in W and s’ in S \ W
Either W =  or no vacancies in any dorm-group t (qt students are assigned to dorm-group t)
Note: waiting lists of all stable outcomes are ordered by set inclusion
Stable Matchings for Assigning Students to Dormitory-Groups

14. McVitie-Wilson Algorithm
At each stage:
An unassigned (non single) student is picked
Proposal of student to his highest dorm-group that have not yet rejected him
If there is an empty bed or a less preferred student – accept
In the latter case, the dorm-group rejects the less preferred student
Stable Matchings for Assigning Students to Dormitory-Groups

15. The dormitory assignment algorithm (DorAA)
Data Structure:
W - Waiting list
P - In process
R - Refugees
Idea:
Iteratively run MV-W over the set of students in P
replace students that are destined to be refugees by students from W with highest merit-score
Stable Matchings for Assigning Students to Dormitory-Groups 

16. Properties
The output is independent of the selection of proposing students
Students can be moved from W to P in blocks
Each execution of the iterative step may start with the assignment generated in the previous iteration.
Stable Matchings for Assigning Students to Dormitory-Groups

17. Main Results
Let (µ, W, R) be the output of DorAA:
(µ, W, R) is stable
R is a minimal set among all stable outcomes
W is a maximal set among all stable outcomes
Each student in S \ W gets the best outcome he can get over all stable outcomes
R contains no student who finds all dorm-groups acceptable, and is acceptable over all dorm-groups (like in the case with credit-score)
Incentive compatibility: A student cannot submit a false preferences list and gain while all other students give true preferences list
Stable Matchings for Assigning Students to Dormitory-Groups

18. Open questions and current research topics
Incentive compatibility for a group of students
Incentive compatibility for a joint set
Linear programming for the dormitory- group assignment model
Students applying as “groups”
Stable Matchings for Assigning Students to Dormitory-Groups

19. Summary Case study of dormitory assignment at the Technion
Variant of the (GS) stable matching model which includes an “entrance criterion”
Modification of the (MV-W) Deferred Acceptance Algorithm and some of its properties
Unresolved issues and current research topics
Stable Matchings for Assigning Students to Dormitory-Groups

20. Nitsan Perach Nitsan.perah@gmail.com 054-9779427

21. The dormitory assignment algorithm (DorAA)
Initialization:
Let P= , R= and let W be the set of all students in S, ordered by their merit-score. Also, let μ be the empty assignment.
Stable Matchings for Assigning Students to Dormitory-Groups

22. The dormitory assignment algorithm (DorAA)
Iterative step:
If W is empty, STOP.
Move the first student (having the highest merit-score) from W to P.
Apply the Student-Courting version of the McVitie-Wilson Algorithm on data which considers only students in P. Let µ be the outcome assignment.
If for each tT then STOP. Otherwise, run another iterative step.
Stable Matchings for Assigning Students to Dormitory-Groups

23. The dormitory assignment algorithm (DorAA)
Output:
Return (µ, W, R), where R is the set of single students under µ when stopping.
Stable Matchings for Assigning Students to Dormitory-Groups 