Properties of matching algorithm in college admission problem with variable capacity quotas Robert Golański Warsaw School of Economics Social Choice and Welfare Moscow, July 2010 1
College admission problem: S = {s 1,...,s n } a set of students C = {c 1,...,c m } a set of colleges R =(R s1,..., R sn, R c1,..., R cm ) a list of preferences of students and colleges q =(q 1,...,q m ) a vector of college capacities C, S finite, disjoint, non-empty sets 2
Gale-Shapley student-optimal college admission algorithm: Step 1: Each student proposes to her first choice. Each college tentatively assigns its seats to its proposers one at a time following their priority order. Any remaining proposers are rejected. 3
Gale-Shapley student-optimal college admission algorithm: Step 1: Each student proposes to her first choice. Each college tentatively assigns its seats to its proposers one at a time following their priority order. Any remaining proposers are rejected. Step k: Each student who was rejected in the previous step proposes to her next choice. Each school considers the students it has been holding together with its new proposers and tentatively assigns its seats to these students one at a time following their priority order. Any remaining proposers are rejected. 4
Example: 3 colleges (A, B, C), 6 students Students preferences: 1 A B C 2 A C B 3 A B C 4 B A C 5 B C A 6 C A B Each college prefers 1 to 2 to to 6. Each quota set to 2. 5
Step 1 1, 2, 3 apply to A 4 and 5 apply to B 6 applies to C. All but 3 are tentatively accepted Step 2 3 applies to B (his second choice) as a result B rejects student 5 (who was already tentatively applied); Step 3 5 applies to C and is accepted. 6
Properties: - individual rationality; - stability (no blocking pair); - (group-)strategy proofness of student-optimal mechanism (Dubins- Freedman 1981) 7
Modification preference-induced quotas Why it matters student assignment problems In what follows we ll assume that each college has the same preferences over students + no indifference in preferences. So the timing of the problem is as follows: - the problem is defined by (C, S, R) only; - each students submits the list of preferences; - based on the students preferences, the quotas are determined; - given the quotas the students are assigned to colleges. 8
In what follows we ll assume responsiveness in college preferences and that having any student is preferred to not having them. Different possibilities of choosing quotas based on preferences. Quota = number of top preferences not very interesting (everybody will get assigned to their optimal college; equivalent to setting q i = S in the original problem, so clearly stable, optimal and non-manipulable) 9
We ll have a look at two different rules: - quotas decided based on top preferences with restrictions; - quotas based on scoring rules (Borda). 10
We ll have a look at two different rules: - quotas decided based on top preferences with restrictions; - quotas based on scoring rules (Borda) (in both cases we need some additional rules on rounding and tiebreaking. Consider for example the case in which top preferences for students are A for 7, B for 2 and C for 2 and the maximum capacity is set to 6. The natural assignment of quotas would be to assign 6 to A (since it already exceeds the maximum capacity) and equal number to B and C since they have equal numbers of students top choices. That would result in the quotas of (6; 2.5; 2.5) which is clearly not feasible). 11
Top preferences with restrictions: Consider the following problem: 3 colleges (A, B, C), 11 students with following preferences: 1 A to B to C 6 A to B to C 2 A to B to C 7 B to A to C 3 A to B to C 8 B to A to C 4 A to B to C 9 A to B to C 5 A to B to C 10 C to A to B 11 C to A to B 12
Set the maximum capacity to 5. The top-preference induced quotas will thus be q A = 5; q B = 3; q C = 3. 1 A to B to C 6 A to B to C 2 A to B to C 7 B to A to C 3 A to B to C 8 B to A to C 4 A to B to C 9 A to B to C 5 A to B to C 10 C to A to B 11 C to A to B 13
Set the maximum capacity to 5. The top-preference induced quotas will thus be q A = 5; q B = 3; q C = 3. 1 A to B to C 6 A to B to C 2 A to B to C 7 B to A to C 3 A to B to C 8 B to A to C 4 A to B to C 9 A to B to C 5 A to B to C 10 C to A to B 11 C to A to B 14
Set the maximum capacity to 5. The top-preference induced quotas will thus be q A = 5; q B = 3; q C = 3. 1 A to B to C 6 A to B to C 2 A to B to C 7 B to A to C 3 A to B to C 8 B to A to C 4 A to B to C 9 A to B to C 5 A to B to C 10 C to A to B 11 C to A to B Student optimal assignment is 1 through 5 to college A, 6 through 8 to college B, 9 through 11 to college C. 15
Stability needs redefining (is having one more student feasible?) Optimality holds if all colleges have the same preferences. 16
Stability needs redefining (is having one more student feasible?) Optimality holds if all colleges have the same preferences. Bad news the rule is manipulable. Suppose student 9 (with preferences A B C; assigned to college C) submits the preference ordering of B A C. In the standard Gale-Shapley problem nothing changes (9 is now tentatively assigned to college B in step 2, but eventually ends up in C anyway) 17
Now quotas change: With the falsely submitted preferences we now have: - 6 students with preferences A B C, - 3 students with preferences B A C, - 2 students with preferences C A B. Top-preference restricted quotas should thus be (5; 3.6; 2.4), rounded to (5; 4; 2) rather than (5; 3; 3). With new quotas the only change in the final assignment is that student 9 is now assigned to B (preferred to originally received C). 18
Does breaking the problem into two rounds of voting help? - in the second round the quotas are set and decided => standard Gale- Shapley problem (so stability and strategy-proofness follow); - in the first round still potential for manipulability 19
Does breaking the problem into two rounds of voting help? - in the second round the quotas are set and decided => standard Gale- Shapley problem (so stability and strategy-proofness follow); - in the first round still potential for manipulability Consider the previous example given others behave truthfully, student 9 still has an incentive to falsely present his preference in the quota-setting stage to increase q B. 20
Scoring rule (Borda) Again, we can show Pareto-optimality but the rule is manipulable 21
Scoring rule (Borda) Again, we can show Pareto-optimality but the rule is manipulable 1 C to A to B 6 B to A to C 2 C to A to B 7 B to A to C 3 C to B to A 8 B to A to C 4 C to B to A 9 A to B to C 5 B to C to A 10 A to C to B Borda scores A 9; B 11, C 10 Proportional quotas A 3; B 3.(6); C 3.(3) Rounded to q A = 3; q B = 4; q C = 3 22
Scoring rule (Borda) Again, we can show Pareto-optimality but the rule is manipulable 1 C to A to B 6 B to A to C 2 C to A to B 7 B to A to C 3 C to B to A 8 B to A to C 4 C to B to A 9 A to B to C 5 B to C to A 10 A to C to B Borda scores A 9; B 11, C 10 Proportional quotas A 3; B 3.(6); C 3.(3) Rounded to q A = 3; q B = 4; q C = 3 23
Suppose student 8 misrepresents his preferences as B C A: 1 C to A to B 6 B to A to C 2 C to A to B 7 B to A to C 3 C to B to A 8 B to C to A 4 C to B to A 9 A to B to C 5 B to C to A 10 A to C to B 24
Suppose student 8 misrepresents his preferences as B C A: 1 C to A to B 6 B to A to C 2 C to A to B 7 B to A to C 3 C to B to A 8 B to C to A 4 C to B to A 9 A to B to C 5 B to C to A 10 A to C to B New Borda scores are A 8; B 11, C 11 25
Suppose student 8 misrepresents his preferences as B C A: 1 C to A to B 6 B to A to C 2 C to A to B 7 B to A to C 3 C to B to A 8 B to C to A 4 C to B to A 9 A to B to C 5 B to C to A 10 A to C to B New Borda scores are A 8; B 11, C 11 New quotas: q A = 2; q B = 4; q C = 4 26
Suppose student 8 misrepresents his preferences as B C A: Final assignment becomes: 1 C to A to B 6 B to A to C 2 C to A to B 7 B to A to C 3 C to B to A 8 B to C to A 4 C to B to A 9 A to B to C 5 B to C to A 10 A to C to B New Borda scores are A 8; B 11, C 11 New quotas: q A = 2; q B = 4; q C = 4 27
Suppose student 8 misrepresents his preferences as B C A: Final assignment becomes: 1 C to A to B 6 B to A to C 2 C to A to B 7 B to A to C 3 C to B to A 8 B to C to A 4 C to B to A 9 A to B to C 5 B to C to A 10 A to C to B Now every student gets their best college! Student 8 gets to college B even though q B didn t change. (It s not a Pareto improvement though college A lost) 28
So even more problems manipulability now occurs even without setting additional maximum capacity rules. 29
Do stable non-manipulable rules exist? For some quota-setting rules they may (unrestricted first preferences; scoring rules with equal weights = always assigns equal quotas to each college). 30