UniversityAdmissions
“Shortlist Matching”
Challenges in the Light of Matching Theory
and Current Practises
24 May 2011
HSE
Ahmet Alkan
Sabancı University
Matching Theory
Gale Shapley, ‘College Admissions and Stability of
Marriage’ American Mathematical Monthly, 1962
• A matching is an allocation of students to universities.
• stable if there is no student s who would rather be at
university U and university U would rather replace
one of its students or an empty slot with s.
• Solution Concept , Benchmark Model :
most successful
Institutions
Decentralized “university admissions” Market
U.S.A
Centralized Marketplace Institution
“Student Selection Assesment and Placement”
Turkey, China, …
Semi-Centralized Marketplace Institution
“National Intern Resident Matching Program”
U.S.A
Two-Stage
Decentralized ‘Shortlist’ + Centralized Final Matching
“National Intern Resident Matching Program”
and similar marketplace institutions
studied intensively
Alvin Roth and collaborators
Hailed over decentralized markets for mainly 2 efficiency attributes:
All Together (Scope)
All at Same Time (Coordination)
Inefficiencies in decentralized matching :
bounded search
congestion
unravelling
Turkey
Students : 1 800 000 take Exam
800 000 qualify to submit rankings (up to 24 departments)
Universities : Exam Score-Type (80%) + GPA (20%)
5 Exam Score-Types
Quota Total = 200 000 + 200 000 + 200 000
Placement : Gale-Shapley U-Optimal Algorithm
Full Scope, Binding
China : 35 000 000
Placement : “School Choice” Algorithm : Priority to Students whoTop Rank
Turkey
EXAM woes :
Incentives on Pre-University Education
Poor / Narrow
“You get what you measure”
‘Classroom’ Drilling Sector
twice the budget of all universities
Equity
How to restore quality in Pre-University Schools
Centralized Two-Stage Matching
Shortlist + Final
Incentives on Pre-university Education : restore domain
where middle and high schools can perform and compete
for excellence
Avoid Inefficiencies inherent in Decentralization
save further on search
Control for Corruption
restrict match to shortlist
or shortlist plus all others
or some higher (lower) ranked
or no corruption control & only suggestive
Model
( M ,W , P)
m
P on W
P
~
(a, z )
exam score
gpa
age
location
endowment
~
P
P w on M
~
a
~
( M ,W , P)
depth
maturity
drive
warmth
beauty
Centralized Two-Stage Matching
Shortlist + Final
σ
Find many-to-many matching
σ(m)
Invite
m, w
shortlist of
to submit
Find stable matching on
P
short
on
~
( M ,W , P).
m
 (P
( M ,W , P
w
 ( m)
short
w
 ( w)
,P
).
)
Objections :
not constitutional
all the extra work for University Admissions Offices
corruption
but : “why not decentralize completely as in the US”
how to shortlist : instability ?
shortlisted but unmatched
Proposition : Pure Strategy Nash Equilibrium
holds in very special cases.
1
2
1
2
1
1
1
2
2
2
1
2
1
2
1
No Pure Strategy Equilibrium
•
If m3 does not interview, then m3 gets
w3 with probability ¾, it is better for m4
to interview (w4,w5).
•
If m4 interviews (w4,w5), it is better for
m3 to interview (w3,w4).
(because m4 will toplist w3 so m3 can
get w4+ when w3-.)
•
If m3 interviews (w3,w4), it is better for
m4 to interview (w3,w4).
(because then with prob ¼, m4 gets w4+
when w3- but if m4 interviews (w4,w5),
with prob 3/16 he will get w5+ when w4-.)
•
If m4 interviews (w3,w4), it is better for
m3 not to interview.
w1 w2 w3 w4 w5
m1 1
m2
m3 2
m4
1
2
1
2
2
2
1
1
No Pure Strategy Nash Equilibrium : Idiosyncratic Case
Em4(w4w5|m3(w4w5)) = α + (1- α) p(1-p) ≥ (1- α) (1-p)
Em3(w3w4|m4(w3w4)) =
α ≥ (1- α) (1-p(1-p))
Em4(w5w6|m3(w3w4)) = (1-α) + α p(1-p) ≥ α (1-p(1-p))
Em3(w3w4|m4(w4w5)) =
(1-α) ≥ α
= Em4(w5w6|m3(w4w5))
= Em3(w4w5|m3(w3w4))
= Em4(w4w5|m3(w3w4))
= Em3(w2w3|m4(w4w5))
p=1/2, α =7/16
w1
w2
m1
1
1
m2
2
2
m3
m4
w3
w4
w5
1
2
2
1
w6
1
2
1
shortlisted but unmatched
~
One can continue and match the unmatched with P.
Question : Likelihood of being matched within
Benchmark:
~
 on ( M ,W , P)
Say
such that
 ( m)  k
for all m  M . (hence
 ( w)  k
for all w  W ).

P short
is
k - listing
is k - regular
P short ?
say M  W  n.
Proposition:
For k  regular P
short
and  a stable matching of ( M , W , P

k

n.
2k  1
short
),
M   (W ), W   ( M )
A( M , W ) 

edges in P short

by stability (maximalit y)
M  M \ M , W  W \W
Say   m.
A( M  W )  
so A( M  W )  A( M  W )  k (n  m)
  A( M  W )  kn  2k (n  m)  k (2m  n)
k
  m  m 
n
2k  1
m    mk  nk
 midway

3
m  n
4
minimum maximal matching
for
k-regular bipartite graph
B(n,k)
cardinality
 k

  
n
 2k  1 
Yannakakis and Gavril 1978
NP-hard
even when max degree is 3
sharp
W
worst case
M
M
n=15
k=3
W
circular
2
 n
3
circular
2
 n
3
Proposition
The minimum maximal matching cardinality
for circular B(n,3) is
2/3 n
for n multiple of 3(k-1).
k

n
k 1
worst case
n=12
k=3
circular
circular
k =3
n
almost decomposing
circular
worst case
12
9
8
8
60
45
40
36
Concluding Remarks
Two-Stage Mechanism to improve efficiency
scope
coordination
information acquisition
incentives for pre-university education
with levers to control for corruption.