The Nobel Prize in Economic Sciences 2012 and Matching Theory
Tınaz Ekim
Department of Industrial Engineering, Bogazici University, Bebek, 34342, Istanbul, Turkey
Keywords:
Stable Matching, Maximal Matching, Graph Classes, Computational Complexity.
Abstract:
The Nobel Prize in Economic Sciences 2012 was awarded jointly to A. E. Roth and L. S. Shapley “for the the-
ory of stable allocations and the practice of market design." The reason why it was awarded to A. E. Roth and
L. S. Shapley is two-fold: their extremely valuable efforts in applying scientific findings to very important real
life problems such as kidney exchange and student placement problems, and their contribution to the theory
of stable matchings.
In this mini survey, we will first present the theory of stable matchings starting from the basics such as the
Gale-Shapley Algorithm, and then discuss some variations encountered in various contexts. Two important
applications, namely student placement and kidney exchange problems, will be given special consideration.
The main focus of the survey will be the role of graph theory in the study of stable matchings. In particular,
the links between stable matchings and the problem of finding an inclusion-wise maximal matching of min-
imum size will be explored. As a natural consequence of this link, the field of graph classes which became
increasingly important, will be presented and illustrated with examples from matching theory.
1 INTRODUCTION
When the Nobel Prize in Economic Sciences was
awarded to A. E. Roth and L. S. Shapley in 2012, this
became the subject of a number of articles in the pop-
ular media. This popularity was indeed due to the im-
portance of the two important applications of stable
matchings for the whole human kind: placing can-
didates to institutions in a competitive environment
and the allocation of donors to patients for kidney
transplantation. Most of these articles emphasized
the tremendous efforts A. E. Roth and L. S. Shap-
ley spent for communicating their findings into insti-
tutions and the extremely positive outputs resulting
from these hands-on experience. However, not sur-
prisingly, the contributions of both Nobel prize recipi-
ents to the mathematical background of stable match-
ings was neglected in these popular articles. In this
mini survey, while the subtlety of these applications
will be covered thoroughly, we aim at shedding light
on the developments in matching theory which led A.
E. Roth and L. S. Shapley to the Nobel Prize from the
graph theoretical point of view.
Our paper is organized as follows. In Section 2,
we present the basics on stable matchings. Then, we
consider the problem of finding a stable matching un-
der various assumptions that can be encountered in
real life applications such as student placement and
kidney exchange problems. The existence of a sta-
ble matching, and in case it exists, the computational
hardness of finding one in each one of these contexts
are discussed. Various challenges encountered in stu-
dent placement and kidney exchange applications are
analyzed in more depth in Section 3. In Section 4,
we turn our attention to graph theoretical aspects of
stable matchings and related problems. We present
the notion of graph classes and discuss to what extent
it is helpful when considering hard-to-solve problems
in general. We illustrate the use of graph classes on
the problem of Minimum Maximal Matching which
is closely related to stable matchings.
2 STABLE MATCHINGS
The recent book by the Nobel recipient A. E. Roth en-
titled “Who Gets What â
˘
A¸T and Why: The New Eco-
nomics of Matchmaking and Market Design” (Roth,
2016) gives an extremely rich range of situations
where stable matchings play a key role in our daily
lives.
Economists study how societies allocate re-
sources. In market economics, most of the alloca-
tion problems are solved by the price system: high
wages attract workers into a particular occupation,
Ekim, T.
The Nobel Prize in Economic Sciences 2012 and Matching Theory.
DOI: 10.5220/0009459600050016
In Proceedings of the 9th International Conference on Operations Research and Enterprise Systems (ICORES 2020), pages 5-16
ISBN: 978-989-758-396-4; ISSN: 2184-4372
Copyright
c
2022 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
5
high energy prices induce consumers to conserve en-
ergy, etc. In many instances, however, using the
price system would encounter legal and ethical ob-
jections. Consider, for instance, the allocation of
public-school places to children, or the allocation of
human organs to patients who need transplants. This
is the territory of matching markets, where ¸Sseller-
ˇ
T and ¸SbuyersÂ
ˇ
T must choose each other, and
price is noŠt the only factor determining who gets
what. In such situations, typically, there are some
more desired agents/items than others such as high
ranked universities, jobs with higher wages or health-
ier and richer partner to marry. Markets where con-
ventional price-making does not apply such as mar-
kets for kidneys, job markets, assignment of users to
servers in a large distributed Internet service, student
placement problems and on-line dating services are
the topic of matchmaking where a “good" matching
is sought. A. E. Roth explains in his book in a very
fluent and accessible way how preferences are taken
into account in matchmaking by using the principals
of stable matchings, illustrated with examples from
our daily lives.
The notion of stable matching was first introduced
by D. Gale and L. S. Shapley (Gale and Shapley,
1962). In this seminal paper, the authors considered
a matching problem between n women and n men,
each one having a total preference list over the other
set; this is where the alternative term stable marriage
comes from. Let us first introduce the notion of stable
matchings through a placement problem which will
help us to give a better illustration of other variations
in upcoming sections.
2.1 Gale-Shapley Algorithm
Consider the problem of placing n candidates
C
1
, C
2
, . . . , C
n
into n institutions I
1
, I
2
, . . . , I
n
. Each
candidate has a total preference order over the insti-
tutions and each institution has a total preference or-
der over the candidates. The aim is to find a “good”
assignment of candidates to institutions which takes
into account the preferences of both sides. This prob-
lem can be modeled using a bipartite graph; one part
of the bipartition contains vertices representing candi-
dates and the other part contains vertices representing
institutions. In case each side has a total preference
list over the other set, every vertex in one side is ad-
jacent to every vertex in the other side and we obtain
what we call a complete bipartite graph. A placement
is an assignment of each candidate to an institution.
This corresponds to a matching, that is, a set of edges
sharing no common end-vertex, in the related bipar-
tite graph.
Clearly, there are many matchings corresponding
to different placements in such a bipartite graph.
What are the “natural” properties we can require for
a “good” placement / matching? Let us consider
an example (taken from (Sciences Prize Committee
of the Royal Swedish Academy of Science, 2012))
where candidates 1, 2, 3 and 4 are to be placed into
institutions S, O, D and P. The preferences of every
candidate and every institution are expressed as lists
where the notation B C is used to indicate that
institution B is preferred to institution C.
1 : SODP S: 4321
2 : SDOP O: 4132
3 : SOPD D: 1243
4 : DPOS P: 2143
D. Gale and L. S. Shapley formalized the notion
of a “good” matching as follows. A pair of candi-
date and institution not matched to each other, but
mutually prefers each other to their current matches
forms an unstable pair. An assignment is called a sta-
ble matching if it contains no unstable pair. In the
above example, assume there is a matching with as-
signments 1-P and 2-D. It can be seen that candidate
1 prefers D to P, and institution D prefers candidate
1 to 2. It follows that 1 and D constitute an unstable
pair and therefore such a matching is not stable.
D. Gale and L. S. Shapley proposes the Deferred
Acceptance Algorithm, thereafter called the GS Algo-
rithm (shorthand for Gale-Shapley Algorithm), which
finds a stable matching in a setting with n institutions,
n candidates and preference lists of each agent over
the other set. Before going into detail, let us present
the GS Algorithm in its original form of marriages
between men and women, and then come back to the
problem of placing candidates to institutions for fur-
ther discussion.
In the first round, first each unengaged man pro-
poses to the woman he prefers most, and then each
woman replies “maybe” to her suitor she most prefers
and “no” to all other suitors. She is then provisionally
“engaged” to the suitor she most prefers so far, and
that suitor is likewise provisionally engaged to her.
In each subsequent round, first each unengaged
man proposes to the most-preferred woman to whom
he has not yet proposed (regardless of whether the
woman is already engaged), and then each woman
replies “maybe" if she is currently not engaged or
if she prefers this man over her current provisional
partner (in this case, she rejects her current provi-
sional partner who becomes unengaged). The provi-
sional nature of engagements provides the right of an
already-engaged woman to “trade up".
ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems
6
This process is repeated until everyone is engaged,
which yields the final set of marriages (a matching
between men and women). It should be noted that,
once each side reveals their preferences, GS Algo-
rithm works as a black box in the sense that no propo-
sition occurs in reality and the algorithm simply pro-
duces the resulting matching.
Theorem 2.1. (Gale and Shapley, 1962) In a setting
where n candidates and n institutions express their
preferences over the other set, the deferred accep-
tance algorithm always finds a stable matching that
places all candidates.
When applied to the above example, the GS Al-
gorithm yields the following matching: (1D, 2P, 3S,
4O). This matching is an institution-optimal stable
matching in the sense that no institution is better off
in another stable matching. One can also note that the
roles of the men (candidates) and the women (institu-
tions) are perfectly symmetric in the algorithm. Con-
sequently, by exchanging their roles in the algorithm,
we can obtain the candidate-optimal stable matching.
Theorem 2.2. (Gale and Shapley, 1962) GS algo-
rithm where men propose finds a men-optimal match-
ing, GS algorithm where women propose finds a
women-optimal matching. The optimal matching for
one is the worst matching for the other, but both are
stable matchings.
In (Gale and Shapley, 1962) where D. Gale and
L. S. Shapley established the theory of stable match-
ings, they wrote that they hope this hypothetical prob-
lem of stable marriage between men and women finds
real and useful applications in the future. What they
did not know by the time was that, the very same
GS Algorithm has already been used in 1952 to as-
sign medical school graduates to hospitals in Illinois
(Roth, 2008). After all, isn’t it natural to think that
one should not be an engineer or a mathematician
to suggest a similar algorithm when confronted with
such a problem. Later on, it was noted that the GS
Algorithm has been rediscovered and used indepen-
dently over and over in various contexts (Roth, 2008).
On the other hand, as we will see in forthcoming sec-
tions, handling new types of constraints and various
assumptions will require much deeper mathematical
skills.
2.2 Incentive Compatible Strategies
Is it possible that candidates, knowing that the place-
ment is made with the institution-proposing GS Algo-
rithm, announce their preferences erroneously on pur-
pose and be better off; i.e. they are placed in a more
preferred institution? Even if this seems to be pretty
unlikely, a simple example shows that this can hap-
pen. In our example, assume the institution-proposing
GS Algorithm is applied and the institution-optimal
stable matching (1D, 2P, 3S, 4O) is obtained. Now
assume that candidate 4 misreports its preference list
on purpose as DPSO instead of her real pref-
erence order DPOS. Now, institution-proposing
GS Algorithm will give the following matching which
is stable with respect to the announced preferences:
(1O, 2D, 3S, 4P). Candidate 4 is now matched to P
which she prefers to O in reality. In other words, by
misreporting her preferences, she is matched to a bet-
ter choice for her. This is called a manipulation and
it is indeed not a desired property in such a setting.
A natural question is then, whether there is an algo-
rithm which is immune to any kind of manipulation,
a so-called incentive proof algorithm? The answer is
unfortunately negative as noted by the following “im-
possibility theorem":
Theorem 2.3. (Roth, 1982) No stable matching
mechanism (algorithm) exists for which stating the
true preferences is a dominant strategy for every
agent.
As stated in Theorem 2.3, the GS Algorithm is not
incentive proof. In other words, it does not motivate
all agents to express their true preferences in every
possible setting. Nonetheless, there are some good
news. May be D. Gale and L. S. Shapley were not
aware of it when they published their original paper
in 1962 (Gale and Shapley, 1962) but their algorithm
turned out to be more robust than expected as shown
later on by A. E. Roth:
Theorem 2.4. (Roth, 1984) The GS Algorithm gives a
stable matching with respect to real preferences even
in presence of manipulations.
It can be checked that the matching (1O, 2D, 3S,
4P) obtained under manipulation is a stable matching
with respect to real (original) preferences. As sug-
gested by Theorem 2.3, the institution-proposing GS
Algorithm is open to candidate’s manipulation (and
candidate-proposing GS Algorithm is open to institu-
tion’s manipulation). Theorem 2.4 states in turn that,
after all, preferences manipulated by candidates can
eventually place them into better institutions but never
into worse ones. Both in (Gale and Shapley, 1962)
and (Sciences Prize Committee of the Royal Swedish
Academy of Science, 2012), it is recommended that in
order not to encourage manipulations, and also keep-
ing in mind that institutions exist basically for candi-
dates, the use of candidate-proposing GS Algorithm
should be preferred in such placement problems. This
choice will ensure that truth telling is the best option
for candidates. As a consequence, the preferences
The Nobel Prize in Economic Sciences 2012 and Matching Theory
7
will reflect a more realistic value of the institutions,
and one will avoid any kind of mistrust in the mecha-
nism due to candidates gaming the system (see exam-
ples in Section 3.1).
Let us now focus on some variations of the place-
ment problem that occur in real life situations and dis-
cuss how we can deal with them.
2.3 Incomplete Preference Lists with
Ties
In a placement problem, what happens if a candidate
does not list all the institutions in his/her preference
list? Can we still guarantee the existence of a sta-
ble matching? If yes, how to find it and how many
candidates can be placed in a stable matching? This
problem, called stable matching with incomplete pref-
erence lists, is considered by D. Gale and M. So-
tomayor:
Theorem 2.5. (Gale and Sotomayor, 1985) In a
stable matching problem with incomplete preference
lists, the GS Algorithm modified in such a way that
only candidates/institutions existing in a preference
list receive propositions always gives a stable match-
ing. Moreover, in every stable matching, candidates
and institutions which are not assigned are the same.
It follows from Theorem 2.5 that an instance of
stable matching with incomplete lists has all its stable
matchings of the same size and a slightly modified
version of the GS Algorithm can find one.
Assume now that in addition to incomplete prefer-
ence lists, we also allow ties, that is, equally preferred
candidates or institutions in the preference lists. Such
a situation is very likely to happen when institutions
order candidates according to their scores at some
exam. In a stable matching problem with incomplete
lists with ties, if we assume that equally preferred can-
didates or institutions do not cause an unstable pair,
then there is always a stable matching and one can
be found in polynomial time. However, these stable
matchings do not necessarily have the same size and
finding one having the largest size (which is clearly
more desirable in all applications) is an NP-hard prob-
lem (Iwama and Miyazaki, 2008). On the other hand,
if we assume that equally preferred candidates or in-
stitutions may cause a pair to be unstable, then the ex-
istence of a stable matching is not guaranteed. How-
ever, it can be decided whether there is a stable match-
ing or not, and if any, a stable matching can be found
in polynomial time. Furthermore, in case there is a
stable matching, every stable matching has the same
size, thus seeking for a largest stable matching is not
an issue (Iwama and Miyazaki, 2008).
2.4 Is There a “Best" Stable Matching?
In Section 2.1, we have already mentioned two spe-
cial stable matchings, namely candidate-optimal and
institution-optimal stable matchings. These two sta-
ble matchings are in some sense the two extremes
and there can be many others. It has been shown that
in a stable matching problem, there can be exponen-
tially many stable matchings in the number of candi-
dates/institutions (Iwama and Miyazaki, 2008). This
observation arises the question of whether other opti-
mization criteria could be considered for the quality
of stable matchings. Let us mention three classical
criteria from the literature. Let p
I
(C) denote the posi-
tion of candidate C in the preference list of institution
I, and similarly p
C
(I) denote the position of institu-
tion I in the preference list of candidate C. Clearly,
every candidate desire to be matched to an institution
with higher rank in his/her preference list, and vice
versa.
For a stable matching M, the regret cost measures
the worst assignment in M as follows:
r(M) = max
(C,I)M
max{p
I
(C), p
C
(I)}.
The egalitarian cost c(M) and the sex-equalness
cost d(M) of a stable matching M are defined as fol-
lows:
c(M) =
(C,I)M
p
C
(I) +
(C,I)M
p
I
(C).
d(M) =
(C,I)M
p
C
(I)
(C,I)M
p
I
(C).
The minimum regret stable matching problem (the
minimum egalitarian stable matching problem and
the minimum sex-equal stable matching problem, re-
spectively) is to find a stable matching M with mini-
mum r(M) (c(M) and |d(M)|, respectively). Polyno-
mial time algorithms to find a minimum regret stable
matching and a minimum egalitarian stable matching
have been derived in the literature, whereas the min-
imum sex-equal stable matching problem has been
shown to be NP-hard (Iwama and Miyazaki, 2008).
As illustrated in these examples, the computational
complexity of finding the “best" stable matching de-
pends on the nature of the optimization criteria.
More details on variations of stable matching
problems and their algorithmic and computational
hardness issues can be found in (Iwama and
Miyazaki, 2008). Now, we will consider two impor-
tant applications of stable matchings and focus on
both practical and theoretical challenges encountered
during their implementations.
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8
3 IMPORTANT APPLICATIONS
As we already noted, the first application of the GS
Algorithm was a centralized clearinghouse, a match-
ing mechanism, to produce a matching of medical
school graduates with hospitals from the preference
lists. Note that this centralized clearinghouse, nowa-
days called the National Resident Matching Program,
was subject to an antitrust suit challenging the use of
the matching system which was claimed to be a con-
spiracy to hold down wages for residents and fellows,
in violation of the Sherman Antitrust Act. Yet, fol-
lowing studies showed that, empirically, the wages
of medical school graduates with and without cen-
tralized matching in fact do not differ. As a result,
the deferred acceptance algorithm has been explicitly
recognized as part of a pro-competitive market mech-
anism in American law (Roth, 2008).
In (Sciences Prize Committee of the Royal
Swedish Academy of Science, 2012) and (Roth,
2016), disadvantages of decentralized systems com-
pared to centralized matching mechanisms are ex-
plained with examples from our daily lives. In mar-
kets for skilled labors (such as doctors or lawyers),
offers are made to specific individuals rather than “to
the market" or a central clearinghouse. When an of-
fer is rejected, it is often too late to make other of-
fers. Thus, employers impose strict deadlines which
force students/applicants to make decisions without
knowing what other opportunities would later become
available to them. In a decentralized market, the prob-
lem of coordinating the timing of offers can result in
an unstable outcome. This phenomena is called con-
gestion. A stunning example of congested markets is
about law students hired by judges as early as second
year of law school. This is clearly not beneficial for
neither employers nor applicants but the congestion
causes such an undesired situation (Roth, 2016). An-
other result of a bad market design addressed in (Roth,
2016) is about the admission of kindergarten students
in Boston public schools, which consequently forced
parents to go to playgrounds to “collect information"
about other parent’s preferences in order to manipu-
late the system by making strategic choices (instead
of expressing their real preferences).
Let us now focus on two important applications
along with their practical and theoretical challenges.
3.1 School Admission
Whenever there are more than one school option for
a given student, various placement systems are used
throughout the world in order to place students to
schools. Students (or their parents) are usually the
main actors expressing their preferences. However,
schools may also have preferences over students; an
applicant may be given higher priority if for instance
she lives close to the school, has a sibling who at-
tends the school or has a higher score on a centralized
exam. This student placement problem can be seen as
a stable matching problem in a bipartite graph, where
students are applicants and schools are institutions.
One important point to note is that, unlike the exam-
ple of applicants and institutions, the preferences of
schools over students are required by regulation to be
based on objectively verifiable criteria. It means that
the incentive compatibility does not apply on the part
of schools. In other words, it is out of question for
schools to manipulate their preferences. It is there-
fore recommended to use the applicant-proposing GS
algorithm in school admission problems. This would
avoid the applicants to game the system by mak-
ing strategic choices since they would know that, by
telling their true preferences, they would be placed in
their best choice, among all possible stable matchings.
Two implementations of the GS algorithm in
school admission systems are given special consid-
eration in (Sciences Prize Committee of the Royal
Swedish Academy of Science, 2012); the New York
City public high schools and Boston public schools
which started to use a version of the GS Algorithm in
2003 and 2005, respectively.
The New York City public high schools were
asking applicants to list ve most preferred schools.
Then, a three-round acceptance/rejection/wait-list
procedure was applied. Those students who were not
placed after the third round were assigned via an ad-
ministrative process to a school for which they have
not expressed any preference. This system suffered
from congestion and resulted about 30,000 students
per year to end up via the administrative process (Ab-
dülkadiro
˘
glu et al., 2005a). As soon as a version of
the applicant-proposing GS Algorithm adjusted for
regulations and customs of New York City was im-
plemented in 2003, only about 3,000 students ended
up with the administrative process, a 90% of reduc-
tion compared to the previous years.
Unlike the New York City where the schools
were active players expressing their choices by the
acceptance/rejection/wait-list, in Boston, the place-
ment system did not let the schools to express their
preferences. The placement system, known as the
“Boston mechanism", was a central clearinghouse
that can be seen as a priority based system which is
still widely used in the world. It first matches as many
applicants as possible with their first-choice schools,
then tries to match the remaining applicants with their
second-choice school, and so on. When a school is
The Nobel Prize in Economic Sciences 2012 and Matching Theory
9
demanded by too many students, some students are
rejected using some priority criteria (e.g. sibling al-
ready attending the school, geographical proximity,
etc.) previously fixed by the school authority. In or-
der to be placed in a more preferred school, appli-
cants had to identify which schools were realistic op-
tions for them and falsify their reported preferences,
or else they would suffer from poor outcomes. In (Ab-
dülkadiro
˘
glu et al., 2005b), the authors gave evidence
of how the applicant-proposing GS algorithm would
eliminate the need for making strategic choices, thus
manipulating the system. After these successful im-
plementations, other school systems in the US have
followed New York and Boston by adopting similar
algorithms.
3.2 Kidney Exchange
We now focus on the kidney exchange problem which
was at the heart of the Scientific Background docu-
ment compiled by the Economic Sciences Prize Com-
mittee of the Royal Swedish Academy of Sciences
(Sciences Prize Committee of the Royal Swedish
Academy of Science, 2012). It is a known fact that
the number of patients waiting for kidney transplanta-
tion is increasing every year. According to the United
Network for Organ Sharing / Organ Procurement and
Transplantation Network data, around 95,000 patients
were in the waiting list for kidney transplant in 2019
in the US. Only 13,400 of them could be transplanted
a kidney, 9,300 of which from deceased donors and
the rest from living donors. Despite the fact that
many patients die while waiting and many others be-
come too ill to be eligible for transplantation, the
waiting lists get longer and longer every year. In-
deed, this makes any progress in the kidney trans-
plantation problem extremely valuable for the human
kind. L. S. Shapley and A. E. Roth, along with all
the authors cited in (Sciences Prize Committee of the
Royal Swedish Academy of Science, 2012), made gi-
ant steps towards improved solutions for the kidney
transplantation problem via their contributions to the
theory of stable matchings, and their extremely valu-
able efforts in implementing their findings in various
institutions. In (Roth, 2012), A. E. Roth describes
these developments as “a history of victory after vic-
tory in a battle that we are loosing" since the waiting
lists become longer and longer every year despite all
progress.
The conventional procedure for kidney donation
is as follows. Some patients may have a willing kid-
ney donor. However, a direct donation may be ruled
out for medical reasons such as blood type mismatch.
Still, if patient P
1
has a willing (but incompatible)
donor D
1
, and patient P
2
has a willing (but incompat-
ible) donor D
2
, then if P
1
is compatible with D
2
and
P
2
with D
1
, an exchange is possible: D
2
donates to P
1
and D
1
to P
2
. Such a two-pair exchange is illustrated
in Figure 1.
Figure 1: Kidney exchange between two incompatible
donor-patient pairs.
In many countries, non-profit organizations signif-
icantly increase kidney transplantation rates by cen-
trally collecting donor and patient data and finding
compatible donor-patient matches to perform kidney
exchanges (in addition to direct donations from de-
ceased or living donors). Using various medical in-
dicators, doctors evaluates the compatibility of each
donor with each patient. Below some threshold value,
a donor is said to be incompatible with a patient. Let
us represent each pair of patient P
i
and donor D
i
(who
is willing to donate to his/her patient P
i
but not com-
patible with) by a vertex labeled D
i
P
i
. If donor D
i
is
compatible with some patient P
j
for i 6= j then add
an arc from vertex D
i
P
i
to vertex D
j
P
j
. It is impor-
tant to note that the graph obtained in this way is di-
rected; moreover it is not necessarily bipartite as in
the case of student placement problem (see Figure 2
for an example). Each patient P
i
has a list of com-
patible donors represented by the end-vertices of the
incoming arcs to vertex D
i
P
i
, and P
i
can possibly ex-
press his/her preference order over these donors.
Let us observe what a feasible kidney exchange
solution corresponds to in this graph. One major dif-
ference with respect to the stable matching problem
is that, exchange chains involving more than two ver-
tices are possible: Donor D
1
donates to patient P
2
,
donor D
2
donates to patient P
3
and so on until donor
D
k
donates to patient P
1
. In the graph model, such a
chain corresponds to a directed cycle since every in-
coming arc to a vertex should be followed by an out-
going arc (recall that a donor is only willing to donate
his/her kidney if his/her patient can receive a kidney
from another donor). See Figure 2 for an example of
a graph modeling the kidney exchange problem and a
possible solution.
Since a matching edge in the stable matching
problem can be seen as a directed cycle between two
vertices, these exchange chains are a generalization
of matchings. Accordingly, A. Abdülkadiro
˘
glu, A.
ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems
10
Figure 2: A graph modelling a kidney exchange relation
(with compatibility estimates on arcs which are not shown)
and a solution shown with bold arcs.
E. Roth and T. Sönmez extended previous works on
kidney exchange where only two donor-patient pairs
are allowed to the case of exchange chains by de-
veloping an algorithm called Top Trading Cycle (Ab-
dülkadiro
˘
glu and Sönmez, 1999; Roth et al., 2004).
Unfortunately, this nice theoretical achievement had
very limited impact in real life due to several practi-
cal constraints. For example, for technical and eth-
ical reasons, a kidney exchange requires simultane-
ous surgeries of all donors and patients. This implies
six simultaneous surgeries (thus six surgical teams,
six operating rooms, etc.) for an exchange chain of
size three; which makes such exchanges impossible
in practice due to infrastructural shortcomings and the
hardness of scheduling medical staff. This fact im-
plied the need for new methods which allow exchange
chains involving only two donor-patient pairs. The
Economic Sciences Prize Committee of the Royal
Swedish Academy of Sciences emphasizes the valu-
able contributions of A. E. Roth, T. Sönmez and M.U.
Ünver to show that one can still come up with efficient
outcomes even if we restrict the size of the chains to
two (Sciences Prize Committee of the Royal Swedish
Academy of Science, 2012).
Note that the kidney exchange problem where
only two-pair exchanges are allowed is modeled as a
stable matching problem in a general graph (which is
not necessarily bipartite) where edges are not directed
since the two-pair condition implies that if donor D
i
donates to patient P
j
then necessarily donor D
j
do-
nates to patient P
i
. However, preference lists are not
necessarily symmetric in the sense that the vertex D
i
P
i
may prefer D
j
P
j
the most whereas D
i
P
i
is the least
preferred neighbor of D
j
P
j
. This new problem is
called the stable roommate problem in the literature.
Unlike the bipartite case, a simple example shows that
a stable matching does not necessarily exist in such
a setting (Gale and Shapley, 1962). Let 1, 2, 3 and
4 be four students who will be placed in two rooms.
As roommate, Student 1 prefers most Student 2, Stu-
dent 2 prefers most Student 3 and Student 3 prefers
most Student 1. Let also Student 4 be the least pre-
ferred roommate of all the others. Independently from
the preferences of Student 4, no matching would be
stable: let i be the student matched with Student 4.
Then the student whose most preferred roommate is
i and Student i form an unstable pair since they pre-
fer one another compared to their current roommates.
This observation shows that the structure of the graph
modeling the stable matching problem (here bipartite
versus general) plays a key role in the solution of the
problem. So, a natural question is the following: Un-
der which circumstances in terms of the graph struc-
ture, the existence of a stable matching is guaranteed?
We will come back to this question in Section 4.2
where we will discuss how graph classes play a key
role in the computational complexity of various opti-
mization problems.
Although the existence of a stable matching is not
guaranteed in a general graph, it is possible to de-
cide in polynomial time if a graph (with given pref-
erence lists of every vertex over its neighbors) admits
a stable matching or not, and find a stable matching if
any (Iwama and Miyazaki, 2008). On the other hand,
in the context of kidney exchange (and also in other
circumstances where a stable matching is sought), if
there is no stable matching, it is desirable to find a
matching which is as large as possible (to ensure a
maximum number of kidney transplants) whilst the
number of unstable pairs is kept minimum. Unfor-
tunately, this problem turns out to be NP-hard and
therefore related research is mainly focused on (sub-
optimal) approximation algorithms. In addition to ex-
change chains, taking into account volunteer donors
with no associated (incompatible) patient and trans-
plants from deceased donors makes kidney exchange
a very complex problem both theoretically and in
practice. On the other hand, in practice, doctors ex-
press that preferences of patients over donors can be
(and should be for ethical and efficiency reasons) ne-
glected and considered as 0/1 preferences (Roth et al.,
2015). This basically makes the kidney exchange
problem a maximum (cardinality) matching problem
(in case only two donor-patient pair exchanges are al-
lowed).
Another interesting challenge in kidney exchange
problem arose due to hospitals’ profit maximization
motive which led them try to arrange as many ex-
changes as possible in their premises. Large trans-
plantation centers, becoming “individually rational"
players, tend to withhold information about their
easy-to-match (over-demanded) donor-patient pairs
(with high compatibilities) and expose only their
(under-demanded) hard-to-match donor-patient pairs
(with low-compatibilities) to a centralized clearing-
house. They think that they may loose by show-
ing their over-demanded pairs in the kidney market
and thus prefer to match locally those over-demanded
pairs before or after the exchange clears. In order
to convince hospitals to the value of fully participat-
The Nobel Prize in Economic Sciences 2012 and Matching Theory
11
ing to the kidney market, experimental studies are
conducted using randomly generated compatibility
graphs (see e.g. (Ashlagi and Roth, 2012; Toulis and
Parkes, 2011)). The random graph models in (Ash-
lagi and Roth, 2012) also took into account the “jel-
lyfish structure" of real compatibility graphs; over-
demanded pairs are highly connected between them
and under-demanded pairs form rather sparse com-
ponents. Studies on Erdös-Renyi random graphs al-
lowed them to derive the following:
Theorem 3.1. (Ashlagi and Roth, 2012) In almost
every large graph without non-directed donors, there
exists an efficient allocation with cycles of size at most
3 where all over-demanded pairs are matched.
In (Roth, 2012), A. E. Roth notes that this re-
sult is in line with the so-called Gallai-Edmonds de-
composition (Plummer and Lovász, 1986): vertices
of a graph G are uniquely decomposed into three
sets D(G), A(G), C(G) where D(G) contains all ver-
tices which are not matched by at least one maxi-
mum matching (under-demanded vertices), A(G) is
the set of neighbors of D(G) in V (G) \ D(G) (over-
demanded vertices), and C(G) is the rest of vertices
which are perfectly matched in between by every
maximum matching. Main results from simulations
on random graphs show that in the worst case, the cost
of fully participating to the market place can be “very
high" for hospitals as compared to being individually
rational players. However, it is observed that on aver-
age, sharing full information with the kidney market
has “almost no cost" to hospitals. This empirical ev-
idence supports the full participation of hospitals to
the kidney market, rendering this later more efficient
in practice.
As outlined in this section, several practical and
theoretical challenges are faced in the kidney ex-
change problem. This explains why matching related
problems in the context of kidney exchange continue
to be a major research area in economics, computer
science and mathematics.
4 MATCHING THEORY
In this section, we will discuss matchings in more
depth from a graph theoretical point of view. We will
first introduce the notion of maximal matchings in re-
lation with stable matchings. Then, we will give spe-
cial emphasize to the approach of graph classes for
solving hard problems, in particular related to match-
ing problems.
4.1 Minimum Maximal Matchings
A matching is called (inclusion-wise) maximal if
there is no other matching which properly contains
it. In other words, no new edge can be added to a
maximal matching. Clearly, a maximal matching is
not necessarily of maximum size, unlike a maximum
matching (with respect to its size) which is necessar-
ily maximal (or else its size could have been increased
by adding a new edge). In particular, a graph can ad-
mit maximal matchings of different sizes as illustrated
in Figure 3.
Figure 3: Maximal matchings of size 2 and 3 shown by bold
edges.
It can be noted that a stable matching is necessar-
ily maximal by definition; otherwise, there would be
a pair of vertices which are adjacent (thus, eligible
for each other in the sense that they are in their
respective preference lists), yet not matched to any
vertex and would prefer to be matched to each other
rather than not being matched at all. Consider the
following example with candidates 1, 2 and 3 and in-
stitutions A, B and C with incomplete preference lists:
1: AC A: 123
2: A B: 3
3: BCA C: 31
The graph in Figure 4 models this stable match-
ing problem and the application of the GS Algo-
rithm (where institutions propose) gives the matching
shown with bold edges. Clearly, this stable matching
is maximal but not maximum as there is a matching
of larger size; namely, (1C, 2A, 3B) forms a matching
of size 3 (but it is not stable because 1 and A form an
unstable pair).
Figure 4: A stable matching is necessarily maximal but not
necessarily of maximum size.
Remind that in presence of incomplete preference
lists with ties there can be stable matchings of dif-
ferent sizes. Since large stable matchings are more
desirable in all applications, a natural question is how
small can a stable matching be in the worst case. By
the above observation, the size of a stable matching
is bounded below by the size of a maximal matching
of minimum size. In other words, a stable matching
ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems
12
can not be smaller than a smallest maximal match-
ing. We call the problem of finding a maximal match-
ing of smallest size the Minimum Maximal Matching
problem (MMM for short). It is well known that,
given a graph, a maximum (size) matching can be
found in polynomial time by the so-called Edmond’s
Augmenting Path Algorithm (Plummer and Lovász,
1986). However, the problem of finding a minimum
maximal matching (equivalent to a minimum edge
dominating set in terms of optimization problem) is
known to be NP-hard (Garey and Johnson, 1979).
Note that the student placement problem is mod-
eled as a stable matching problem in a bipartite graph.
So, it is crucial to know whether the bipartite struc-
ture of the graph makes MMM any easier to solve, so
that a lower bound on the size of a stable matching
can be efficiently computed. Unfortunately, it turns
out that MMM is NP-hard even when the input graph
is restricted to be a 3-regular bipartite graph, that is,
a bipartite graph where every vertex has three neigh-
bors (thus, every candidate lists 3 institutions and ev-
ery institution lists 3 candidates for each quota) (De-
mange and Ekim, 2008). It has been shown that if
the structure of the 3-regular bipartite graph is further
restricted in such a way that the vertices in each side
can be ordered so that every vertex is adjacent to three
consecutive vertices in the other side, then MMM can
be solved in polynomial time (Alkan and Alio
˘
gulları,
2015).
As illustrated by these examples, the structure of
the graph that we obtain when we model a real life
problem is crucial in deciding how efficiently we can
solve it. In other words, it does not really matter that
a problem is NP-hard in general graphs if the real
life application implies a structure on the graph that
makes the problem easier to solve. This notion is
best captured and formalized by the notion of “graph
classes" which is the topic of the next section.
4.2 Graph Classes
We will first address the approach of graph classes
in general, and then focus on MMM from this point
of view. Many optimization problems in graph the-
ory are NP-hard and thus do not admit efficient so-
lution procedures unless P=NP. A classical approach
to solve these problems consists of taking into ac-
count the structural properties of the graphs arisen
from the related applications and developing efficient
algorithms by the help of these properties. The set
of graphs defined by a common property is called a
graph class. Various graph classes obtained by mod-
eling several real life applications and the complex-
ity of the related problems in these graph classes are
nicely illustrated in (Golumbic, 2004). In this ap-
proach, whenever a real life problem is modeled with
graphs, it is crucial to understand the structure of the
graphs obtained in this way and whether this structure
is of any help in solving the related problem. An NP-
hard problem can sometimes be solved efficiently in
some special graph classes; in such cases polynomial-
time exact algorithms can be derived. On the contrary,
sometimes an NP-hard problem remains NP-hard
even when the input graph is restricted to a special
graph class. A systematic analysis of such polyno-
mial and NP-hard cases gives important insight about
the real difficulty of the problem under consideration.
According to (http://www.graphclasses.org/, 2019),
there are more than 1600 graph classes in the liter-
ature and there is a huge literature about polynomial-
time exact algorithms or NP-hardness proofs for var-
ious optimization problems in special graph classes.
Motivated by the fact that structural properties of var-
ious graph classes are at the heart of the complexity
studies, many books consider the structure of special
graph classes and the relationship between different
classes (see e.g. (Brandstädt et al., 1999; Golumbic,
2004)). The complexity analysis of an optimization
problem with respect to various graph classes is based
on the following observation.
Let G and H be two graph classes such that G
H , that is G is a subclass of H , or equivalently H is
a superclass of G. Then the following hold:
1. If problem Π is NP-hard in G then Π is also NP-
hard in H .
2. If problem Π can be solved in polynomial-time in
H , then it can also be solved in polynomial-time
in G (using for instance the same algorithm as for
H , but hopefully a more efficient one using the
additional properties of G with respect to H ).
It follows from this observation that it is crucial
to decide whether a graph model falls into one of
the known graph classes. The problem of deciding
whether a given graph belongs to a graph class or
not is called the recognition problem for this class.
It should be noted that this decision problem is not
always an easy one. Although recognizing bipartite
graphs or planar graphs (modeling the famous map
coloring problem) can be done in polynomial time,
the recognition of disk graphs (modeling frequency
assignment problems), for instance, turns out to be
NP-complete (http://www.graphclasses.org/, 2019).
We note that the above observation does not al-
ways allow us to conclude about the complexity situ-
ation of our problem in a given graph class G. This
might be the case when G has no containment rela-
tion with other graph classes where the complexity of
the problem is known; that is, either G is completely
The Nobel Prize in Economic Sciences 2012 and Matching Theory
13
disjoint from such a class, or G intersects with such a
class but they have non-empty symmetric differences.
Alternatively, this can happen when all graph classes
for which the problem is known to be polynomial time
solvable are subclasses of G, and all graph classes
where the problem is known to be NP-hard are super-
classes of G; we note that nothing can be concluded
in these cases.
Let now G be a graph class where the complex-
ity of the problem under consideration is open. In
light of the above discussion, if we fail to find a
polynomial-time algorithm in some graph class G,
then we either consider a subclass of G in order to
derive a polynomial-time algorithm, or search for an
NP-hardness proof for the problem in G. On the con-
trary, if we obtain a polynomial-time algorithm in G,
then, as a next step, we can consider a superclass of G
to generalize our polynomial-time algorithm, keeping
in mind that the problem can possibly become NP-
hard in this larger class. An NP-hardness result would
have analogous effects in guiding our research.
In light of these facts, we use diagrams to sum-
marize the complexity situation of an optimization
problem in various graph classes. Such a diagram
for MMM is given in Figure 5 where an arc from a
graph class H to another graph class G means that
G H . Note that this diagram is not comprehen-
sive in terms of graph classes; it should be rather seen
as a snapshot of complexity results in some graph
classes. See (http://www.graphclasses.org/, 2019) for
definitions of various graph classes used in Figure 5.
Bold framed graph classes indicate a result shown in
an original research paper whereas the complexity re-
sults in graph classes without bold frames are directly
implied by these results and the relations between
the graph classes. For instance, the NP-hardness of
MMM in bipartite graphs with maximum degree 3
( 3) (Yannakakis and Gavril, 1980) implies di-
rectly that MMM is also NP-hard in (general) bipar-
tite graphs, and thus also in perfect graphs, by the
containment relation. In a similar way, the existence
of a polynomial time algorithm for MMM in unit in-
terval graphs (Boyacı et al., 2017) directly implies
that MMM is also polynomial time solvable in trees
(which are contained in unit interval graphs). Other
results represented in Figure 5 by bold framed rect-
angles are the NP-hardness of MMM in k-regular bi-
partite graphs for every k 3 (Demange and Ekim,
2008), in planar graphs of maximum degree 3 (Yan-
nakakis and Gavril, 1980), in 3-regular planar graphs
(Horton and Kilakos, 1993) and in induced subgrids
(Demange and Ekim, 2013); and the polynomial time
solvability of MMM in bipartite permutation graphs
(Srinivasan et al., 1995), in series-parallel graphs
(Richey and Parker, 1988), in unit interval graphs
(Boyacı et al., 2017), in block graphs (Hwang and
Chang, 1995) and in trees (Mitchell and Hedetniemi,
1977). Such a diagram guides researchers by display-
ing graph classes where the complexity of a problem
is open. For instance, it can be seen in Figure 5 that it
is not known to date whether MMM is NP-hard or not
in interval graphs, in grid graphs, or in chordal bipar-
tite graphs. Each one of these questions is open for
investigation.
Figure 5: Complexity of MMM in various graph classes.
4.3 A Reverse Question
As illustrated with examples in the previous section,
when a real life problem is modeled using graphs,
some structural properties on graphs are implied. Sev-
eral graph classes are defined in this way. Some, how-
ever, are defined by asking a reverse question with
respect to a problem. Remind that MMM is NP-
complete in general. So, a reverse question is, what
are the graphs for which MMM can be solved in a
“trivial way"? This trivial way is usually formalized
by a greedy algorithm, which depends on the prob-
lem under consideration. In case of MMM, it would
be natural to think about an algorithm which greed-
ily constructs a maximal matching by adding edges
(to an initially empty set) until no more edges can be
added. Then the question is, what are the graphs for
which this greedy algorithms always yields a maximal
matching of minimum size? As any maximal match-
ing is likely to be produced by this greedy algorithm, a
minimum size will only be guaranteed if every maxi-
mal matching has the same size. The family of graphs
having this property is called equimatchable graphs.
An example of an equimatchable graph along with all
of its maximal matchings is given in Figure 6.
We note that if the input graph of a stable match-
ing problem is equimatchable, then every stable
matching has the same size (as they are all maximal).
Equimatchable graphs are first considered inde-
pendently in (Grünbaum, 1974), (Lewin, 1974), and
(Meng, 1974). They are formally introduced by (Lesk
ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems
14
Figure 6: An equimatchable graph with all of its maximal
matchings shown by bold edges.
et al., 1984) where their characterization with respect
to the Gallai-Edmonds Decomposition (see Section
3.2) has been presented and a polynomial time recog-
nition algorithm is derived from this characterization.
A more efficient recognition algorithm is then given
in (Demange and Ekim, 2014). Structural properties
of equimatchable graphs such as connectivity, forbid-
den subgraphs and girth have also been studied exten-
sively in the literature (see e.g. (Favaron, 1986; Eiben
and Kotrb
ˇ
cík, 2015; Dibek et al., 2016; Akbari et al.,
2018)).
Several graph classes are defined in a similar
way as equimatchable graphs with respect to other
NP-hard problems. Some well-known examples of
such classes are well-covered graphs where every
(inclusion-wise) maximal independent set has the
same size, and well-dominated graphs where every
(inclusion-wise) minimal dominating set has the same
size (http://www.graphclasses.org/, 2019).
Now, let us turn our attention back to the stable
matching problem. Remind that Theorem 2.5 guar-
antees the existence of a stable matching whenever
the input graph is bipartite, unlike the general case
(called the stable roommate problem) where a stable
matching might not exist. So, a similar question in the
framework of stable matchings can be formulated as
follows: what are graphs for which every preference
list admits a stable matching? It turns out that these
graphs are not more general than bipartite graphs as
stated in the following.
Theorem 4.1. (Abeledo and Isaak, 1991) A graph G
admits a stable matching for every possible prefer-
ence lists (of a vertex over its neighbors) if and only
if G is bipartite.
Theorem 4.1 implies in particular that whenever
the input graph is not bipartite, there exists a prefer-
ence list for which no stable matching can be found.
5 CONCLUSIONS
In their seminal paper where they introduced the no-
tion of stable matchings in the literature, D. Gale and
L. S. Shapley expressed their hope for their new the-
ory to find real applications (other than the hypotheti-
cal marriages between men and women) in the future.
We can see that their wish became true quite rapidly.
As illustrated with examples, different challenges
are faced when searching for a stable matching in
various applications. Each one of these challenges
motivates the development of new methods. For in-
stance, the practical reasons which implied that only
two-paired exchanges can be allowed in the kidney
exchange problem motivated new studies in this area.
Likewise, theoretical advances allow us to solve more
and more complicated problems in practice. The
Economic Sciences Prize Committee of the Royal
Swedish Academy of Science expresses that A. E.
Roth has received the Nobel Prize in Economic Sci-
ences 2012 for his valuable contributions in both di-
rections of this process. Even if sometimes the theory
and the applications seem to progress independently
from each other, the story of the Nobel Prize in Eco-
nomic Sciences 2012 shows that sooner or later con-
tributions in both directions complete each other.
ACKNOWLEDGEMENTS
I am grateful to Tayfun Sönmez and Utku Ünver for
being extremely available in responding my questions
about the kidney exchange problem. I would also like
to thank Ahmet Atıl A¸sıcı for structuring the text and
proofreading.
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