universityadmissions “shortlist matching”
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/1.jpg)
UniversityAdmissions
“Shortlist Matching”Challenges in the Light of Matching Theory
and Current Practises
24 May 2011
HSE
Ahmet AlkanSabancı University
![Page 2: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/2.jpg)
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
![Page 3: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/3.jpg)
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
![Page 4: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/4.jpg)
“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
![Page 5: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/5.jpg)
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
![Page 6: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/6.jpg)
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
![Page 7: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/7.jpg)
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
![Page 8: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/8.jpg)
),,( PWM
Wm onP Mw onP
P ~ ) ,( za
exam scoregpa age location endowment
depthmaturitydrive warmth beauty
P~
~ a
)~
,,( PWM
Model
![Page 9: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/9.jpg)
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)
![Page 10: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/10.jpg)
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
![Page 11: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/11.jpg)
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
![Page 12: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/12.jpg)
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
![Page 13: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/13.jpg)
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
![Page 14: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/14.jpg)
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
![Page 15: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/15.jpg)
.say nWM
.12
),,,(
nk
k
shortshort WMofmatchingstableaandregularkFor
PP
Proposition:
![Page 16: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/16.jpg)
y)(maximalit stability
P
nmmidway
nkmkm
nk
kmm
nmkmnkknWMA
mnkWMAWMAso
WMAby
mSayinedgesWMA
WWWMMMMWWMshort
4
3
12
)2()(2)(
)()()(
)(
.),(
\,\)(),(
![Page 17: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/17.jpg)
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
![Page 18: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/18.jpg)
M
M
WWworst case
n=15k=3
![Page 19: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/19.jpg)
n 3
2
circular
![Page 20: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/20.jpg)
circular
![Page 21: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/21.jpg)
![Page 22: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/22.jpg)
n 3
2
![Page 23: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/23.jpg)
Proposition
The minimum maximal matching cardinality for circular B(n,3) is
2/3 n
for n multiple of 3(k-1).
![Page 24: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/24.jpg)
![Page 25: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/25.jpg)
![Page 26: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/26.jpg)
nk
k
1
![Page 27: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/27.jpg)
worst case
n=12k=3
![Page 28: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/28.jpg)
![Page 29: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/29.jpg)
circular
![Page 30: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/30.jpg)
circular
![Page 31: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/31.jpg)
n almost decomposing circular worst case 12 9 8 8
60 45 40 36
k = 3
![Page 32: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/32.jpg)
![Page 33: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/33.jpg)
![Page 34: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/34.jpg)
![Page 35: UniversityAdmissions “Shortlist Matching”](https://reader035.vdocuments.net/reader035/viewer/2022062321/56813bec550346895da51a6a/html5/thumbnails/35.jpg)
Concluding Remarks
Two-Stage Mechanism to improve efficiency
scope
coordination
information acquisition
incentives for pre-university education
with levers to control for corruption.