UniversityAdmissions “Shortlist Matching”
description
Transcript of UniversityAdmissions “Shortlist Matching”
UniversityAdmissions
“Shortlist Matching”Challenges in the Light of Matching Theory
and Current Practises
24 May 2011
HSE
Ahmet AlkanSabancı 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
),,( PWM
Wm onP Mw onP
P ~ ) ,( za
exam scoregpa age location endowment
depthmaturitydrive warmth beauty
P~
~ a
)~
,,( PWM
Model
Centralized Two-Stage MatchingShortlist + Final
Find many-to-many matching σ on ).~
,,( PWM
shortlist of m
Invite wm, to submit ),( )()(ww
wm
short PPP
Find stable matching on ).,,( shortWM P
σ(m)
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 1
2 2
1 1
2 2
1 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 1
m2 2
m3 2 1 2 2
m4 2 1 1
No Pure Strategy Nash Equilibrium : Idiosyncratic Case
Em4(w4w5|m3(w4w5)) = α + (1- α) p(1-p) ≥ (1- α) (1-p) = Em4(w5w6|m3(w4w5)) Em3(w3w4|m4(w3w4)) = α ≥ (1- α) (1-p(1-p)) = Em3(w4w5|m3(w3w4)) Em4(w5w6|m3(w3w4)) = (1-α) + α p(1-p) ≥ α (1-p(1-p)) = Em4(w4w5|m3(w3w4)) Em3(w3w4|m4(w4w5)) = (1-α) ≥ α = Em3(w2w3|m4(w4w5))
p=1/2, α =7/16
w1 w2 w3 w4 w5 w6
m1 1 1m2 2 2 1m3 2 1 2m4 2 1 1
shortlisted but unmatched
Benchmark:
that such on )~
,,( PWM
henceMmkm (.)( all for
). all for Wwkw )(
regular-k P
listing-k
is
isSay short
One can continue and match the unmatched with
Question : Likelihood of being matched within ?
.~P
shortP
.say nWM
.12
),,,(
nk
k
shortshort WMofmatchingstableaandregularkFor
PP
Proposition:
y)(maximalit stability
P
nmmidway
nkmkm
nk
kmm
nmkmnkknWMA
mnkWMAWMAso
WMAby
mSayinedgesWMA
WWWMMMMWWMshort
4
3
12
)2()(2)(
)()()(
)(
.),(
\,\)(),(
minimum maximal matching for
k-regular bipartite graphB(n,k)
n
k
k
12 sharp
Yannakakis and Gavril 1978NP-hard
even when max degree is 3
cardinality
M
M
WWworst case
n=15k=3
n 3
2
circular
circular
n 3
2
Proposition
The minimum maximal matching cardinality for circular B(n,3) is
2/3 n
for n multiple of 3(k-1).
nk
k
1
worst case
n=12k=3
circular
circular
n almost decomposing circular worst case 12 9 8 8
60 45 40 36
k = 3
Concluding Remarks
Two-Stage Mechanism to improve efficiency
scope
coordination
information acquisition
incentives for pre-university education
with levers to control for corruption.