Stable Matchings for Assigning Students to Dormitory-Groups Nitsan Perach 22.1.2018

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

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 36:519-535, 2007 N. Perach and U. G. Rothblum, Incentive compatibility for the stable matching model with an entrance criterion, International Journal of Game Theory 39: 657-667, 2010 Stable Matchings for Assigning Students to Dormitory-Groups

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

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

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

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

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

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

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

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

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

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

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

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 

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

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

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

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

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

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

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

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 