Matching in Practice:
Junior Doctor Allocation
in Scotland
David Manlove
Joint work with
Rob Irving, Augustine Kwanashie,
and Iain McBride
Centralised matching schemes

Intending junior doctors must undergo training in hospitals

Applicants rank hospitals in order of preference

Hospitals do likewise with their applicants

Centralised matching schemes (clearinghouses) produce a
matching in several countries
–
US (National Resident Matching Program)
– Canada (Canadian Resident Matching Service)
– Japan (Japan Residency Matching Program)
– Scotland (Scottish Foundation Allocation Scheme)
• typically around 750 applicants and 50 hospitals

Stability is the key property of a matching
–
[Roth, 1984]
1
Hospitals / Residents problem (HR)

Underlying theoretical model: Hospitals / Residents problem (HR)

We have n1 residents r1, r2, …, rn1 and n2 hospitals h1, h2, …, hn2

Each hospital has a capacity

Residents rank hospitals in order of preference, hospitals do
likewise

r finds h acceptable if h is on r’s preference list, and
unacceptable otherwise (and vice versa)

A matching M is a set of resident-hospital pairs such that:
1. (r,h)M  r, h find each other acceptable
2. No resident appears in more than one pair
3. No hospital appears in more pairs than its capacity
2
HR: example instance
r1: h2 h1
r2: h1 h2
Each hospital has capacity 2
r3: h1 h3
r4: h2 h3
h1: r1 r3 r2 r5 r6
r5: h2 h1
h2: r2 r6 r1 r4 r5
r6: h1 h2
h3: r4 r3
Resident preferences
Hospital preferences
3
HR: stability

Matching M is stable if M admits no blocking pair
– (r,h) is a blocking pair of matching M if:
1. r, h find each other acceptable
and
2. either r is unmatched in M
or r prefers h to his/her assigned hospital in M
and
3. either h is undersubscribed in M
or h prefers r to its worst resident assigned in M
4
HR: stable matching
r1: h2 h1
r2: h1 h2
Each hospital has capacity 2
r3: h1 h3
r4: h2 h3
h1: r1 r3 r2 r5 r6
r5: h2 h1
h2: r2 r6 r1 r4 r5
r6: h1 h2
h3: r4 r3
Resident preferences
Hospital preferences
M = {(r1, h2), (r2, h1), (r3, h1), (r4, h3), (r6, h2)} (size 5)
r5 is unmatched
h3 is undersubscribed
5
HR: classical results

A stable matching always exists and can be found in linear time
[Gale and Shapley, 1962]

There are resident-optimal and hospital-optimal stable
matchings

Stable matchings form a distributive lattice [Conway, 1976;
Gusfield and Irving, 1989]

“Rural Hospitals Theorem”: for a given instance of HR:
the same residents are assigned in all stable matchings;
2. each hospital is assigned the same number of residents in all
stable matchings;
3. any hospital that is undersubscribed in one stable matching is
assigned exactly the same set of residents in all stable matchings.
1.
– [Roth,
1984; Gale and Sotomayor, 1985; Roth, 1986]
6
Hospitals / Residents problem with Ties

In practice, residents’ preference lists are short

Hospitals’ lists are generally long, so ties may be used –
Hospitals / Residents problem with Ties (HRT)

A hospital may be indifferent among several residents

E.g., h1: (r1 r3) r2 (r5 r6 r8)

Matching M is stable if there is no pair (r,h) such that:
1. r, h find each other acceptable
2. either r is unmatched in M
or r prefers h to his/her assigned hospital in M
3. either h is undersubscribed in M
or h prefers r to its worst resident assigned in M

A matching M is stable in an HRT instance I if and only if M is
stable in some instance I of HR obtained from I by breaking the
ties [M et al, 1999]
7
HRT: stable matching (1)
r1: h1 h2
r2: h1 h2
Each hospital has capacity 2
r3: h1 h3
r4: h2 h3
h1: r1 r2 r3 r5 r6
r5: h2 h1
h2: r2 r1 r6(r4 r5)
r6: h1 h2
h3: r4 r3
Resident preferences
Hospital preferences
8
HRT: stable matching (1)
r1: h1 h2
r2: h1 h2
Each hospital has capacity 2
r3: h1 h3
r4: h2 h3
h1: r1 r2 r3 r5 r6
r5: h2 h1
h2: r2 r1 r6(r4 r5)
r6: h1 h2
h3: r4 r3
Resident preferences
Hospital preferences
M = {(r1, h1), (r2, h1), (r3, h3), (r4, h2), (r6, h2)} (size 5)
9
HRT: stable matching (2)
r1: h1 h2
r2: h1 h2
Each hospital has capacity 2
r3: h1 h3
r4: h2 h3
h1: r1 r2 r3 r5 r6
r5: h2 h1
h2: r2 r1 r6(r4 r5)
r6: h1 h2
h3: r4 r3
Resident preferences
Hospital preferences
M = {(r1, h1), (r2, h1), (r3, h3), (r4, h3), (r5, h2), (r6, h2)} (size 6)
10
Maximum stable matchings

Stable matchings can have different sizes

A maximum stable matching can be (at most) twice the size of a
minimum stable matching

Problem of finding a maximum stable matching (MAX HRT) is
NP-hard [Iwama, M et al, 1999], even if (simultaneously):
–
each hospital has capacity 1 (Stable Marriage problem with Ties and
Incomplete Lists)
– the ties occur on one side only
– each preference list is either strictly ordered or is a single tie
– and
• either each tie is of length 2 [M et al, 2002]
• or each preference list is of length 3 [Irving, M, O’Malley, 2009]

Minimization problem is NP-hard too, for similar restrictions!
[M et al, 2002]
11
MAX HRT: approximability

MAX HRT is not approximable within 33/29 unless P=NP, even if
each hospital has capacity 1 [Yanagisawa, 2007]

UGC-hard to approximate MAX HRT within 4/3- [Yanagisawa,
2007]

Trivial upper bound of 2 for MAX HRT

Succession of papers gave improvements, culminating in:
–
MAX HRT is approximable within 3/2 [McDermid, 2009; Király, 2013;
Paluch 2012] and within 19/13 (~1.4615) for ties on one side only [Dean
and Jalasutram, 2014]

Experimental comparison of approximation algorithms and
heuristics for MAX HRT [Irving and M, 2009]

Integer programming models for MAX HRT [Kwanashie and M,
2014]
12
Couples in HR

Pairs of residents who wish to be matched to geographically close
hospitals form couples

Each couple (ri,rj) ranks in order of preference a set of pairs of
hospitals (hp,hq) representing the assignment of ri to hp and rj to hq

Stability definition may be extended to this case [Roth, 1984;
McDermid and M, 2010; Biró et al, 2011]

Gives the Hospitals / Residents problem with Couples (HRC)

A stable matching need not exist:
(r1,r2): (h1,h1)
r3:
h1:2: r1 r3 r2
h1
13
Couples in HR

Pairs of residents who wish to be matched to geographically close
hospitals form couples

Each couple (ri,rj) ranks in order of preference a set of pairs of
hospitals (hp,hq) representing the assignment of ri to hp and rj to hq

Stability definition may be extended to this case [Roth, 1984;
McDermid and M, 2010; Biró et al, 2011]

Gives the Hospitals / Residents problem with Couples (HRC)

A stable matching need not exist:
(r1,r2): (h1,h1)
r3:

h1:2: r1 r3 r2
h1
Stable matchings can have different sizes
14
Couples in HR

The problem of determining whether a stable matching exists in a
given HRC instance is NP-complete, even if each hospital has
capacity 1 and:
–
there are no single residents
[Ng and Hirschberg, 1988; Ronn, 1990]
–
there are no single residents
each couple has a preference list of length ≤2
each hospital has a preference list of length ≤2
[Biró, M and McBride, 2014]

Heuristic (Algorithm C) described in [Biró et al, 2011]

Integer programming models for HRC [Biró, M and McBride, 2014]
15
Scottish Foundation Allocation Scheme

Set of applicants and programmes (residents and
hospitals)

Each applicant:
– ranked 10 programmes in strict order of preference
– had a score in the range 40..100
– from 2009, two applicants could link their applications

Each programme:
– had a capacity indicating its number of posts
– had a preference list derived from the above scoring function
– so ties were possible
16
Empirical results for SFAS instances (1)

SFAS datasets from the 2006-2008 matching runs were modelled
and solved [Kwanashie and M, 2014]
–
no couples; ties exist only on hospitals’ preference lists
–
sizes (|M|) were compared against those obtained from
approximation algorithms and heuristics due to
[Irving and M, 2009] (|M|)
Year
Residents Hospitals
|M|
|M|
SFAS
Time
(sec)
2008
748
52
709
705
705
75.5
2007
781
53
746
744
736
21.8
2006
759
53
758
754
745
93.0
17
Empirical results for SFAS instances (2)

For SFAS datasets from the 2010-2012 matching runs
–
approx. 5% of applicants involved in a couple – IP model
extended to HRC [Biró, M and McBride, 2014]
–
maximum stable matching sizes (|M|) were compared against
those obtained from heuristics due to [Biró et al, 2011] (|M|)
Year
Residents Couples Hospitals Posts
|M|
|M| SFAS Time
(sec)
2012
710
17
52
720
683
681
683
9.6
2011
736
12
52
736
?
688
684
?
2010
734
20
52
735
681
681
681
33.9
18
Open problems

Approximation algorithm for MAX HRT with performance
guarantee < 3/2?
– consider special cases:
• ties on one side only (< 19/13?)
• master lists

To make an HRC instance solvable, try to find “perturbation” of
the hospital capacities. A solvable instance can be found by:
– varying the capacity of each hospital by at most 4
– increasing the total number of posts by at most 9
[Nguyen and Vohra, 2014]
– is it sufficient to vary the capacity of each hospital by a smaller
number?
19
Book on Matching Under Preferences
20
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Nobel prize in Economic Sciences, 2012
27