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.