Two-Sided Matching with (almost) One-Sided
Preferences
Guillaume Haeringer, Vincent Iehlé
To cite this version:
Guillaume Haeringer, Vincent Iehlé. Two-Sided Matching with (almost) One-Sided Preferences.
2017.
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Two-Sided Matching with (almost) One-Sided
Preferences⇤
Guillaume Haeringer†
Vincent Iehlé Iehlé‡
April 25, 2017
Abstract: In a two-sided matching context we show how we can predict stable matchings
by considering only one side’s preferences and the mutually acceptable pairs of agents.
Our methodology consists of identifying impossible matches, i.e., pairs of agents that can
never be matched together. We use our results to analyze data from the French academic
job market and show that the strict preferences of candidates over positions, unobserved
and submitted at the final stage of the market, matter very little in the final matching.
JEL classification: C78, J41, C81
Keywords: Stable matchings, Hall’s marriage theorem, French academic job market
⇤
We are grateful to Atila Abdulkadiroğlu, Itai Ashlagi, Francis Bloch, Françoise Forges, Nicole Im-
morlica, Jacob Leshno, Vikram Manjunath, Thayer Morrill, Juan Pereyra, Baharak Rastegari, Al Roth,
Daniela Saban, Jay Sethuraman, Olivier Tercieux, Peter Troyan, and seminar audiences at Boston College, UC Davis, UCSD, Microsoft Research Cambridge, Rochester, ITAM, UCSB, Berkeley, Duke, Oslo,
USC, Paris School of Economics, Vanderbilt, Bielefeld, Salerno, Barcelona, Dauphine, Caen, York and
the NYC market design group for helpful comments. Special thanks to Mar Reguant, Tayfun Sönmez,
Utku Ünver and Alexander Westkamp for their numerous comments and suggestions. The first author
acknowledges the support of the Barcelona GSE Research Network and of the Government of Catalonia,
Spanish Ministry of Science and Innovation through the grant “Consolidated Group-C” ECO2008-04756
and the Generalitat de Catalunya (SGR2009-419). Part of this research has been made when the first author was visiting the Stanford Economics department, whose hospitality is gratefully acknowledged. The
second author gratefully acknowledges the support of ANR Dynamite-13BSH1-0010 and Risk Foundation
(Chaire Dauphine-ENSAE-Groupama “Les particuliers face au risque”).
†
Haeringer: Baruch College, [email protected].
‡
Université de Rouen Normandie, CREAM, [email protected].
1.
Introduction
This paper is aimed at understanding further the implications of stability in two-sided
matching markets. There is now a large consensus in the matching literature that stability,
which can be understood as the equilibrium notion for matching markets, is one of the
most desirable properties when designing matching institutions.1 Stability, in one-to-one
matching problems, is a conjunction of two requirements: each agent is matched to an
acceptable partner (individual rationality), and no pair of agents who would prefer to be
matched to one another rather than to their current mate (absence of blocking pairs).2
Evidence collected by Roth (1990), and later by Kagel and Roth (2000) indeed suggests
that the lack of stability can be a cause of market failure. As the second requirement
shows, stability hinges on the preferences of both sides of the market. We challenge this
interpretation by showing how to identify stable matchings by considering the preferences
of only one side of the market.
Our goal is to investigate, when considering the preferences of one side, whether there
exists some preference profile of the other side so that a particular pair of agents can
be matched at some stable matching (for that profile). If there is no such profile, the
pair is called an impossible match. Identifying such pairs is valuable because, as we
shall show, the more there are the more we are able to narrow down the set of stable
matchings that may emerge. This insight provides a new tool to analyze the performance
of a stable matching mechanism. In particular, it allows us to measure the extent to which
the preferences of one side of the market matter in the determination of stable matchings,
or equivalently the extent to which the market can be cleared whereas some of the strict
preferences are not yet declared.
To streamline the analysis we consider job market problems, that are otherwise standard many-to-one matching problems between candidates and academic departments. In
our model departments have one or several job openings (positions) to fill, and are assumed to have responsive preferences over candidates.3 For most of the paper we depart
from this model by considering broader environments that we call pre-matching prob1
See Shapley and Scarf (1974) and Quinzii (1984) for early approaches to the link between market
competitive equilibrium and stable matchings, or Azevedo and Leshno (2012) for a more recent account
on the matter.
2
For many-to-one matching problems the definition of a stable matching changes slightly.
3
See Roth and Sotomayor (1989).
2
lems. Such problems consist of a list of mutually acceptable pairs candidate-position,
and departments’ preferences over candidates. Candidates’ strict preferences over departments, however, are left unspecified. There are mainly two ways to obtain a pre-matching
problem. A first case is when we have access to preferences from both sides, but we simply
“ignore” how candidates rank their acceptable positions (see Section 2.3 for the relation
between a job market problem and a pre-matching problem). A second case is when we do
have access to departments’s preferences but not to candidates’ preferences and we know
or deduce the mutually acceptable pairs, by considering for instance the job interviews
conducted at a preliminary stage of the market (see Section 5).
The existence of impossible matches is strongly related to the fact that at least one
side of the market submits short preference lists. If there are enough positions to
accommodate all candidates and if all candidates and departments’ preference lists are
complete then no candidate is an impossible match for any position.4 So, although we
do not state it explicitly, our results are of particular relevance when the set of mutually
acceptable pairs candidate-position is small compared to the size of the market, which is
a salient feature of most (if not all) real-life matching markets.5 It is important, however,
not to over-interpret our results. In our model we use the term “non-acceptable” to simply
describe the fact that a candidate (department) does not appear in the preference list of
a department (candidate). Whether a candidate is indeed unacceptable for a department
or a department strategically opted for not consider that candidate makes no di↵erence
in our analysis. Put di↵erently, we analyze the consequences of short preference lists, not
4
Take any candidate i and any position d, and construct a matching where i is assigned to d and all
other candidates are assigned to some position. The resulting matching will be stable for the candidates’
preference profile such that each candidate is matched to his most preferred position.
5
There are various explanations for this stylized fact. For instance, some matching markets involve
a large number of participants, making it virtually impossible to expect complete preference lists by
all participants, e.g. if employers must interview before matching as in Rastageri et al. (2012). Also,
some matching institutions may constrain agents with respect to the length of their preference lists
(e.g., position choice in New York City, university admission in Spain). See Kojima et al. (2013) for a
discussion for the medical match. Abdulkadiroğlu, Pathak and Roth (2005), Haeringer and Klijn (2009),
Calsamiglia, Haeringer and Klijn (2010), Pathak and Sönmez (2013) or Bodoh-Creed (2013) constitute
additional accounts on the length of submitted preference lists in centralized matching markets. It is
also the case on the French academic market where the departments could rank at most 5 candidates per
position until 2008 (see Section B of the Web Appendix or Haeringer and Iehlé (2010)).
3
what causes preference lists to be short.
Our first contribution consists of characterizing impossible matches in pre-matching
problems. In a pre-matching problem, the only way to know with certainty that a matching cannot be stable for any realization of candidates’ preferences is when a candidate is
not matched and can form a mutually acceptable pair with a position that is either unfilled or matched to a candidate ranked lower in the preferences of the department holding
that position. It is this second requirement that is crucial to characterizing impossible
matches. If a candidate i is matched to a position d at some stable matching then all the
candidates that are ranked higher than i for position d must be matched to some other
position, for otherwise the matching would not be stable (i and d could block it). Those
candidates must pass the same test, i.e., all the candidates ranked higher than them (at
the positions where they are matched) must be matched to a position. Matching those
additional candidates may in turn bring additional candidates to our problem, and so
on and so forth. Candidate i will be an impossible match for position d when, for any
possible way we match the other candidates, there is always at least one candidate in
excess: a candidate that we must match, but all his acceptable positions are matched to
other candidates. In other words, the impossibility to match a candidate to a position
will arise when there is a set of candidates ranked “sufficiently high” that outnumbers the
total number of positions acceptable to them.
This condition is reminiscent of one of the very first results of matching theory, namely
Hall’s (1935) marriage theorem. Hall’s condition is not sufficient for our purposes,
however, as it provides no link to stability. Part of our contribution consists precisely of
making precise what being ranked “sufficiently high” means, by carefully selecting the set
of individuals for which we have to check Hall’s condition (Theorem 5). As a by-product,
our result establishes a formal connection between stability and Hall’s marriage condition.
In very rough terms, our characterization is as follows. A candidate-position pair (i, d)
is an impossible match if, and only if, there is a set of candidates that can exhaust all
the positions acceptable to them, including position d (i.e., a set of candidates for which
Hall’s condition holds), but if we match i to d then the set of candidates cannot exhaust
the remaining positions while also respecting their rankings (i.e., Hall’s condition does
not hold anymore).6
6
Our condition relies on the relative rankings of the candidates in the set, not their absolute rankings.
4
Our contribution can also be related to the matching literature analyzing the e↵ects of
competition on the size of the core (i.e., the set of stable matchings). Our results parallel
a recent result by Ashlagi, Kanoria and Leshno (2017). Unlike us, they assume that every
pair candidate-position is acceptable (i.e., no short preference lists) and specify each side’s
preferences (random, iid).7 They show that the slightest imbalance between the number
of agents on the two sides of the market is sufficient to get a core that is essentially unique.
Our identification of impossible matches goes in the same direction. The main di↵erence
relies on the notion of market imbalance, which is more demanding on the primitives of
the market in our context of short preference lists. In our paper the market imbalance is
captured by the above-mentioned Hall’s marriage conditions, which require not only the
imbalance between the sizes of each side of the market but also specific overlaps of the
mutually acceptable pairs. Our main result (Theorem 5) shows that this strong version
of market imbalance is necessary to obtain impossible matches in our environment.
In the second part of the paper we use the concept of impossible match to show how
the market can be cleared, even though the preferences of one side of the market are not
observed or not yet declared. The basic intuition is the following. Given a pre-matching
problem, we first identify the candidate-position pairs that are impossible matches. Once
all such pairs are identified we simply delete all the candidates that are impossible matches
from the preferences of the positions for which they are impossible. Suppose that after
this step there is a candidate who appears in the preferences of only one department and
furthermore he is the top candidate for that position. We may predict that this candidate
is necessarily matched to that position. Similarly, if a candidate has been eliminated
from the rankings of all positions that used to rank him we may also predict that that
candidate is necessarily unmatched. We show that such predictions are necessarily correct
(Proposition 7), that is, no matter what candidates’ submitted preferences are, they are
indeed matched as predicted (and their match is the same for any submitted preferences).
These predictions then allow us to pinpoint candidates whose submitted preferences have
an impact on the final matching.
To demonstrate that some candidates’ preferences may not have any impact on the
final matching, we use data from the French academic job market, which is, unlike most
7
In contrast, our approach to candidates’ potential preferences over positions is distribution-free, i.e.,
we do not make any assumption on the distribution over candidates’ preference profiles.
5
academic job markets, centralized. In this market, departments having open positions
submit, to a clearinghouse, a ranking of candidates (those who applied to the positions)
after interviewing them. A few weeks after departments have ranked candidates, candidates also submit a preference list over positions and an algorithm, which turns out to be
equivalent to Gale and Shapley’s (1962) Deferred Acceptance algorithm, then determines
the matches.8 Our data covers the years 1999–2013, for junior positions in mathematics
and fully describes the preliminary stage of the market, which is a perfect example of
a pre-matching environment: we do observe the rankings of candidates established by
the departments, from which we deduce the set of mutually acceptable pairs, but not
the preferences submitted to the clearinghouse by the candidates later on.9 On average,
there are about 100 open positions every year, with about 220–230 candidates competing for those positions. The identification of impossible matches permits us to correctly
predict, on average, nearly 45% of the positions. Besides, the match of about 60% of the
candidates does not depend on the realization of candidates’ preferences. Our findings
thus show that candidates’ submitted preferences have, in this market, little impact on
the final matching, or equivalently that most of the market clears early at the preliminary stage. A comparison with random markets show that the high level of prediction is
mostly due to the overlap between departments’ rankings, and not the way departments
rank candidates.
The paper is structured as follows. In Section 2 we outline the basic model we work
with. Section 3 contains the first main contribution of this paper, i.e., the characterization of impossible matches. In Section 4 we show how the identification of impossible
matches can be used to make predictions about matchings, and Section 5 illustrates our
findings with an empirical analysis of the French academic job market. The main proofs
are relegated to the Appendix. The Web Appendix provides additional results on the
characterization of impossible matches (Section A) and details on the French market (B)
and on the data (C). The main steps of the program for finding impossible matches are
in the Supplementary Web Appendix.
8
9
See Haeringer and Iehlé (2010).
We explain in details in Section 5 how we can deduce the set of mutually acceptable pairs candidate-
position.
6
2.
Preliminaries
1.
Matchings
We consider a (finite) set I of candidates and a (finite) set D of departments, where
each department d is endowed with a positive capacity qd . The problem studied here is
that of matching candidates to departments subject to their capacities. A matching is
a mapping µ : I [ D ! 2I [ D such that, for each i 2 I and each d 2 D,
• µ(i) 2 D [ {i},
• µ(d) 2 2I ,
• µ(i) = d if, and only if, i 2 µ(d), and
• |µ(d)|  qd .
For v 2 D [ I, we call µ(v) agent v’s assignment. For i 2 I, if µ(i) = d 2 D then
candidate i is matched to department d under µ. If µ(i) = i then candidate i is said to
be unmatched under µ.
2.
Job Market Problem
A job market problem is a 5-tuple (I, D, (
candidates, D a set of departments,
d
d , qd )d2D , (
i )i2I ),
where I is a set of
a preference ordering for department d, qd a
capacity constraint for department d, and
i
a preference ordering for candidate i.
Each department d 2 D has a strict preference over the candidates, i.e.,
d
is a
linear ordering over I. We say that candidate i is ranked higher than candidate j at
department d if i
d
j.
Each candidate i 2 I has a strict preference relation
the option of remaining unassigned, i.e.,
d
i
i
i
over the departments and
is a linear ordering over D [ {i} The notation
d0 means that candidate i prefers to go to department d than department d0 . A
department d is acceptable for a candidate i under the preferences
matched to d to being matched to himself, i.e., d
we shall simply use
v ⌫i v , v
i
0
if i prefers being
i. When there is no risk of confusion
to denote a job market problem.
Given a preference relation
0
i
i
0
i
we denote by ⌫i the weak relation associated to it, i.e.,
v or v = v . For a set I 0 ✓ I, we denote by
7
I0
the profile (
i )i2I 0 .
Given
a job market problem
we denote by Ai the set of departments that are acceptable to
i, and for a set I 0 of candidates AI 0 denotes the set of all departments that are acceptable
to candidates in I 0 , i.e., AI 0 = [i2I 0 Ai .If a department d 2
/ Ai then d is unacceptable
for i.
Candidates’ preferences over departments can be straightforwardly extended to preferences over matchings. We say that candidate i prefers the matching µ to the matching
µ0 if he prefers his assignment under µ to his assignment under µ0 . Formally µ
and only if, µ(i)
i
µ0 (i), and µ ⌫i µ0 if, and only if, ¬ (µ0
i
i
µ0 if,
µ).
A matching is stable if, for each candidate, all the departments he prefers to his
assignment have filled their capacities with candidates who are ranked higher, and he is
matched to an acceptable department. Formally, a matching µ is stable for a job market
problem
if
(a) it is individually rational, i.e., for all i 2 I, µ(i) ⌫i i,
(b) it is non wasteful (Balinski and Sönmez, 1999), i.e., for all i 2 I and all d 2 D,
d
i
µ(i) implies |µ(d)| = qd , and
(c) there is no justified envy, i.e., for all i, j 2 I with µ(j) = d 2 D, d
j
d
i
µ(i) implies
i.
Given a job market problem
, the core is the set of stable matchings and is denoted by
⌃( ). For a matching µ, the pair (i, d) is a blocking pair (or i and d block the matching
µ) if either i prefers d to his assignment and |µ(d)| < qd , or candidate i has justified-envy
towards a candidate j matched to a department d (i.e., µ(j) = d, i
3.
d
j, and d
i
µ(i)).
Pre-matching problems
A pre-matching problem is similar to a job market problem except that preferences are summarized only by a list of mutually acceptable pairs from both sides of the
market and the departments’ preferences over their acceptable candidates. Formally,
a pre-matching problem is given by a 5-tuple (I, D, (Pd , qd )d2D , A), where I is a set of
candidates, D a set of departments, Pd a preference of department d, qd a capacity of
department d and A ✓ I ⇥ D a set of mutually acceptable pairs (i, d) of candidates and
departments. In a pre-matching problem, the preference of department d, Pd , is a linear
ordering over Ad , which is the subset of candidates that form mutually acceptable pairs
8
with d, i.e. Ad := {i 2 I : (i, d) 2 A}.10 With a little abuse of terminology, we also say
that Ad is the set of acceptable candidates for d. Similarly, Ai := {d 2 D : (i, d) 2 A} is
the set of acceptable departments for i. For a set I 0 of candidates, AI 0 denotes the set of
all acceptable departments with some candidates in I 0 . For a set D0 of departments, AD0
denotes the set of all acceptable candidates with some departments in D0 .
Whenever there is no risk of confusion we shall use the shorthand P to denote a
pre-matching problem.
The concept of stability, as defined for job market problems, is not applicable prematching problems, because candidates’ preferences are not completely defined. Stability
has a bite, however, whenever a candidate is matched to an unacceptable department or
when a candidate is unmatched and an acceptable department is matched to a candidate
ranked lower. We summarize these two situations with the following concepts.
• A matching µ is feasible for P if for each i 2 I, µ(i) 6= i implies (i, µ(i)) 2 A.
• A matching µ is comprehensive for P if it is feasible and, for each department
d 2 D, if µ(i) = d for some i 2 I, jPd i implies µ(j) 2 D.
We sometimes consider comprehensiveness among a subset of candidates only. In this
case we say that a matching µ is comprehensive for P among candidates in I 0 ✓ I
if it is feasible and if for each department d 2 D, if µ(i) = d for some i 2 I 0 , jPd i and
j 2 I 0 implies µ(j) 2 D.
Remark 1 Comprehensiveness shares some similarities with the concept of stability in
a job market problem, and is similar to the absence of blocking pairs in a job market
under dichotomous preferences for the candidates. The feasibility constraint is akin to
individual rationality. The second condition for comprehensiveness is simply a condition
on the absence of blocking pairs, but for unmatched candidates only. We can thus intuit
that if a matching is not comprehensive for a pre-matching problem it cannot be stable
for any compatible job market problem. The reader will notice, however, that a concept
related to non-wastefulness is missing in the definition of comprehensiveness. This is
10
This contrasts with a job market problem where a department’s preference is an ordering over the
set of all candidates.
9
deliberate because such a requirement will not be needed for our purposes.11,12
Given a pre-matching problem P , a job market problem
with the same sets of
candidates and departments and identical departments’ capacities is P -compatible if
(a) for each candidate i, Ai = Ai
(b) for each department d, and each pair of candidates i, i0 2 Ad , i
d
i0 if, and only if,
iPd i0 .13
Note that the same pre-matching problem can generate di↵erent job market problems
(and thus generate di↵erent sets of stable matchings). We denote by ⇥(P ) the set of job
market problems that are P -compatible.
Finally, notice that when representing a pre-matching problem (I, D, (Pd , qd )d2D , A)
it is sufficient to describe only the profile of departments’ preferences over the acceptable
candidates (and their capacities), since by construction i 2 Ad if, and only if, d 2 Ai .
That is, all the relevant information can be summarized on one side of the market (here
the departments).
3.
Impossible matches
The purpose of this section is to define and characterize the candidates that are impossible matches for some specific department in a given pre-matching problem. The
question addressed by the concept of impossible match is the following: upon observing one side of the markets’ preferences (i.e. a pre-matching problem), is there a stable
matching with respect to some realization of the other side’s preferences such that a particular candidate is matched to a particular department? If the answer is “no” then that
candidate is said to be an impossible match for that department.
11
Our main objective is to characterize to which departments a candidate can be matched at a stable
matching. If there is a matching µ such that a candidate, say, i, is matched to a department, say d, and µ
satisfies individual rationality and there is no justified envy but µ is wasteful, then it is easy to construct
a stable matching µ0 such that µ0 (i) = d by filling departments’ capacities with unmatched candidates
(starting with those with the highest rank).
12
Comprehensiveness in a pre-matching problem P is in fact equivalent to individual rationality and
the absence of justified envy in the job market problem where each candidate is indi↵erent among all his
acceptable departments.
13
In a job market problem, the set of mutually acceptable pairs is determined by the candidates only.
Given P , a P -compatible job market problem leaves this set unchanged by construction.
10
Definition 1 Given a pre-matching problem P , a candidate i is an impossible match
for department d 2 Ai if, for each job market problem
µ 2 ⌃( ) such that µ(i) = d.
1.
2 ⇥(P ), there is no matching
Introductory elements
To understand how the preferences of only one side of the market can suffice to check
the stability of a matching consider the following instances of job market problems.
Example 1 The simplest case occurs if qd candidates ranked higher than i at department
d have no other acceptable department. Then for any job market problem
2 ⇥(P ) and
stable matching µ 2 ⌃( ) either all those candidates are matched to d, or at least one of
them is not matched to any department.14 In either case, i cannot be matched to d at a
stable matching, for otherwise one of the candidates ranked higher than i would form a
⇤
blocking pair with department d.
Example 2 A (slightly) more complex case is the following. For simplicity we assume
that each department has only one position to o↵er. Let i1 , i2 , i3 and i4 be candidates
whose sets of acceptable departments are {d1 , d2 }, {d2 , d3 }, {d3 , d4 }, and {d2 , d4 }, respectively. Table 1 describes the departments’ preferences over these four candidates. For
instance, at department d2 candidate i2 is ranked higher than candidate i1 , who is ranked
higher than candidate i4 .
Table 1: Candidate i1 is an impossible match for department d2 .
Pd 1
Pd 2
Pd 3
Pd 4
i1
i2
i3
i4
i1
i2
i3
i4
We claim that for any preference profile of the candidates that is P -compatible it is
impossible to obtain a stable matching where i1 is matched to d2 . To see this, note that if
14
There could be more than qd candidates ranked higher than i at d and a P -compatible job market
problem such that some of these candidates have department d being their most preferred department.
11
this is the case then i2 must be matched to d3 , for otherwise (i2 , d2 ) would form a blocking
pair. This in turn implies that i3 is matched to d4 , which is only possible if i4 is matched
to d2 , a contradiction. So, there is no preference profile and a stable matching for that
⇤
preference profile where i1 is matched to d2 .
Without entering into much detail, it is important to note that in the previous example the key property of departments d2 , d3 and d4 ’s preferences to ensure that i1 is an
impossible match for department d2 is not, as it may appear, that there is some sort of
cycle with the relative positions of candidates i2 , i3 and i4 . As we shall see, i1 being an
impossible match for d2 is rather due to the more general feature that there are (at least)
3 candidates competing for three departments and such that their relative rankings in the
preferences of the departments allow them to claim the positions over i1 .15
Finally, note that to establish that candidate i1 is an impossible match for department
d1 only candidates i2 , i3 and i4 were needed. That is, if there were other candidates who
also find one of those departments acceptable candidate i1 would remain an impossible
match for d2 . For instance, we could have a candidate, say, i5 ranked higher than i3 at
department d3 , or a candidate, say, i6 , ranked between i4 and i3 at department d4 , without
altering our conclusion about candidate i1 .
Given a pre-matching problem P , for any P -compatible matching problem
and any
matching µ 2 ⌃( ), µ is also comprehensive for the pre-matching problem P . However,
the reverse may not be true. Hence, given a pre-matching problem P , if a candidate can
be matched to a department at a stable matching for some P -compatible problem
then
he can be matched to that department at a matching comprehensive for P . Similarly, if
there is no P -compatible problem for which a candidate i is matched to a department d
at a stable matching (i.e., i is an impossible match for d), then there is no comprehensive
matching for P where i is matched to d.
Proposition 1 A candidate is an impossible match for a department in a pre-matching
problem P if, and only if, there is no comprehensive matching for P such that they are
matched together.
Proof
Let P be a pre-matching problem and suppose that candidate i is not an impos-
sible match for d. So, there exists
15
2 ⇥(P ) and µ 2 ⌃( ) such that µ(i) = d. Clearly, µ
See Example 1 for an example where no cycle is involved.
12
is a feasible matching for P , and since µ satisfies no justified envy, it is also comprehensive
for P , the desired result.
Conversely, let µ be a comprehensive matching in P such that µ(i) = d. Consider the
departments that do not exhaust their capacities under µ and fill these departments taken
in any particular order with the best ranked and unmatched remaining candidates (if any).
The resulting matching, that we call µ0 , still satisfies comprehensiveness in P . Let (
0
j )j2I
0
be the preference profile such that for each j 2 I with µ (j) 6= j department µ (j) is the
most preferred department under
j
and such that
2 ⇥(P ). By construction, µ0 satisfies
individual rationality and no wastefulness in the job market problem
2 ⇥(P ). Under µ0 ,
each j 2 I with µ0 (j) 6= j gets his top choice, hence envy originates only from candidates
j 2 I such that µ0 (j) = j. But since µ0 is comprehensive in P these candidates are ranked
below the least ranked candidates matched to departments according to departments’
preferences, hence no blocking pair can be formed. The matching µ0 is thus stable in the
job market problem
2 ⇥(P ) and µ0 (i) = d. Therefore i is not an impossible match for
⌅
d.
In the next two propositions we first establish a Rural-Hospital type of result when
some candidates are impossible matches for some departments.
Proposition 2 If a department is part of an impossible match for P then it fills capacity
at any stable matching, for any compatible job market problem.
Proof
Let P be a pre-matching problem and suppose that candidate i is an impos-
sible match for department d. Suppose that for some P -compatible department choice
problem
there exists a matching µ 2 ⌃( ) such that |µ(d)| < qd . Since µ satisfies non
wastefulness it holds that µ(j) 6= j for every j such that d 2 Aj .
Consider the job market problem
0
2 ⇥(P ) such that for each candidate j 6= i matched
to a department under µ, µ(j) is the most preferred department under
d is candidate i’s most preferred department under
0
i.
0
j
and department
0
Let µ be the matching such that
µ0 (j) = µ(j) for each j 6= i, and µ0 (i) = d. In addition, if there exists a candidate for
which µ(i) is acceptable and who is unmatched under µ, match under µ0 the highest
rank candidate among those unmatched candidates. Since |µ(d)| < qd the matching µ0
is feasible. It is easy to check that, since µ 2 ⌃( ), µ0 satisfies individual rationality,
13
no justified envy and non-wastefulness. Finally, observe that
0
2 ⇥(P ), so i is not an
⌅
impossible match for d, a contradiction.
We next show that the impossible match property is not monotonic with respect to
the preferences of the department.
Proposition 3 If a candidate is an impossible match for a department, candidates ranked
lower at that department are not necessarily impossible matches for that department.
Proof
Consider again the pre-matching problem P given in Table 1. It is easy to check
that candidate i1 is also an impossible match for d2 , yet candidate i4 is not an impossible
match for d2 . Indeed, let
2 ⇥(P ) be such that i1 ’s most preferred department is d1 , d2
is the most preferred department of i4 , d3 is the most preferred department of i2 , and d4
is the most preferred department of i3 . The matching µ such that µ(d1 ) = i1 , µ(d2 ) = i4 ,
µ(d3 ) = i2 , µ(d4 ) = i3 is stable for
2.
⌅
.
A characterization
Part of our contribution consists of showing that we do not need to consider all candidates and all possible profiles of candidates’ preferences (and compute all the corresponding stable matchings) to check whether a candidate is an impossible match for a
department. Instead we are able to characterize impossible matches on the basis of the
fundamentals of the pre-matching problem only. Proposition 1 provides a first baseline
for the analysis of impossible matches, at the cost of extreme computations: checking
whether a candidate i is an impossible match for a department d amounts to list every
comprehensive matching of the pre-matching problem P to see whether one of those assigns i to d. In what follows, we will narrow down the set of cases one has to consider,
showing that it is sufficient to find a particular subset of candidates, called a block. In
addition, an interesting feature of the result is that it relies on one of the earliest results
of matching theory, which we rephrase to fit our setting.
Theorem 4 (Hall’s Marriage Theorem, 1935) Given a pre-matching problem P and
ˆ there exists a feasible matching where all candidates are matched to
a set of candidates I,
some department if, and only if,
ˆ |I 0 | 
for each I 0 ✓ I,
14
X
d2AI 0
qd .
(1)
In words, there exists a matching where all candidates can be matched to some department if, and only if, for each subset I 0 of candidates the number of candidates is at
most equal to the total number of (admissible) positions among the departments that are
considered by some of the candidates in I 0 as acceptable. Eq. (1) is known as Hall’s
marriage condition.16
To incorporate the no-justified envy condition, or its counterpart in the pre-matching
framework (comprehensiveness), we shall not apply Hall’s marriage condition to the original pre-matching problem but rather to a collection of restricted and truncated prematching problems.
In order to characterize impossible matches in terms of Hall’s algebraic marriage condition it will be useful to consider particular transforms of department’s preferences profiles
of a pre-matching problem. Given a pre-matching problem P = (I, D, (Pd , qd )d2S , A) and
two sets of candidates J and K, with K possibly empty, the pre-matching problem restricted to J and truncated at K is denoted by P (J, K). In that pre-matching problem
the set of acceptable pairs is
AP (J,K) = {(i, d) 2 A : i 2 J and iPd i0 for each i0 2 K}
P (J,K)
and the preference of department d, Pd (J, K), is the ordering over Ad
with Pd on
(2)
that coincides
P (J,K)
Ad
.
Remark 2 When we consider the pre-matching problem restricted to J and truncated
at K the set of “available” candidates and departments may not be equal to J \ K and
P (J,K)
AJ\K , respectively. Indeed, it may well be that there is a candidate i 2 J\K such that
P (J,K)
there is no department d for which i 2 Ad
may have
P (J,K)
Ad
= ;. For a truncated pre-matching problem P (J, K) we say that a
P (J,K)
candidate i is admissible if i 2 [d2D Ad
P (J,K)
Ad
16
6= ;.
. Similarly, for some department d we
. Similarly, a department is admissible if
17
Hall’s marriage condition was originally stated for the one-to-one matching problem. It is easy to see
that this condition can be easily extended to the case of job market problems when departments have
responsive preferences by making for each department as many copies as its capacity (where each copy
has only one position to o↵er) and making each of its copy acceptable for a candidate if the department
is acceptable for that candidate.
17
Note that the pre-matching problem P (J, ;) is simply the problem P restricted to the set of candidates
J and P (I, K) is the problem P truncated at K.
15
Given a pre-matching problem P , we say that Hall’s marriage condition holds if
Eq. (1) holds when Iˆ is the set of all admissible candidates in P .
For a pre-matching problem P , d 2 D, i 2 Ad and J ✓ I \ {i}, we denote by P̄ d the
same problem as P except that one position of d is removed, i.e. department d’s capacity
is qd
1, and we denote by Ji,d ✓ J the set of candidates in J that are ranked higher than
i by d, i.e., Ji,d = {j 2 J : jPd i}.18
Definition 2 Given a pre-matching problem P = (I, D, (Pd , qd )d2D , A), a block at
(i, d) 2 A is a set J ✓ I\{i} such that the following conditions hold:
1. For some J0 ✓ J, |J0 | =
P
ˆ J
d2A
qdˆ and Hall’s marriage condition holds in P (J0 , ;);
2. For each K ✓ J \ Ji,d such that |K| = |J|
does not hold in P̄ d (J, K).
P
ˆ J
d2A
qdˆ + 1, Hall’s marriage condition
One can check the definition on Example 2. For that pre-matching problem, there is a
block at (i1 , d2 ) formed by the set of candidates {i2 , i3 , i4 }. We state now the main result
of the paper.
Theorem 5 A candidate i is an impossible match for a department d in a pre-matching
problem P if, and only if, P admits a block at (i, d).
The result provides additional insights on impossible matches. First, we prove in
Appendix (Proposition A.1) that the conditions 1. and 2. in the definition of a block
imply there are at least qd candidates in J ranked higher than i at department d, i.e.,
P
|Ji,d | qd . Next, Hall’s condition holds in P (J0 , ;) for some J0 such that of |J0 | = d2A
ˆ J qdˆ
is tantamount to say that there exists a perfect match between the departments AJ and
the candidates J0 , i.e., there exists a matching where all candidates J0 are matched to a
department and all departments exhaust their capacities with candidates in J0 . Second,
condition 2 implies in particular that the ability to obtain a perfect match is tight in the
following sense: If we consider the problem P̄ d (the problem identical to P except that
one position is removed from d), then there is no perfect match between the departments
18
Note that admissibility in a pre-matching problem refers to the notion of being acceptable for a
department and is independent of the departments’ capacities. For instance, if for some department d
P (J,K)
we have qd = 1, in P̄ d (J, K) all candidates in Ad
d
of department d is zero in P̄ (J, K).
16
consider d as acceptable, even though the capacity
AJ and any subset J0 ✓ J that includes all the candidates in Ji,d , such that the matching
is comprehensive among candidates J.19
We extend our analysis of blocks in the Section A of the Web Appendix. We consider
a particular case of blocks where, in the Definition 2, the set J contains exactly as many
members as the total numbers of seats in their acceptable departments. This simple
declination, called exact block, sheds more light on the structure of blocks. Of course,
the existence of an exact block at (i, d) is a sufficient condition for i to be an impossible
match at d (but not necessary). Section A of the Web Appendix contains also the full
description of the algorithm we use to construct the blocks at a pair (i, d). The final issue
we address in the Web Appendix is the computational complexity for finding blocks or
exact blocks.
The next examples show that the requirements of Definition 2 are all independent
from each other.
Example 3 Let I = {i0 , i1 , i2 , i3 } and d = {d0 , d1 , d2 , d3 , d4 }, and assume that for each
department d 2 D, qd = 1. The admissible departments for each candidate are given by
Ai0 = {d0 }, Ai1 = {d0 , d1 , d2 }, Ai2 = Ai3 = {d1 , d2 , d3 , d4 }. The departments’ preferences
are described in Table 2.
Table 2: Condition 1 not satisfied — too few candidates
Pd 0
Pd 1
Pd 2
Pd 3
Pd 4
i1
i2
i3
i2
i3
i0
i3
i2
i3
i2
i1
i1
The matching µ such that µ(i0 ) = d0 , µ(ih ) = dh for h = 1, 2, 3 is comprehensive,
so from Proposition 1 candidate i0 is not an impossible match for department d0 . In
19
As already pointed out in the Introduction, the notion of block echoes a strong version of market
imbalance in the spirit of Ashlagi, Kanoria and Leshno (2013). An implication of condition 1. is that
the candidates outnumbers the positions by at least one in the market consisting of the candidate i, the
candidates J and their acceptable positions AJ . The condition requires in addition that those candidates
can exhaust all the positions in the sub-market, a condition which is trivially satisfied in the case where
all pairs are mutually acceptable as in Ashlagi, Kanoria and Leshno (2013).
17
this example condition 1 of the definition of a block is not satisfied, because for any
P
set J ✓ I\{i0 } it holds that |J| < d2AJ qd . In this case, condition 2 is satisfied but
has clearly no bite, as there cannot be a set K ✓ J\Ji0 ,d0 that satisfies the cardinality
⇤
requirement of condition 2.
Example 4 Let I = {i0 , i1 , . . . , i8 }, D = {d0 , d1 , . . . , d4 }, and assume that qd = 1 for
each d 6= d4 , and qd4 = 3. The departments’ preferences are described in Table 3.
Table 3: Condition 1 not satisfied — no perfect match
Pd 0 Pd 1 Pd 2 Pd 3 Pd 4
i1
i2
i3
i3
i0
i1
i2
i4
i6
i7
i8
i5
Candidate i0 is not an impossible match for department d0 : the matching µ such that
µ(i0 ) = d0 , µ(ih ) = dh for h = 1, 2, 3, is comprehensive. Consider the set J = {i1 , . . . , i8 }.
P
So, |J| = 8 > d2AJ qd = 7. Unlike Example 3, condition 2 of the definition of a block
does bite. For this set J, however, condition 1 is not satisfied because there cannot be
a perfect match between any set J0 ✓ J and the departments in AJ (because for any
matching department d4 does not fill its capacity).
We claim that condition 2 of the definition is satisfied for this set J. To begin with,
P
note that we need to consider all sets K ✓ J\{i1 } such that |K| = |J|
s qd + 1 = 2.
If i2 2 K, then for any other candidate we may add, Hall’s marriage condition does not
hold in P̄ d0 (J, K) because we cannot match i1 . If i3 2 K, then again Hall’s marriage
condition does not hold in P̄ d0 (J, K), for any other candidate in K: either i1 or i2 must
remain unmatched (if K = {i2 , i3 } then we cannot match i1 ).
It remains then to consider all sets K ⇢ {i4 , i5 , . . . , i8 } such that |K| = 2. Routine
examination shows that for any such set K, we cannot match all admissible candidates
⇤
in P̄ d0 (J, K), which proves the claim.
Example 5 By restricting the previous example to candidates I = {i0 , i1 , i2 , i3 , i4 } and
to their acceptable departments (see Table 4), candidate i0 is still not impossible for d0 .
It is an instance where the only possible set J satisfying condition 1 is {i1 , i2 , i3 , i4 }. We
18
consider K = {i4 } to see that condition 2 does not hold. All admissible candidates in
P̄ d0 (J, {i4 }) can be matched to departments at the matching µ such that µ(i1 ) = d1 ,
µ(i2 ) = d2 , µ(i3 ) = d3 . That is, Hall marriage condition is satisfied in P̄ d0 (J, {i4 }).
⇤
Table 4: Condition 2 not satisfied
P d 0 P d 1 Pd 2 Pd 3
4.
i1
i2
i3
i3
i0
i1
i2
i4
Matching prediction
We now investigate what the identification of impossible matches entails. Our claim
is that such an identification can be used to predict matchings, thereby measuring the
extent to which one side’s preferences matter in the determination of a final matching.
Identifying impossible matches will permit to single out two sets of candidates. On the one
hand, we shall have the set of candidates for whom the strict preferences over departments
is irrelevant for their final match. For them, no strict preferences, even partial, can be
deduced from the data. On the other hand, we shall have candidates for whom the strict
ordering in their (submitted) preference list matters for the final match, but only their
(relative) top choice can be sometimes recovered.
Our strategy is the following. Given an observed pre-matching problem, deleting
candidates identified as impossible matches (from the departments’ rankings for which
they are impossible matches) creates a reduced pre-matching problem. In this new problem
we show that a candidate who is ranked top by some department, and ranked only once,
is necessarily matched to that department, irrespectively of the way candidates rank
departments in their preferences.
If the observed data also contains the final (realized) matching we can go one step
further by considering market stars, that is, candidates that are always top ranked by their
acceptable departments in the reduced problem. For those candidates, there is no way
to infer their final match on the basis of the sole pre-matching problem. Note, however,
that their match trivially indicates their top choice (within the remaining choices). Once
market stars’ choices are taken into account new impossible matches may be identified
and so on and so forth. This is the baseline of our general procedure.
19
Let P be a pre-matching problem. The reduced pre-matching of P , denoted by
R(P ), is the pre-matching problem where impossible matches have been deleted from the
rankings of the positions for which they are impossible matches, and where the lists of
acceptable candidates or departments are made mutually consistent. Formally, the set of
mutually admissible pairs is
AR(P ) = A\{(i, d) : i is an impossible match for d in P }
R(P )
and for each pair i, j 2 Ad
, iR(P )d j if and only if iPd j. We shall denote by R(P )d the
ranking list of d in the pre-matching problem R(P ).
The next result motivates the construction of the predicting algorithm that we present
hereafter.
Proposition 6 Let P be a pre-matching problem and let
be any P -compatible job
market problem.
1. Candidate i is matched to department d under any matching that is stable for
and only if,
R(P )
Ai
= {d} and the rank of i in Pd is weakly lower than qd .
2. Candidate i unmatched under any matching that is stable for
R(P )
Ai
if,
if, and only if,
= ;.
3. For any candidate who falls into one of the two above cases, the final match is the
same for any stable matching of any P -comptatible job market problem.
Even though Proposition 6 seems straightforward, deleting matches is not innocuous
for the property of impossible matches. Deleting iteratively impossible pairs one after the
other can alter the matching predictions we can make, by declaring some pairs (candidate,
department) not impossible while they are actually impossible matches. To see this,
consider the pre-matching problem depicted in Table 5. It is easy to see that i3 is an
impossible match for d1 and d2 , and i2 is an impossible match for d1 (and i1 for d2 ). So
R(P )d1 = i1 and R(P )d2 = i2 . The unique comprehensive matching for P that fills the
available positions is µ(i1 ) = d1 and µ(i2 ) = d2 (indicated by the square boxes in Table 5),
thus it is the stable matching of any compatible job market problem. Note, however, that
if we delete i3 at d1 and d2 then i2 is no longer an impossible match for d1 (and i1 for d2 ).
20
Table 5: A predicted matching
P d 1 Pd 2
i1
i2
i3
i3
i2
i1
The reduction procedure can be generalized if the economist has access to the final
match of the candidates. The initial data is a pre-matching problem P and a matching µ
for P . If the matching is not observed, then the matching µ used in the algorithm is the
empty matching, i.e, µ(i) = i for each candidate i.
Predicting Algorithm
Input: A pair (P, µ), where µ is a matching for P .
Step 1 Identify all impossible match pairs in P and construct P 1 := R(P ).
Step h for h
2
h.1 Identify the set S h ✓ {i 2 I | µ(i) 6= i} of all candidates i such that (a) i
has at least 2 acceptable positions in P h ; and (b) i is ranked among the top qd
h
candidates in every department d 2 APi .
h.2 If S h is empty STOP. Otherwise, set Peh := P h and do:
h
for each i 2 S h and d such that µ(i) 6= d and d 2 APi , eliminate i from Pedh
eh
and d from APi .
h.3 Construct P h+1 := R(Peh ) and go to Step h + 1.
Note that if we do not observe the final matching the algorithm stops at step 2.2. The
algorithm implicitly partitions candidates in three sets:
• Candidates for which we will predict their match for the pre-matching problem:
Any candidates who does not belong to the sets S 2 , S 3 , . . . , but there is a problem
P h in which the candidate is ranked by only one department, say, d, and is among
the top qd top candidates in that department’s ranking.
21
• Candidates for which we use their observed match, i.e., they belong to the sets
S 2, S 3, . . . .
• All the other candidates.
For each step h
2, the candidates in S h are the “market stars” of the problem P h ,
and we say that the match of market stars is extracted in the algorithm. By construction,
the algorithm makes sure that the match of such candidadte is only used once, i.e., for
0
any h, h0
2, S h \ S h = ;. Adding the fact that there is a finite number of candidates
and departments, we deduce that the algorithm eventually stops after a finite number of
steps.
Definition 3 In the Predicting Algorithm, the match of a candidate i is predicted at
0
step h if there is no h0 6= h such that i 2 S h and h is the smallest index such that either
h
(a) APi = {d} for some d and the rank of i in Pdh is weakly lower than qd ; or
h
(b) APi = ;.
In case (a) the prediction is d and it is consistent with µ if µ(i) = d. Similarly, in case
(b) the prediction is i and it is consistent with µ if µ(i) = i.
The Predicting Algorithm can use as an input any matching µ. This implies that the
predictions used made through the algorithm are not necessarily consistent. However, if
the matching µ is stable for a job market problem
that is P -compatible, predictions
turn out to be consistent.20
Proposition 7 Let (P, µ) be a pair such that µ is a stable matching of an unobserved
job market problem
consistent with µ.
2 ⇥(P ). All the predictions of the algorithm with input (P, µ) are
Since at Step 1 the Predicting Algorithm does not use market stars we obtain the
following corollary: the final matching of candidates whose match is predicted in Step 1
does not depend on candidates’ preference profile.
20
The market stars are also extracted consistently in the algorithm. That is, if i is a market star at
step h then µ(i) is still an acceptable department for the candidate i in the problem P h , and remains thus
the unique acceptable department in the next steps. It can be deduced from Lemma 16 in the Appendix,
which shows that µ(i) is never deleted from the list of acceptable departments of i.
22
Corollary 8 Let (P, µ) be a pair such that µ is a stable matching of an unobserved job
market problem
2 ⇥(P ). If the match of a candidate is predicted in Step 1 of the Pre-
dicting Algorithm, then the match of that candidate is the same, for any stable matching
of any job market problem in ⇥(P ).
The predictions obtained from the Predicting Algorithm obviously depend on the
match of the market stars, i.e., the candidates from the sets S 1 , S 2 , . . . . We can show
that the predictions of our algorithm remains the same for any preference profile of the
candidates that does not alter the top choice of market stars in a certain way. To see this,
suppose that µ (the input of the algorithm) is a matching stable for some problem
.
h
Observe that if a candidate i is a market star at some step h (i 2 S ) and the matching µ
is a stable matching for a job market problem
2 ⇥(P ), then µ(i) is necessarily candidate
i’s top choice among the departments in APi h . Consider the job market problem
,
i ),
with
0
i
0
=(
0
i
is such that µ(i) is also the top choice among the departments in APi h , and
let µ0 be a stable matching for
0
. As the next proposition shows, running the Predicting
Algorithm with input (P, µ) or (P, µ0 ) yield the same predictions, and the market stars
identified throughout steps 2, 3, . . . are identical.
To prove this, we need to define a subset of the compatible job market problems.
0
Given (P, µ), let ⇥(P, µ) be the set of job market problems
market star j at some step h,
µ(j) ⌫0j d for all d 2 APj
2 ⇥(P ) such that, for every
h
Proposition 9 Let (P, µ) be a pair such that µ is a stable matching of an unobserved job
market problem
for every problem
2 ⇥(P ) and consider the Predicting Algorithm with input (P, µ). Then
0
2 ⇥(P, µ) and matching µ0 stable for
0
,
1. The set of matching stars is the same with input (P, µ) and (P, µ0 ).
2. µ0 (i) = µ(i) for every candidate i whose match is predicted and for any market star
i (i 2 S h for some h
In addition,
2).
2 ⇥(P, µ), that is, every market star j at step h obtains his best department
among those remaining at step h.
The main feature to be deduced from Proposition 9 is stated explicitly in the next
corollary.
23
Corollary 10 If the match of a candidate is predicted in the Predicting Algorithm then
the order in which i ranks the acceptable departments is irrelevant for the final match of
the candidate.
Recall that the Predicting Algorithm does not fully clear the market. Some departments and candidates are not considered at any step of the algorithm. Typically, it means
that none of the remaining candidates is top ranked in the remaining acceptable departments of the final pre-matching problem. Those assignments are the most difficult to
sort out in the market.21 From Proposition 9, we only know that the preferences of the
remaining candidates a↵ect neither the matches of the stars and nor the predictions.
5.
1.
Empirical analysis
The French academic job market
We consider in this paper junior positions, called Maı̂tre de Conférences. They correspond to a lectureship in the U.K., or to an assistant professorship in the US. Junior
positions in France are civil servant, tenured positions.22 We first outline the general
timeline of the French job market, and then comment on several aspects. There is much
to say about this market, but to avoid unnecessary digressions we only present the aspects
that are relevant for this paper (more details can be found in the Web Appendix).
Unlike, for instance, the US academic job market, most of the hiring procedure in
French universities is centralized.23 During the hiring season all departments first submit
to the Ministry of Higher Education an ordered list of candidates (after interviewing
them, usually in May), and then candidates are invited to submit a preference list over
the positions that ranked them (usually in June). The final step is made by the ministry,
where an algorithm is run to compute a match. Haeringer and Iehlé (2010) showed that
the algorithm used in France is equivalent to the Deferred Acceptance algorithm with
candidates proposing. That is, the matching computed is stable, and thus comprehensive
in the context of pre-matching problems.
21
One can cope with those remaining candidates through brute match extractions of non markets stars
and then run again the Predicting Algorithm. But the message on their roles is less clear since their
matches are fully conditioned by the overall profile of preferences.
22
Tenure is made official after one year of service, but it is extremely rare that tenure is not confirmed.
23
Since 2008 there is a “decentralized” market running parallel to the regular, centralized market, but
with much less job openings. See the Web Appendix for more details.
24
2.
Market for mathematicians
In 1998 a small group of young mathematicians set up a web site, Opération Postes,
inviting recruiting committees to announce the lists of candidates to be interviewed as well
as the rankings of candidates that will be submitted to the clearinghouse (the ministry), as
soon as these would be decided.24 The community of mathematicians was very responsive
and the web site quickly became a central tool in the job market.25 The data for each
position (interviewees list and rankings) is usually uploaded by the the chairs of the
recruiting committees themselves (and if not, by a member of the committee). On average,
about 90–95% of the job openings’ interview lists and rankings are available.26 The data of
Opération Postes is public, although not in a format that makes it immediately usable for
any analysis. There are many misspellings, and we sometimes found confusions between
the married and maiden names of some female candidates. But cross-referencing the data
with other sources permitted to have a clean dataset.27
We also collected for each year the assignment that was established by the Ministry
of Higher Education using candidate’s submitted preference lists over the positions and
the rankings of candidates established by the recruiting committees.28
24
25
http://postes.smai.emath.fr/.
The web site is now supported by the main professional mathematical societies in France: the French
Mathematical Society (SMF), the Society for Applied and Industrial Mathematics (SMAI, which hosts
the web site), and the French Society of Statistics (SFdS).
26
The missing data usually concerns overseas departments (e.g, La Réunion, Martinique or New Caledonia). The web site also invites the recruiting committees to submit the date when the interview list
will be decided, the date of the interviews, and the date when the ranking will be decided. We discarded
the positions that are reserved for administrative transfer (postes à la mutation), i.e., positions that are
only open to candidates that are already Maı̂tre de Conférences. Such positions represent only a small
fraction of the total number of openings, most positions being reserved for rookies.
27
To be able to apply to a job opening candidates must first pass an evaluation by a national committee.
The lists of candidates that have passed that evaluation (without spelling errors) is published by the Ministry of Higher Education and is freely available at https://www.legifrance.gouv.fr/initRechJO.do.
28
Unfortunately, the assignment computed by the Ministry is not published. However, all of the assignments could have been easily collected using manually using web searches, looking at the departments’
lists of faculty, submitted rankings (hired candidates have to be ranked by the departments), and candidates résumés (some departments also have an archive of the candidates they hired the previous years).
25
3.
Data analysis
Our data does not contain the preference lists submitted by the candidates in June.
Such lists are considered confidential by the Ministry of Higher Education; access to this
data has been denied. It is nevertheless reasonable to assume that whenever a candidate
appears on the ranking for a position that position is also acceptable for the candidate,
i.e., he would prefer to be matched to that position than not being matched to any
position. There are several observations that uphold this acceptability assumption for
that market.
First, a candidate can only be ranked for a position if he has been interviewed, on
campus, for that position. Since transportation and lodging costs accrue to the candidates
it is unlikely that a candidate would bear the cost of applying for a position he would
eventually refuse.29 The timeline of the job market is another element to sustain our
assumption. Interviews take place in May for a position that starts in September. That
is, candidates still looking for a job have usually few outside options (if any). Finally, the
vast majority of candidates have French degrees (and are French), and mathematics departments in France rank well according to international standards. Adding the fact that
the departments o↵er tenured positions, such jobs are very attractive for mathematicians.
Using the observed matching we can nevertheless easily check whether our acceptability assumption is satisfied. Since we know that the final matching must be comprehensive
(stable), any candidate i ranked for a position above the candidate that has been hired
must be hired by another department. If this is not the case, then we must conclude that
candidate i declared that position unacceptable in June. So from the observation of the
matching we can refine the candidates’ sets of acceptable departments.30
Our analysis in this paper will consider the rankings submitted by the recruiting
committees, where a candidate i ranked by a department d means that (i, d) is a mutually
acceptable pair. So our data is a real-life example of pre-matching problems and we can use
29
Interviews are not “flyouts” where the candidate can meet his (potentially) future colleagues and
have a better knowledge of the department. The interview lasts only 20 or 30 minutes, and candidates
usually do not hang around before or after the interview.
30
Most of the candidates violating the acceptability assumption turn out to be also participating to
the job market for another discipline (e.g., physics or biology), obtaining a job in a research institution
(e.g., CNRS, INRIA, etc.) or getting a job at a foreign institution. The number of violations for every
year are given in the Web Appendix.
26
the Predicting Algorithm from Section 4, once we have identified the candidates violating
our acceptability assumption. Table 6 together with Table 3 of the Web Appendix give
a good picture of the complexity of the problems. In 2013, for instance, there are 73
positions and 193 candidates ranked at least once in one of them; the length of the ranking
is between 5 and 7 in 75% of the positions. For each year in our dataset, we first identify all
the pairs (candidate, position) that are impossible matches. Once all such pairs (c, d) are
identified we simply delete candidate c from the ranking list of department d. From this
reduced market we count the number of positions’ matchings that are predicted. Table 6
reports for the years 1999 to 2013, in this order: the number of job openings, the number of
candidates that appear in the rankings, the number of predictions of candidates’ matches
and the number of predictions of positions’ matches (the percentage in parenthesis relates
the number of predictions to the numbers of candidates and job openings, respectively).
The last two columns (with the “total” term) are obtained when running our Predicting
Algorithm using for the input matching the final matching computed by the Ministry of
Higher Education.31
We are not aiming here at identifying what drives department’s strategies when establishing their rankings of candidates, a future research project will address that question.
Nevertheless, Table 6 is instructing as to the impact that department’s strategies have
on final outcomes (together with the set of mutually acceptable partners). Indeed, recall
that in the French market all departments establish their rankings of candidates before
candidates are called to submit their preferences. Table 6 shows that, on average, nearly
45% of the positions are awarded independently of the way candidates rank the positions
in their preference lists. If we consider the (observed) choices of market stars to refine
our analysis, we end up with almost 60% of the positions’ matches being predicted. In
other words, the way candidates rank their choices have little impact on the final matching. Table 6 is also instructive, from candidates’ viewpoint. On average, we are able to
predict ex ante the matches of 60% of the candidates, be they matched to a department
or not. According to Corollary 10, for 78% of the candidates, the way each of them ranks
his acceptable departments is irrelevant for his final match (all other things remaining
31
We consider in the Predicting Algorithm a less severe definition of stars, that we call pseudo stars
(see Section C.3 of the Web Appendix). Proposition 9 remains valid. Also, across all years, the Predicting
algorithm stops after 5 to 9 steps, the average number of steps is 6.8 (i.e., 5.8 iterations of steps h.1, h.2
and h.3).
27
equal), and up to 91% in 2012.
4.
Mutually acceptable?
As we explained in the Introduction, our goal is to analyze the consequences of short
preference lists and not why preference lists are short. We can nevertheless use the concept of impossible matches to shed some light on the structure of those lists. A natural
hypothesis to explain the length of the departments’ preference lists is that departments
are naturally constrained in their choices. Any candidate that appears in a department’s
submitted preference list must be interviewed by that department. Obviously, in a sufficiently large job market like the one we are analyzing it is not materially possible for
departments to interview all candidates who applied to the positions o↵ered by the departments. But departments are likely to be strategic, too. That is, when establishing the
rankings of candidates departments may focus on a subset of candidates, namely candidates that they may have a chance to hire. In other words, a candidate not appearing in
a department’s (submitted) preference list is not necessarily unacceptable for the department. A reasonable assumption could be then that departments consider in their ranking
only candidates that are achievable, that is, candidates that departments can be matched
with at some stable matching (in the job market problem). The next result shows that if
this were the case then we should not identify impossible matches in the data. Therefore
the presence of impossible matches in our data shows that departments do not rank only
achievable candidates.32
Before going further some definitions are in order. Let
be a job market problem. A
candidate i is achievable for a department d (or d is achievable for i) if there exists a
matching µ stable for
such that µ(i) = d. Given a job market (I, D, (
we define the achievable pre-matching associated to
fd , qd )d2D , A)
e such that
(I, D, (P
32
d , qd )d2D , (
i )i2I ),
the pre-matching problem
Intuitively, this result is not too surprising since matching markets tend to have a unique stable
matchings as the market gets large. In this case, most departments would rank only one candidate per
position, the one who is matched to that position at the unique stable matchings. One issue with that
reasoning is that large markets do not necessarily have a unique stable matchings. Even though we may
suspect that in the French job market there is only one stable matching we cannot show for sure that it
is the case.
28
29
121
89
86
70
97
73
89
128
115
111
127
114
121
83
73
99.8
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
Average
222.2
193
213
273
249
250
213
261
268
207
186
213
164
194
199
250
] Positions ] Candidates
1999
Year
130.2 (59.5%)
147 (76.2%)
152 (71.4%)
144 (52.7%)
125 (50.2%)
121 (48.4%)
115 (54.0%)
169 (64.8%)
140 (52.2%)
122 (58.9%)
131 (70.4%)
132 (62.0%)
105 (64.0%)
121 (62.4%)
128 (64.3%)
101 (40.4%)
43.4 (44.8%)
44 (60.3%)
49 (59.0%)
49 (40.5%)
39 (34.2%)
43 (33.9%)
45 (40.5%)
56 (48.7%)
52 (40.6%)
37 (41.6%)
37 (50.7%)
46 (47.4%)
35 (50.0%)
39 (45.3%)
44 (49.4%)
36 (29.8%)
Table 6: Predictions
] Cand pred. ] Pos. pred
171.7 (77.9%)
169 (87.6%)
194 (91.1%)
223 (81.7%)
150 (60.2%)
157 (62.8%)
141 (66.2%)
221 (84.7%)
209 (78.0%)
157 (75.8%)
161 (86.6%)
171 (80.3%)
141 (86.0%)
168 (86.6%)
159 (79.9%)
154 (61.6%)
] Cand. total Pred.
60 (61.6%)
54 (74.0%)
64 (77.1%)
78 (64.5%)
47 (41.2%)
59 (46.5%)
55 (49.5%)
79 (68.7%)
79 (61.7%)
53 (59.6%)
48 (65.8%)
64 (66.0%)
50 (71.4%)
60 (69.8%)
58 (65.2%)
52 (43.0%)
] Pos. total Pred.
e if, and only if, d is achievable for d in
• for each i 2 I, d 2 D, (i, d) 2 A
• for each d 2 D, and i, i0 2 I, iPed i0 if, and only if, iPd i0 .
Proposition 11 Let
associated to
be a job market problem and let Pe be the achievable pre-matching
. If a candidate i is achievable for a department d in
impossible match for d in Pe.
Proof
Let
be a job market problem and let i be achievable for d in
then i is not an
, i.e., there exists
a matching µ 2 ⌃( ) such that µ(i) = d. Let (I, D, (Pd , qd )d2D , A) be the pre-matching
problem such that 2 ⇥(P ). Since µ is stable, µ is comprehensive for P . Consider now
Pe, the achievable pre-matching associated to . By construction, for each j 2 I, µ(j) = d
e
e
e
implies j 2 AP . Note that for each department d, AP ✓ Ad and for each j, j 0 2 AP , j Ped j 0
d
d
d
if, and only if, jPd j . It follows that µ is also comprehensive for Pe, and thus µ(i) = d
implies that i is not an impossible match for d in Pe.
⌅
0
5.
Random markets
One may wonder what drives the number of impossible matches we can identify and,
more generally, the number of matchings that can be predicted. It turns out that across
the 15 years of data a large number of candidates are ranked only once. Table 7 (first
row) shows the average rank distribution for the years 1999-2015. Undoubtedly, the
high proportion of candidates being ranked only once or twice (almost 70%) has a great
impact on the number of impossible matches present in the ranking profiles. Indeed, a
candidate ranked only once is a trivial block, so all candidates ranked below that him
are impossible matches. Since the number of impossible matches is a key parameter to
determine the number of matches that can be predicted, we may legitimately conjecture
that our prediction rate is due to the skewness of the distribution. If we consider only
hired candidates (second line of Table 7), the distribution is still less skewed, though.
There are two variables that influence the existence of blocks (and thus of impossible
matches), beyond the number of times each candidate is ranked. The first variable is the
relative ranking of each candidate. A profile where most of the candidates being ranked
first are also ranked only once is clearly more predictable than another profile where most
of the candidates being ranked only once are at the bottom of the rank lists. The second
30
Table 7: Rank distribution (average 1999–2013)
] times ranked
1
2
3
4
5
6
7 or more
All candidates
48.3%
20.2%
12.5%
7.2%
4.4%
2.6%
5.7%
Hired candidates
31.7%
20.6%
15.8%
11.2%
6.9%
4.7%
9.2%
variable relates to the overlap between the rank lists in terms of the sets of candidates
being ranked. To see this, consider two candidates being ranked only twice. If these two
candidates are ranked by the same departments they constitute a block for any candidate
ranked below them and for the candidates ranked between them. However, they are less
likely to be part of a block if they are not ranked by the same departments.
To understand the impact of each of these parameters we ran two types of simulations,
one using random ranking profiles and one reshu✏ing (randomly) the observed ranking
profiles. This latter specification maintains the overlaps between the rank lists (i.e., the
set of candidates any pair of rankings have in common), while the former only consider
the impact of the rank distribution (first line of Table 7). For each type of simulations
we consider 15 parameterizations, one for each year. More precisely, for any given year
we took the same number of positions, the same distribution over the lengths of the rank
lists, and the same distribution in terms of how many times each candidate is ranked. So
the only di↵erence with the observed data is how rankings are filled. For each year, we ran
10,000 simulations for each specification (random rankings and reshu✏ed rankings). Figure 1 shows the averages (over 10,000 runs) of the percentage of positions being predicted
for both randomizations protocols and the prediction using the original rankings.33
Reshu✏ed rankings (resp. the original rankings) allow on average about 15% (resp.
22%) more predictions than random rankings. This di↵erence indicates that the high
level of prediction we observe in the original data is mostly due to the overlaps in departments ranking. Perhaps more surprising is the di↵erence in the prediction rates of the
reshu✏ed and original rankings, which is on average only of 5.8%. In other words, the
33
To make the comparison meaningful between observed and simulated rankings we did not withdraw
the candidates that violate the acceptability assumption (we did it for the results in Table 6). Table 8 in
the Web Appendix presents, for each year, the average percentage of positions predicted for the random
and reshu✏ed rankings, and the percentage of predicted positions for the original rankings when we do
not withdraw the candidates violating the acceptability assumption.
31
60
●
50
●
●
30
40
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
Original rankings
Reshuffle rankings
Random rankings
●
●
●
●
●
●
●
●
2006
2008
●
●
●
●
●
●
●
●
20
% positions predicted
●
2000
2002
2004
2010
2012
year
Figure 1: Prediction rate
way departments rank the candidates are not as important as who they rank.
6.
Concluding remarks
Our results show that stability constraints in a two-sided matching problem can be
sufficiently strong to rule out some matchings when we do observe only one side’s preferences and the set of mutually acceptable partners. The existence of impossible matches
relies on the presence of short preference lists compared to the size of the market, and
some overlap in the individuals’ set of acceptable partners; two properties that are likely
to be met in real-life settings. One could thus conjecture that, as the market size grows,
the number of impossible matches increases. This in turn would imply that the number
of stable matchings decreases, in line with existing results in the literature.34
The French academic job market is clearly unbalanced: there are more candidates
34
See Immorlica and Mahdian (2005).
32
than positions available. A recent paper by Ashlagi, Kanoria and Leshno (2017) explores
the impact of such imbalances, showing that agents on the short side of the market (the
departments in our case) tend to be matched with their top choices. This is confirmed by
Table 4 in the Web Appendix, showing that nearly half of the hired candidates are ranked
either top or second by the department that hire them. Another (equivalent) salient
feature of their contribution is that a slight market imbalance is key to get a unique
core in matching markets. This is also in line with our results on the French market.
Indeed the ability to predict most of the junior positions is revealing that the core is
close to be unique. However the imbalance between the number of agents on each side
of the market is not sufficient to obtain our results. In the framework of pre-matching
problems with one-sided preferences and short preference lists, market imbalance merely
materializes through the stronger notion of a block at a pair that involves overlaps of
mutually acceptable pairs. Overall the parallelism with the main findings of Ashlagi,
Kanoria and Leshno (2017) suggests that Hall’s conditions might provide an interesting
and novel tool for proving uniqueness results in market settings with short preference
lists.
From a broader perspective, this paper addresses the question of extracting information from partial matching data. One of the earliest results in that direction is the
so-called Rural Hospital Theorem in the context of matching interns to hospitals (Roth,
1986). This theorem states that if a hospital does not fill its capacity at some stable
matching, then it is matched to the same set of interns at any stable matching.35 From
the observation of one stable matching we can thus extract some information about all
stable matchings. Echenique, Lee and Shum (2013) go one step further and show for aggregate matchings how the observation of stable matchings can be used to estimate agents’
preferences.36 Behind these results is the idea to extract information on preferences or
35
The original Rural Hospital Theorem is stated for a many-to-one matching problem where hospitals
have responsive preferences. Another celebrated result is that in marriage problems, the set of agents
matched at a stable matching is the same for all stable matchings (Roth (1984) and Gale and Sotomayor
(1985)).
36
See also Echenique, Lee, Shum and Yenmez (2013). Agarwal (2013) presents an estimation of preferences in the medical matched based upon the observation of the matching. Another result reminiscent
of the Rural Hospital Theorem is Roth and Sotomayor (1989), who show how the observation of di↵erent
stable matchings for a College Admission problem gives some information about colleges’ preferences over
group of students.
33
(stable) matchings from the observation of some matchings.
Our contribution takes the opposite approach, as we aim at deducing information on
(un)stable matchings from the partial observation of preferences. A recent example of this
approach is Martı́nez et al. (2012) who, starting from a profile of incomplete preferences,
identify all the possible completions thereof that give the same set of stable matchings.
The two closest papers to ours are Rastegari et al. (2012), and Saban and Sethuraman
(2013). Rastegari and her coauthors consider a one-to-one matching market where agents
from both sides do not know their preferences and must conduct interviews to uncover
them. They show that as preferences become more complete, some potential interviews
may be discarded because the parties involved in such interviews can never be matched at
any stable matching, for any possible completion of preferences. Unlike the other works
we have just cited, Saban and Sethuraman focus on efficient assignments. They consider
random serial dictatorship, and address a question similar to ours, i.e., upon observing
agent’s preferences for objects they aim at knowing whether there exists an ordering of
agents such that an agent will get a specific object.
A.
1.
Appendix
Preliminaries
Let P = (I, D, (Pd , qd )d2D , A) be a pre-matching problem. For our purposes it is
convenient to describe it by a non-directed graph GP (V, E) where the set of vertices is
V = I [D and the set of edges is E = {(v, v 0 ) 2 V ⇥V : v 2 Av0 }. A path ⇡ in the graph
GP (V, E) is a (finite) sequence of pairwise distincts vertices (v1 , v2 , . . . , vk ) such that for
each h = 1, . . . , k
1 we have (vh , vh+1 ) 2 E. Given a matching µ, an edge (v, v 0 ) is free
if v is not matched to v 0 under the matching µ. Otherwise the edge is matched. A path
⇡ 2 GP (V, E) is an alternating path for a matching µ if it alternates matched and
free edges. Given a matching µ, a vertex v is neutral if either v 2 I and µ(v) = v, or
v 2 D and |µ(v)| < qv . A path ⇡ = (v1 , . . . , vk ) in GP (V, E) is an augmenting path for
a matching µ if it is an alternating path for µ where both (v1 , v2 ) and (vk 1 , vk ) are free
edges and where both v1 and vk are neutral vertices. A feasible matching µ is maximum
in P if there does not exist a feasible matching µ0 in P that matches more candidates
than µ, i.e., |{i 2 I : µ(i) 6= i}|
|{i 2 I : µ0 (i) 6= i}| for any feasible matching µ0
34
in P .37 A well-known result by Berge (1957) establishes that a matching is maximum if,
and only if, there is no augmenting paths. We rephrase Berge’s theorem in our context.38
Theorem 12 ((adapted from) Berge, 1957) Let P and µ be respectively a pre-matching
problem and a feasible matching in P . The matching µ is maximum for P if, and only if,
there is no augmenting path in GP (E, V ) for µ.
Let µ be a matching and let ⇡ = (i1 , d1 , i2 , d2 , . . . , ik , dk ) be an augmenting path for µ.
Without loss of generality we consider in the remainder only augmenting paths starting
with a (neutral) candidate and ending with a (neutral) department.39 The matching
µ0 = µ
⇡ is the matching obtained when updating µ with the path ⇡, i.e., µ0 (i) = µ(i)
for each i 2
/ ⇡ and µ0 (ih ) = dh for each ih 2 ⇡, h = 1, . . . , k.
Lemma 13 Let µ be a comprehensive matching in P . If µ is not a maximum matching
then there exists an augmenting path ⇡ such that µ0 = µ
Proof
⇡ is comprehensive in P .40
Let µ be a comprehensive matching for P and suppose that µ is not a maximum
matching. So, by Berge’s Theorem there exists an augmenting path for µ. Let ⇡ =
(i1 , d1 , i2 , . . . , dk ) be the smallest augmenting path, and if there are several augmenting
paths of the same length as ⇡ assume that ⇡ is one of those paths such that iPd1 i1 ) µ(i) 6=
i. Let µ0 = µ
⇡. We claim that µ0 is comprehensive for P . To see this, suppose that
µ0 is not comprehensive. So, there exists j such that µ0 (j) = j and j 0 such that jPµ0 (j 0 ) j 0 .
From the choice of ⇡, j 0 6= i1 . Since µ is comprehensive there exists h > 1 such that
j 0 = ih (and thus µ0 (j 0 ) = dh ). Consider now the path ⇡ 0 = (j, dh , ih+1 , . . . , dk ). Clearly,
⇡ 0 is an augmenting path, and since h > 1 it has a smaller length than ⇡, contradicting
⌅
the choice of ⇡.
37
A related concept (albeit di↵erent) is that of a maximal matching: a matching µ is maximal for
the candidates if there does not exist a matching µ0 such that µ0 (i) = µ(i) if µ(i) 6= i, and there exists
i such that µ(i) = i and µ0 (i) 6= i.
38
Berge’s original theorem is stated for one-to-one matching problems. It is straightforward to extend
Berge’s result to a many-to-one matching problem by adequately adapting the definition of a neutral
vertex as we did.
39
Let ⇡ = (v1 , v2 , . . . , vk ) be an augmenting path for some matching µ, and suppose that v1 2 D.
Then it must be that vk 2 I. We can then consider the path ⇡ 0 = (vk , vk
1 , . . . , v2 , v1 ),
definition, an augmenting path.
40
Andersson and Ehlers (2016) use a similar result in a di↵erent context.
35
which is also, by
Remark 3 The proof of Lemma 13 suggests an algorithm to construct a maximum and
comprehensive matching, which works as follows. Start with the empty matching, which
is trivially comprehensive. Now it suffices to identify augmenting paths of the smallest
length and select the smallest path with the same criterion we chose ⇡ in the proof. This
operation can be performed in polynomial time with a Breadth First Search algorithm.41
It follows then that finding a maximum and comprehensive matching can be done in
polynomial time.
2.
Proof of Theorem 5
The next two lemmata will be useful to prove Theorem 5.
Lemma 14 Let J be a block at (i, d) and let µ be a comprehensive matching in P̄ d (J, ;)
where µ(j) 6= j for each j such that jPd i. Then
|{j 2 J : µ(j) = j}| > |J|
Proof
Let r = |J|
thus there are at most
P
P
X
qdˆ + 1
(3)
ˆ J
d2A
ˆ J
d2A
qdˆ + 1. Since µ is comprehensive it is feasible for P̄ d , and
ˆ J
d2A
qdˆ
1 candidates in J that are matched to a department
under µ. It follows that there exists a set K made by r unmatched candidates. Hence, µ
is also feasible for P̄ d (J, K). Since J is a block, Hall’s marriage condition does not hold
for P̄ d (J, K), i.e., for any feasible matching µ0 for the problem P̄ d (J, K) there exists an
admissible candidate j such that µ0 (j) = j. Therefore, there exists j 2 J\K such that
µ(j) = j. So, there are at least r + 1 candidates in J who are unmatched under µ, as was
⌅
to be proved.
Lemma 15 Let J be a block at (i, d) and let µ be a comprehensive matching for P̄ d (J, ;)
such that
|{j 2 J : µ(j) = j}| > |J|
X
qdˆ + 1
(4)
ˆ J
d2A
and µ(j) 6= j for each j such that jPd i. Then there exists a matching µ0 that is comprehensive for P̄ d (J, ;) such that
|{j 2 J : µ0 (j) = j}| = |{j 2 J : µ(j) = j}|
and µ0 (j) 6= j for each j such that jPd i.
41
See Lovász and Plummer (1986).
36
1.
(5)
Proof
Let µ be a matching satisfying the conditions of the Lemma. From condition 1 of
P
P
a block we can match d2A
ˆ J qdˆ candidates of J in P (J, ;). So we can match
ˆ J qdˆ 1
d2A
candidates of J in P̄ d (J, ;). Hence, µ is not a maximum matching in P̄ d (J, ;). Since µ is
comprehensive in P̄ d (J, ;), there exists from Lemma 13 an augmenting path ⇡ such that
µ0 = µ
⇡ is comprehensive in P̄ d (J, ;). Clearly, µ0 satisfies Eq. (5). Since µ(j) 6= j
implies µ0 (j) 6= j, we have also µ0 (j) 6= j for each j such that jPd i.
⌅
Necessity
Let i0 be an impossible match for d0 in the pre-matching problem P . We need to show
that there exists a block J at (i0 , d0 ), with i0 2
/ J. Let Pb be the problem that is identical
to P except that in Pb candidate i0 finds only acceptable d0 .42 Observe that if µ is such
that µ(i0 ) = d0 and µ is comprehensive for P (resp. Pb) then µ is also comprehensive for
Pb (resp. P ). So from Proposition 1 i0 is an impossible match for d0 in P if, and only
if, i0 is an impossible match for d0 in Pb. So it is without loss of generality that we can
assume P being such that Ai0 = {d0 }.
Let µ be any maximum comprehensive matching in P . So by Proposition 1, µ(i0 ) = i0 .
The next procedure constructs explicitly an exact block at (i0 , d0 ). To start with, define
the following sets D0 and J 1 ,
D0 = {d 2 D : |µ(d)| < qd }
J1 = µ(d)\AD0 .
In other words, D0 is the set of departments that do not fill their capacity under
the maximum and comprehensive matching µ, and the set J1 is the set of all matched
candidates minus the candidates that consider acceptable some department in D0 . For
h
1, we define recursively the following sets,
Dh = {d 2 AJh : |µ(d) \ Jh | < qd }
Jh+1 = Jh \ADh .
So, the set D1 is the set of departments such that are acceptable by candidates in
J1 that do not fill their capacity with candidates in J1 . The set J2 is then the set of
candidates in J1 minus the candidates that consider acceptable some departments in D1 .
b
b
P
The problem Pb is such that Pbd0 = Pd0 (and thus AP
d0 = Ad0 ), and for each d 6= d0 , Ad = Ad \{i0 }
and for each i, i0 6= i0 , iPd i0 if, and only if, iPbd i0 .
42
37
Observe that at each step we withdraw candidates, i.e., Jh+1 ✓ Jh , for h
1. Since
the number of candidates is finite we eventually reach a step ` such that J`+1 = J` .43
Let K be the set of candidates, except i0 , who are matched to themselves under µ but
consider acceptable some department in AJ` ,
K = {i 2 I\{i0 } : µ(i) = i and i 2 AJ` }
Finally, let J = J` [ K. We claim that J is an exact block at (i0 , d0 ).
Claim: AK ✓ AJ`
Suppose by way of contradiction that AK * AJ` . So, there exists a candidate ih+1 2 K
and dh 2 Aih+1 such that dh 2
/ AJ` . Since µ is a maximal matching and µ(ih+1 ) = ih+1 it
must be that |µ(dh )| = qdh , and thus d 2
/ D0 . Hence, there is a step 1  h < ` such that
dh 2 D h
1
0
0
and dh 2
/ Dh for each h0 < h
and |µ(dh ) \ J
h 1
1. So, |µ(dh ) \ Jh | = qdh for each h0 < h
| < qdh .
1
It follows that there is at least one candidate, say, ih , such that ih 2 µ(dh ) with
ih 2
/ Jh
1
0
and ih 2 Jh for h0 < h
say, dh 1 , such that ih 2 Adh
2
1. Since Jh
and dh
1
2 D
1
h 2
= Jh 2 \ADh
2
there is a department,
0
. From dh 2
/ Dh for h0 < h
deduce dh 6= dh 1 . Hence, (ih+1 , dh , ih , dh 1 ) is an alternating path. From dh
deduce that there exists a candidate, say, ih
Therefore, ih
1
1
such that ih
6= ih , ih+1 . Note that we also have ih
there exists a department, say, dh
2
such that ih
1
1
2 Ad h
1
2 Dh
2 µ(dh 1 ) and ih
0
2 Jh for h0 < h
2
1
and dh
2
1 we
1
2
we
2
/ Jh 2 .
2.44 Hence,
2 Dh 3 . Continuing
this way we eventually end up with a path ⇡ = (ih+1 , dh , ih , dh 1 , ih 1 , dh 2 , . . . , i2 , d1 )
with µ(ih0 ) = dh0 , ih0 2 Adh
1
and dh0 2 Dh
0
1
. Observe however that d1 2 D0 implies
that |µ(d1 )| < qd1 . So ⇡ is an augmenting path and thus from Berge’s Theorem µ is not
maximum. Applying Lemma 13 we deduce that µ is not a maximum and comprehensive
⇤
matching, a contradiction. So, AK ✓ AJ` .
Claim: µ(d0 ) ✓ Jh , for each h  `.
Let ih+1 2 µ(d0 ) such that j 2
/ J` . So j 2 ADh for some h < `. Let h be such
0
that Dh \ µ(d0 ) = ; for h0 < h, i.e., ih+1 is among the first candidates in µ(d0 ) to
43
44
This occurs when there is a step ` 1 such that D` = ;.
0
0
0
If ih 2 2 Jh \Jh +1 for some h0 < h 2 then we would have j 2
/ Jh +2 , which would contradict
j 2 Jh
1
.
38
be withdrawn.45 Like for the previous Claim, we can deduce that there exists a path
⇡ = (ih+1 , dh , ih , dh 1 , . . . , i2 , d1 ) with d1 2 D0 . Let j0 be the candidate with the highest
rank for d0 such that µ(j0 ) = j0 (such a candidate exists since µ(i0 ) = i0 ). Observe
that the path ⇡ 0 = (i0 , d0 , ih+1 , dh , ih , . . . i2 , d1 ) is an augmenting path since d1 2 D0 .
Again applying Berge’s Theorem and Lemma 13 we deduce that µ is not a maximum and
comprehensive matching, a contradiction. So, µ(d0 ) ✓ Jh , for each h  `.
⇤
From the previous claim we deduce that J 6= ;. By construction |µ(d)| = qd for each
d 2 AJ` . Since AK ✓ AJ` = AJ , we have then all positions of departments in AJ are filled
under µ. That is, the first condition for a block is met with the set J.
It remains to show that condition 2 of a block also holds. Suppose by way of contradicP
tion that there exists K̄ ✓ J\Ji,d with |K̄| = |J|
d2AJ qd + 1, such that Hall’s condition
is satisfied in P̄ d0 (J, K̄). This is tantamount to saying that there exists a matching µ̃ for
P̄ d0 (J, K̄) such that all admissible candidates are matched to a department. Obviously, µ̃
is also comprehensive for P . Let µ0 be the matching such that µ0 (i0 ) = d0 , µ0 (j) = µ̃(j) for
each j 2 J\K̄, and µ0 (j) = µ(j) for each j 2 I\(J [ {i0 }). Notice that by construction for
each candidate j such that µ(j) 6= j and j 2
/ J, Aj \ AJ = ;, so µ0 is a feasible matching
for P . Observe that µ0 (j) 6= j for each j such that jPd0 i0 because K̄ \ {i : iPd0 i0 } = ;.
Therefore µ0 is comprehensive for P . From Proposition 1 we deduce then that i0 is not
an impossible match for d0 , a contradiction.
Sufficiency
Let J be a block at (i0 , d0 ). Suppose that there exists a matching µ̃ comprehensive for
P such that µ̃(i0 ) = d0 . Let µ be the matching such that µ(i0 ) = i0 and µ(i) = µ̃(i) for
each i 6= i0 . Thus, µ is a comprehensive matching in P̄ d0 (J, ;), and in particular µ(i) 6= i
for each i such that iPd0 i0 .
P
Let r = |J|
d2AJ qd + 1. For any matching, let K(µ) = {i 2 J : µ(i) = i}.
From Lemma 14 we have |K(µ)| > r. So we can invoke Lemma 15 to deduce that there
exists a matching, say, µ1 , such that µ1 is comprehensive for P̄ d0 (J, ;) and µ1 (i) 6= i for
each i such that iPd0 i0 and |K(µ1 )| < |K(µ)|. Note that by Lemma 14 |K(µ1 )| > r, so
we can invoke Lemma 15 and deduce that there exists a matching, say, µ2 , that satisfies
45
Although it is not needed for the proof, note that if there is j 2 µ(d0 ) such that j 2
/ J` then
µ(d0 ) \ J` = ;. Indeed, j 2
/ J` implies that there is a step h such that j 2 ADh . So we have d0 2 Dh+1
and thus µ(d0 ) ✓ ADh+1 , which implies µ(d0 ) \ Jh+2 = ;.
39
the properties required by Lemma 14 and 15. Again by Lemma 14, |K(µ2 )| > r, so by
Lemma 15 there exists µ3 that satisfies the properties required by Lemma 14 and 15, and
such that |K(µ3 )| < |K(µ2 )|. Repeating the process yields |K(µ)| = 1, a contradiction
with I being a finite set. Hence there is no comprehensive matching µ̃ in P such that
µ̃(i0 ) = d0 . From Proposition 1 we deduce then that i0 is an impossible match for d0 ⌅
3.
Proof of Proposition 6
R(P )
Proofs of Section 4
1 . (()
Suppose by way of contradiction that there
= {d} for some department d, yet for some e 2 ⇥(P ) and matching µ̃ stable
is Ai
R(P )
for e , µ
e(i) 6= d. Suppose first that µ
e(i) = i. Note that d 2 Ai
implies d 2 Ai . Since
µ
e is stable for e , it is obviously comprehensive for P because e 2 ⇥(P ). Furthermore,
i’s rank in Pd being lower or equal to qd implies that |e
µ(d)| < qd , which contradicts that
µ
e is non-wasteful. So µ
e(i) = d0 for some d0 6= d (and thus d0 2 Ai ). By construction,
R(P )
d0 2
/ Ai
())
implies that i is an impossible match for d0 , which contradicts µ
e(i) = d0 .
Suppose that for any e 2 ⇥(P ) and any matching µ
e stable for e , µ
e(i) = d.
To begin with, suppose that i is not among the top qd candidates in Pd . Let
2 ⇥(P ) be
such that d is the top choice of each candidate j whose rank in Pd is lower or equal to qd .
Clearly, if µ is stable for
then µ(j) = d for any candidate j ranked lower or equal to qd
in Pd . So i is not always matched to d for any stable matching of any P -compatible job
R(P )
market problem. Suppose now that there exists d0 6= d such that d0 2 Ai
0
an impossible match for d , and thus there exists by definition
µ stable for
R(P )
2 . If Ai
. So i is not
2 ⇥(P ) and a matching
such that µ(i) = d0 , which contradicts our initial assumption.
= ; then i is an impossible match for each department d 2 Ai . Hence,
candidate i cannot be matched to any department in Ai , under any stable matching for
2 ⇥(P ). Conversely, if i is always unmatched under any stable matching then i is an
R(P )
impossible match for any d 2 Ai , which implies Ai
= ;.
3 . Since the construction of the reduced problem R(P ) is the same for any P -compatible
job market problem, the result follows directly from the definition of impossible matches.
⌅
The next lemma will be key to prove the proposition.
40
Lemma 16 Let P be a pre-matching problem and
µ be a matching stable for
2 ⇥(P ) a job market problem. Let
. The following two statements hold, for any step h
1 of
the Predicting Algorithm with input (P, µ):
1. If a candidate i is deleted from Pdh or Pedh , then d 6= µ(i).
2. The matching µ is comprehensive for P h and quasi non wasteful for P h , i.e., for
h
any d such that |µ(d)| < qd , there is no candidate i such that µ(i) = i with d 2 APi .
Proof
Comprehensiveness for step h = 1 directly follows from Proposition 1. Quasi
non-wastefulness for step h = 1 follows immediately from the non-wasteful property of µ.
Observe that if µ is a comprehensive and quasi non-wasteful matching for P , then µ is
obviously a comprehensive and quasi non-wasteful matching for any P 0 that is obtained
from P by deleting from Pd , for some department d, a candidate i 2 Ad such that µ(i) 6= d.
Repeating for all impossible matches in P gives statement (2) for h = 1.
Consider the beginning of Step h and suppose as induction hypothesis that µ is comprehensive and quasi non-wasteful for the pre-matching P h . By construction if a candidate
i 2 S h is deleted from Peh then µ(i) 6= d. It also follows that µ is comprehensive and quasi
d
non-wasteful for Peh after Step h.2. It remains to check Step h.3. A candidate i is deleted
from the department d’s ranking if, and only if, i is an impossible match for d in Peh . So µ
h
being comprehensive for Peh implies that d 6= µ(i). Since i is thus never deleted from Pµ(i)
or Peh , it also follows that µ is still a feasible matching for P h+1 , and also comprehensive
µ(i)
⌅
and quasi non-wasteful, as was to proved. Statement (2) is thus proved.
Suppose that µ(i) = d, yet the prediction for i is d0 6= d or
Proof of Proposition 7
h
i. Since µ(i) = d it must be the case that d 2 APi for every h by Lemma 16 (no deletion).
Since the prediction is either d0 or i, there exists some iteration h0 where APi
APi
h0
h0
h0
= {d0 } or
= {i}. Hence, d 2
/ APi , which is a contradiction. Suppose now that µ(i) = i, yet
the prediction is d, predicted at some step h. Since d is a prediction, the department is
h
acceptable at step h for candidate i, d 2 APi , and the rank of i is weakly lower than
qd in Pdh . By Lemma 16, µ is comprehensive for the problem P h , so µ(i) = i implies
that no candidate ranked below i in Pdh is matched to d under µ. Again by Lemma 16,
0
candidates j such µ(j) = d cannot be deleted from Pdh , at any step h0  h. So we must
have |µ(d)| < qd , which contradicts that µ is quasi non-wasteful for P .
41
⌅
Proof of Proposition 9
Denote by P h and P 0h the pre-matching problems obtained
at step h in the algorithm with input (P, µ) and (P, µ0 ), respectively. Similarly, for h
2,
let S h , and T h , denote the set of market stars for (P, µ) and (P, µ0 ), respectively. We
first show by induction that P h = P 0h for every h
1. By construction it is clear that
P 1 = P 01 since neither the realized matching, nor the realization of the preferences are
used at this step. Suppose that P h = P 0h , for some h
2. We show that P h+1 = P 0h+1 .
At step h.1, notice that S h = T h , for otherwise there would be i 2 S h \T h or i 2 T h \S h .
In the former case, the construction of the sets S h and T h , together with P h = P 0h , imply
that µ0 (i) = i. But since i 2 S h the rank of i is weakly lower than qd in Pdh = Pd0h for any
h
0h
d 2 APi = APi . Therefore, there are strictly less than qd candidates ranked higher than
i in Pd and matched to d under µ0 . So µ0 is wasteful, which contradicts the stability of µ0 .
For the latter case i 2 S 0h \ S h , the reasoning is exactly the same.
h
At step h.2, since µ(j) 2 APj whenever j 2 S h (see footnote 20 in Proposition 7),
h
stability implies that j can claim any of the seats in the departments APj and that µ(j)
h
is the top choice in the set APj according to
to the problem
0
j.
The exact same argument should apply
(recall that P h = P 0h ). Since the top choice is the same department
by construction, j obtains the same department under µ0 . So µ(j) = µ0 (j) for every
j 2 S h . It also follows that the deletions are the same in both cases. So Peh = Pe0h . As
we proceed with step h.3, the impossible matches are clearly the same in both problems,
which implies So P h+1 = R(Peh ) = R(Pe0h ) = P 0h+1 , as was to be proved.
Since the sequence of sets of market stars and reduced pre-matching problems are the
same with inputs (P, µ) and (P, µ0 ). So any prediction obtained with (P, µ) is obtained
with (P, µ0 ), and vice-versa, that is, µ(i) = µ0 (i) for any candidate i whose match is
⌅
predicted.
42
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45
Web appendix
For Online Publication
Tuesday 25th April, 2017
Contents
A Blocks: additional results
2
A.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
A.2 A simple sufficient condition: exact block . . . . . . . . . . . . . . . . . . . .
4
A.3 Identifying impossible matches . . . . . . . . . . . . . . . . . . . . . . . . . .
6
B The French academic job market
B.1 Organization
9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
B.2 Changes during the time framework . . . . . . . . . . . . . . . . . . . . . . .
11
C Data analysis
14
C.1 On the acceptability assumption . . . . . . . . . . . . . . . . . . . . . . . . .
14
C.2 A remark on reshu✏ed profiles . . . . . . . . . . . . . . . . . . . . . . . . . .
14
C.3 Market stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
C.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1
A
Blocks: additional results
A.1
Basic Properties
The next proposition establishes several properties of a block. First, we show that if conditions 1 and 2 hold, then condition 2 also holds when we consider all possible sets K ✓ J\Ji,d .
This stronger statement of the block condition will be used to deduce the second and third
properties of Proposition A.1. If there are less than qd candidates ranked higher then i cannot be an impossible match. So, a necessary condition for i to be an impossible match is that
there are qd or more such candidates. The second statement of Proposition A.1 states that
this property is implied by the existence of a block at (i, d), but also that at least qd such
candidates must be part of the block. Finally, we can strengthen slightly the sufficient condition given in Theorem 5. We show that in the presence of a block, whenever candidate i is
matched to d it must be that one of the candidate j from the block must remain unmatched,
and is ranked higher than a candidate matched at one of the departments acceptable for j.
Proposition A.1 Every block J at (i, d) satisfies the following properties:
1. For each nonempty K ✓ J \ Ji,d , Hall’s marriage condition does not hold in P̄ d (J, K).
2. There are at least qd candidates in J ranked higher than i at department d, i.e., |Ji,d |
qd .
3. If µ is a feasible matching such that µ(i) = d then there exist k 2 J and j 2 J [ {i}
such that kPµ(j) j and µ(k) = k.
To prove Propositions A.1 we need the following lemma.
Lemma A.2 Let P be a pre-matching problem and i 2 Ad . Let J ✓ I \ {i} be a subset of
P
candidates such that for some J0 ✓ J with |J0 | = d2A
ˆ J qdˆ, Hall’s condition is satisfied in
P (J0 , ;). The following three conditions are equivalent,
(a) For each nonempty K ✓ J \ Ji,d , Hall’s marriage condition does not hold in P̄ d (J, K).
(b) For each K ✓ J \ Ji,d , with |K| = |J|
d
not hold in P̄ (J, K).
P
2
ˆ J
d2A
qdˆ + 1, Hall’s marriage condition does
(c) For each K0 ✓ J\Ji,d such that |K0 | = |J|
P
ˆ J
d2A
qdˆ + 1, there is no perfect match
between J\K0 and AJ in the problem P̄ d (J, K0 ).
Proof
Obviously, (a) ) (b) )(c). Thus it just remains to show (c) ) (a). To this end,
suppose there exists a set K ✓ J\Ji,d such that Hall’s condition is satisfied in P̄ d (J, K).
So, there exists a matching µ such that each candidate admissible in P̄ d (J, K) is matched
to a department. Define the matching µ0 that leaves unchanged the match of admissible
candidates in P̄ d (J, K) under µ and such that µ0 (j) = j for every other candidate j. Then
µ0 is also comprehensive in P̄ d (J, ;). Note also that every candidate in Ji,d is matched to a
department under µ0 since these candidates are all admissible in P̄ d (J, K).
Applying successively Lemma 13 we obtain a comprehensive and maximum matching in
d
P̄ (J, ;), where every candidate j 2 Ji,d is matched to a department. Since, by assumption,
P
J0 is such that |J0 | =
ˆ J qdˆ, each department in AJ fills its capacity at a maximum
d2A
matching in P̄ d (J, ;). Thus we have obtained a comprehensive matching µ̃ for P̄ d (J, ;)
where every candidate in Ji,d is matched to a department and every department in AJ fills
P
its capacity. Consider now K0 = {j 2 J : µ̃(j) = j}. Clearly, |K0 | = |J|
ˆ J qdˆ + 1,
d2A
and there is a perfect match between J\K0 and AJ in the problem P̄ d (J, K0 ), as was to be
⌅
proved.
Proof of Proposition A.1
1 . See Lemma A.2.
2 . One can use the alternative to condition 2 given in (a) of Lemma A.2. The set Ji,d
cannot be empty, for otherwise Hall’s marriage condition trivially holds in P̄ d (J, K) with
K = J, which is a contradiction. If Ji,d = J then condition 1 of a block implies |Ji,d |
qd .
For otherwise, consider the nonempty set K = J \ Ji,d . The set of admissible candidates
in P̄ d (J, K) is Ji,d . Since Hall’s marriage condition does not holds in P̄ d (J, K), one cannot
assign all candidates in Ji,d to department d at any feasible matching in P̄ d (J, K). That is,
|Ji,d | > qd
1.
3 . Let K = {j 2 J : µ(j) = j}. Since µ(i) = d, we have that K 6= ; from condition
1 of the definition of a block. Let µ0 be such that µ0 (i) = i, and µ0 (j) = µ(j) if j 2 J,
and µ0 (j) = j if j 2
/ J. Suppose that the conclusion of 3 . is not satisfied. Then µ is a
comprehensive matching for P (J [ {i}, ;). Since µ(i) = d we have necessarily K ✓ J\Ji,d .
In addition µ0 is a feasible matching in P̄ s (J, K), where, by construction, every admissible
3
candidate is matched to a department under µ0 . This contradicts the statement (a) of Lemma
⌅
A.2.
A.2
A simple sufficient condition: exact block
To illustrate further the potential structure of a block one may consider a particular case of
the definition of a block where the number of candidates in J is equal to the total number
of positions in the acceptable departments. In that case, conditions 1 and 2 specialize
respectively into a perfect match between J and AJ , and Hall’s marriage conditions with
singleton sets K.
Definition A.3 Given a pre-matching problem P = (I, D, (Pd , qd )d2D , A), an exact block
at (i, d) 2 A is a set J ✓ I\{i} such that:
1. |J| =
P
ˆ J
d2A
qdˆ and Hall’s marriage condition holds in P (J, ;);
2. For each k 2 J \ Ji,d Hall’s marriage condition does not hold in P̄ d (J, {k}).
The above tractable conditions are sufficient for the existence of impossible matches.
In the next section, we will show that those conditions are also necessary under a mild
restriction on the pre-matching problem.
The following result is deduced from Theorem 5 in the paper.
Corollary A.4 If P admits an exact block at (i, d) then i is an impossible match for d.
The next two examples illustrate the interplay between conditions 1 and 2 (for blocks or
exact blocks) in order to get impossible matches.
Example A.5 Let d0 and d1 be two departments and i0 , i1 and i2 three candidates. Consider
the problem P depicted in Table 1, where Ai1 = {d0 , d1 }, Ai2 = {d1 }, qd0 = qd1 = 1.
Clearly, i0 is an impossible match for d0 , and it is easy to see that J = {i1 , i2 } is an exact
block at (i0 , d0 ). Condition 1 holds trivially. As for condition 2, K = {i2 } is the only set K
for which we have to consider condition 2. So we only have to check whether we can match
i1 in P̄ d0 (J, {i2 }). This is impossible since no department is admissible. So J is a block. ⇤
4
Pd 0
Pd 1
i1
i2
i0
i1
Table 1: Impossible match and exact block
An exact block J does not always exist. A counter-example, where an impossible match
does not admit any exact block, is the following.
Example A.6 Consider four departments, d0 , d1 , d2 and d3 , with one position at each department, and six candidates ih , h = 0, . . . , 5. The problem P is depicted in Table 2. It is
easy to see that candidate i0 is an impossible match for d0 .
Pd 0
Pd 1
Pd 2
Pd 3
i1
i2
i2
i3
i0
i3
i4
i4
i5
i1
i5
i1
Table 2: Impossible match and no exact block
To begin with, notice that any block must contain at least 4 candidates, for otherwise
condition 1 cannot be satisfied.1 The set J1 = {i1 , i2 , i3 , i4 } satisfies condition 1, but not
condition 2. To see this, consider the matching µ (feasible for P̄ d0 ) defined as µ(i1 ) = d1 ,
µ(i2 ) = d2 and µ(i3 ) = d3 . Candidate i4 is ranked lower than the matched candidates for
all his acceptable departments. So, condition 2 is not satisfied for the block J1 , since Hall’s
condition is satisfied in P̄ d0 (J1 , {i4 }).
A similar reasoning applies for the sets J2 = {i1 , i2 , i3 , i5 }, J3 = {i1 , i2 , i4 , i5 }, and J4 =
{i1 , i3 , i4 , i5 }. That is, there is no block J at (i0 , d0 ) such that |J| = 4,, hence no exact block,
while the reader can check that J = {i1 , i2 , i3 , i4 , i5 } is the (unique) block at (i0 , d0 ).
1
⇤
Candidate i1 must be part of the block, and thus condition 1 implies that we must include at least two
other candidates to fill departments d1 and d2 . So any block must contains at least candidate i3 , or i4 or
i5 . If any of these three candidates is added to the block we span all four departments, and thus any block
must contains at least four candidates.
5
A.3
Identifying impossible matches
According to Proposition 1 in the paper we would have to consider all possible comprehensive
matchings to check whether a candidate is an impossible match for a department. Theorem
5 in the paper greatly simplifies this task, for we only have to consider all the sets K ✓ J of
a particular size. The set J turns out to be easy to construct; the proof of the necessity part
of Theorem 5 suggests a procedure to identify impossible matches, which we now detail.
Let P be a pre-matching problem and let i 2 Ad .
Algorithm: Impossible Matches Algorithm
1. Delete candidate i from the preference of all departments but department d, without changing the relative ranking of the other candidates.
2. Construct a maximum and comprehensive matching µ.2
3. If µ(i) = d then i is not an impossible match for d. Otherwise go to step 4.
4. Constructing the set J:
4.1 Let J0 be the set of all candidates matched to a department, and D0 all the departments that do not fill their capacity at µ. If D0 is empty then go to Step 5. Otherwise
iterate through h until obtaining an empty set Dh :
4.2 Define Jh as the set Jh
1
without all the candidates that consider acceptable some
department in Dh 1 ,
Jh = {j 2 Jh
1
: j2
/ AD h 1 } ,
and let Dh be the set of departments that do not fill their capacities with candidates
in Jh ,
ˆ \ Jh | < q ˆ} .
Dh = {dˆ : |µ(d)
d
2
Set h := h + 1 and repeat step 4.2 until D` = ; for some h = `. Let J̄ = J` .
We describe in the proof of Lemma 13 in the Appendix a way to construct a maximum and comprehensive
matching for a pre-matching problem. This construction consists of an iterative search of augmenting paths
starting from a comprehensive matching (which could be empty) and can be performed in polynomial time —
See the Section A for a definition of augmenting paths.
6
5. If d 2
/ AJ̄ then i is not an impossible match for d. Otherwise check condition 2 of Definition
2 at J:
5.1 Let K̄ be the set of unmatched candidates at µ (except i) that consider acceptable
some department in AJ̄ . Let J = J̄ [ K̄.
5.2 For each K ✓ J\Ji,d such that |K| = |J|
d
P
ˆ J
d2A
qdˆ + 1, find a maximum matching
in the problem P̄ (J, K). If there is no set K such that all admissible candidates of
P̄ d (J, K) are matched to a department then i is an impossible match for d. Otherwise
i is not an impossible match.
We show in the proof of Theorem 5 that the set J constructed in Step 5.1 satisfies
conditions 1 and 2 of the definition of an exact block, and Step 5.2 consists of checking
condition 3. If there is a set K satisfying the conditions set in Step 5.2 and a matching µ
such that all admissible candidates in P̄ d (J, K) can be matched to a department then i is not
an impossible match for d. This is so because µ is comprehensive for P among the candidates
in J. By construction no candidate better ranked than i at d is in K, so these candidates
are all matched. Also, in the problem P̄ d (J, K) the capacity of d is qd
1, so it then suffices
to match i to d to have a comprehensive matching for P , the original pre-matching problem.
On the other hand, if no such set K can be found then Hall’s condition is not satisfied for
any set K, which simply establishes that the second condition of a block holds.
Note that the set J0 constructed in Step 3 is clearly easy to obtain, so the complexity of
our algorithm is determined by Step 5.2. This is where the added value of Theorem 5 comes,
for we only have to truncate by taking sets K that have the size specified in condition 2, a
clear improvement compared to Proposition 1. For a set J of size n and setting k the size of
the sets K the maximal number of cases one has to consider in Step 5.2 is then
n
k
, i.e., our
procedure in Step 5.2 runs in O(nk ) time, which is maximum for k = n/2. The following
result shows that, in general, we cannot hope for a better bound.
Proposition A.7 (Saban and Sethuraman (2015)) Finding whether a candidate is an
impossible match is NP-complete.
In practice, it is however possible to reduce further the sets K one has to consider by not
including the candidates who are necessarily matched at a comprehensive matching.
7
Definition A.8 Given a pre-matching problem P , a candidate i is prevalent if for each
job market problem
2 ⇥(P ) and each µ 2 ⌃( ), it holds that µ(i) 2 D.
Finding whether a candidate i is prevalent is easy. It suffices to consider the profile P
restricted to the departments Ai truncated at i, and then check whether there is at least
one department in Ai that does not fill its capacity at a maximum matching. Given a prematching problem P , let J⇤ denote the set of prevalent candidates. We can then re-write
the second condition of a block in the following way,
2’. For each K ✓ J \ (Ji,d [ J⇤ ), with |K| = |J|
does not hold in P̄ d (J, K).
P
ˆ J
d2A
qdˆ + 1, Hall’s marriage condition
Proposition A.9 Let J ✓ I \ {i} be a subset of candidates satisfying condition 1. of a
block. Then conditions 2 and 2’ are equivalent.
Clearly 2 implies 20 . To show the converse implication suppose that there exists
P
K ✓ J \ Ji,d with |K| = |J|
ˆ J qdˆ + 1, such that Hall’s marriage condition holds in
d2A
Proof
P̄ d (J, K). From Lemma A.2 (c), it means that there exists a perfect match between J\K
and AJ in the problem P̄ d (J, K), i.e., where each department fills its capacity. Let j be
a prevalent candidate. If j 2 K, then it is possible to fill all the departments in Aj with
candidates that are ranked higher than j at these departments. This would contradict the
fact that j is prevalent. So j 2
/ K. It follows that K ✓ J \ (Ji,d [ J⇤ ).
⌅
By imposing a slight restriction on the pre-matching problem our algorithm yields a
computationally easy task. We assume that there exists a matching for P such that all
candidates are matched to a department.
Assumption A.10 Hall’s Marriage condition is satisfied in P .
Proposition A.11 Under Assumption A.10, candidate i is an impossible match for department d if, and only if, P admits an exact block at (i, d). Furthermore, finding whether a
candidate is an impossible match can be done in polynomial time.
Proof
From Corollary A.4, the existence of an exact block J at (i, d) implies that i is an
impossible match for department d.3
3
Note that Assumption A.10 is not needed here.
8
To show that the exact block condition is also necessary and can be checked in polynomial
time, it suffices to reconsider the Impossible Matches Algorithm. Let µ be any feasible
matching for P such that every candidate j 2 I is matched to a department. Such a
matching exists under Assumption A.10. Consider the matching µ0 be defined as follows:
µ0 (j) = µ(j) for all j 6= i, and µ0 (i) = i. Then µ0 is feasible for the problem described at
step 1 of the Impossible Matches Algorithm. It follows that the maximal matching used at
step 4 is such that all candidates but candidate i are matched to a department. Hence at
P
step 5.1, K̄ = ; and |J| = d2A
ˆ J qdˆ. At Step 5.2, the maximal number of cases one has to
consider is thus equal to |I|
⌅
1.4
Finally, note that the identification of impossible matches can be sped up when scrutinizing the blocks. The impossible match property is not monotone with respect to departments’ preferences (Proposition 3), but if we identify an exact block J at some pair
candidate–department (i, d), then all the candidates ranked lower than i at d who are not
part of the block J are also impossible matches for d.
Corollary A.12 Let J be a block at (i, d) and let j be such that iPd j (so d 2 Aj ) and j 2
/ J.
Then j is an impossible match for d.
B
The French academic job market
We provide here more details about the French academic job market.
B.1
Organization
Disciplines in French academics are divided into sections, that correspond to fields. For
instance the section corresponding to economics is section ] 5, section ] 10 corresponds to
comparative literature. There are 77 di↵erent such sections, but medical positions sections
are often split into sub-sections corresponding to sub-specialities, bringing the total to about
120.
For candidates, the job market begins in the Fall. Candidates first request an evaluation
by a national committee, the Conseil National des Universités. There is a committee for
4
We only have to check that for each candidate j 2 J \ Ji,d , Hall’s marriage condition is not satisfied in
the problem P̄ d (J, {j}).
9
each section. A positive evaluation by a committee is valid for four years, after which the
candidate must ask for another evaluation, should he or she want to participate again to the
job market.
Positions are advertised by the Ministry of Higher Education in winter. Shortly after
the publication of job openings the candidates send their applications to the departments
where they want to apply. By law a candidate need only to have a positive evaluation by
one section to apply to any job. That is, a candidate having a positive evaluation from the
committee for section, say, 17 (philosophy), can apply to a position advertised in section,
say, 34 (astronomy and astrophysics). In practice, recruiting committees reject candidates
who do not have a positive evaluation for the section corresponding to the job opening.
To each open position is assigned a recruiting committee of about 15–20 people. The
composition of the committee is constrained by a double parity on the members’ origin (internal or external) and the type of position (junior or senior). Upon receiving the candidates’
applications the recruiting committees decide first which candidate to interview. The interviews usually take place in May, and usually last between 20 and 30 minutes. In most cases
all the interviews are made the same day. Contrary to, for instance, the (international) job
market for economists, interviews in the French job market are held at the universities. That
is, candidates must travel to attend their interviews. So a candidate may have an interview
scheduled in Paris and Marseilles (900 km away) the same day, at the same time. Committees try to be flexible to permit candidates to tune their schedules in case of conflicting
interviews.
After the interviews recruiting committees rank the candidates (usually the same day)
and the rankings are submitted to the Ministry of Higher Education after validation by
the University board.5 Once the interview season is over, candidates are asked to submit
(in June) their preferences over the positions for which they are ranked. When logging in,
candidates see all the positions for which they are ranked (and their rank), independently
of the section (i.e., field) in which the positions were advertised, and have to rank them.
For instance, if a candidate was ranked for positions that were advertised for sections, say,
26 and 34, the candidate will have to submit a unique ranking, not one for each field. In
5
The university board (“Conseil d’administration” is composed by (mostly) faculty, sta↵ and students’
representative. The board has power to alter the ranking decided by the recruiting committee. In mathematics this does happen, albeit rarely. We consider in the paper the rankings decided by the recruiting
committees.
10
other words, the Ministry of Higher Education is considering one job market encompassing
all sections.
The Ministry then computes a matching, which is announced about a week after.6 Most
positions start on September 1st of the same year.
Remark B.1 Parallel to the university job market there is another job market involving
research institutions (CNRS, INRA, INSERM, INRIA, etc.). These positions (also tenured)
are research-only positions and are usually preferred by the candidates. These institutions
usually announce which candidate is o↵ered a job in April, i.e., before the interviews. However, candidates have little control over the institution where they will be assigned. The
assignment is usually announced in June, when candidates must submit their choice lists for
the assistant professor positions. That is, a candidate hired by the CNRS (in April) does
not know before June where (e.g., Paris, Marseilles or Toulouse) he will be assigned.
Remark B.2 There are each year between 80 and 130 positions open in sections 25 and
26. This high number of positions is rather unusual. For instance, in economics there are on
average about 40–50 positions a year. This high number is partly explained by the fact that
mathematics departments often provide the mathematics courses taught in other disciplines.
Each year, there are on average between 400 and 500 candidates having passed the evaluation
of the CNU of sections 25 or 26, so this gives an average of 5 candidates per opening.7 On
average about 30–40% of the candidates do not pass the CNU evaluation. The main reasons
of reject are misfit (the field of specialization of the candidate does not match the field of
the section), incomplete application and weakness of the scientific production (about 5% of
the applicants).
B.2
Changes during the time framework
Our observations cover 15 years (from 1999 to 2013) of the market for assistant professors in Mathematics. We observe four major institutional changes along those years. In
chronological order:
6
For 2016 for instance, the deadline for submitting the preferences over the positions is June 16 and the
match is announced on June 20.
7
There are slightly more candidates qualified for section 26 than for section 25. The former often includes
candidates from other disciplines (e.g., economics, physics, biostatistics, chemistry, computer science)
11
no more linked positions - since 2008 It can happen that a department o↵ers two or
three identical positions at the same faculty (i.e., positions with identical job descriptions). Until 2007, those positions were considered as being “linked,” which implied
that the recruiting committee had to establish a unique ranking for all the linked
positions.8
a new decentralized market Since September 2008, departments must choose between
two separate methods of recruiting before entering the market. The first one is a
“synchronized session,” where the timeline for recruiting is imposed by the Ministry
of Higher Education. Departments recruiting in this market are competing with each
other at the same time. This is this market we consider in this paper. The other
market is not coordinated. That is, departments can choose the dates of the interview
and make job o↵ers to candidates immediately thereafter. This market is called “au fil
de l’eau” (on the fly). A cause for concern is merely the existence of overlaps between
the decentralized and centralized markets for the candidates who matter in our data.
But the few positions of the uncoordinated market are usually published well before
with the respect to the synchronized market, so that the candidates accepting such
early positions do not appear in that latter market.9
no more second session – since 2009 The Ministry used to organize a second job market in September which would, sometimes, include positions that were not filled in the
first job market session (although in general the second session positions were new positions). This session had a very small number of positions and will not be considered
in this paper (e.g., for mathematics, there were 6 positions in 2006, 2 in 2007, 1 in
2008).
no more constrains for the ranking length – since 2009 Until 2008 departments could
rank at most 5 candidates per position. In case of “linked positions” (until 2007), the
length of the rank list was 5⇥] linked positions. For instance if a department had 3
8
In this case the maximal number of candidates that can be ranked was 5 multiplied by the number of
linked positions.
9
The co-existence of a second market has important strategic implications for both departments and
candidates. Some of them are already well documented in the matching literature, including strategic
unraveling and the problem of early resolution of uncertainty (e.g., Halaburda (2010), Echenique and Pereyra
(2016)). We do not discuss those issues here.
12
linked positions it could rank up to 15 candidates. The Ministry of Higher Education
dropped the constraint on the number of candidates that could be ranked. Our data
shows that it took some time for the departments to fully integrate this new possibility
in their strategies. By 2013, most of the departments rank more than 5 candidates per
position, suggesting that the constraints on the length of the rankings was binding.
However, the length of the rankings have not increased substantially. For the last year
in our data (2013), the distribution of the length of the rankings are given in Table 3.
When looking at the rank of hired candidates (Table 4), we can see that there is no
substantial impact of the abandon of the constrain on the length of the rankings.10
Length
3
4
5
6
7
8
9
10
] positions
1
9
17
27
12
4
2
2
1.4%
12.2%
23%
36.5%
16.2%
5.4%
2.7%
2.7%
% of positions
Table 3: Distribution of ranking lengths (year 2013)
Rank
1
2
3
4
5
6
7
8
9 and more
] cand. (year 2013)
26
15
11
11
2
3
1
0
2
% cand. (year 2013)
36.6%
21.1%
15.5%
15.5%
2.8%
4.2%
1.4%
0%
2.8%
avg. years  2008
32.8%
20.9%
16.7%
10.4% 7.2%
3.6%
3.7%
2.2%
2.5%
29.3%
20.2%
13.8%
12.7%
6.9%
5.0%
3.1%
2.9%
avg. years > 2008
6.2%
Table 4: Rank of hired candidates (for the position that hired them)
The e↵ects of those institutional changes are globally ambiguous. On the one hand, we
observe that the joint abandon of quotas and linked positions leads to a thicker centralized
market, all things being equal. But on the other hand, the e↵ect is mitigated by the creation
of a second decentralized market that deprives the centralized market of candidates and
positions.
10
The average distribution for the years prior to 2009 in Table 4 is calculated not taking into account the
linked positions. If two positions are linked then the candidate ranked 2nd could be considered as ranked
first (he is sure to get either position if he desires so).
13
Year
Number Violations
Number Violators
1999
4
4
2000
3
3
2001
16
15
2002
9
6
2003
6
4
2004
6
5
2005
7
6
2006
11
11
2007
6
6
2008
12
8
2009
19
15
2010
5
4
2011
16
13
2012
12
9
2013
9
6
Table 5: Number of violation of acceptability assumption
C
Data analysis
C.1
On the acceptability assumption
See Table 5.
C.2
A remark on reshu✏ed profiles
In this case the ranking of a department simply consists of a random reshu✏ing of the
original ranking. Doing this way ensures that the distribution of number of times candidates
are ranked is identical to the original distribution.
Until 2007 some positions were linked, i.e., recruiting committees had to give a unique
ranking for all the linked positions. In that case we opted for reshu✏ing in the same way all
linked positions. For instance, if two positions are linked and rank (in this order) candidates
a, b, c, d and e, the reshu✏ed ranking of the two positions would be the same (e.g., c, e, a,
14
d and b).
C.3
Market stars
In the main paper the definition of a star given a pre-matching problem P is a candidate
that is ranked more than once and always first (at some stage of the Predicting Algorithm).
In the program we use an additional definition of stars, that we call pseudo stars. Such a
candidate is a candidate who is always ranked weakly higher than the hired candidate (but
not necessarily always first). Table 6 illustrates this, where the candidates in boxes are the
matched (hired) candidates.
Table 6: Pseudo stars.
Pd 1
Pd 2
Pd 3
Pd 4
i2
i2
i5
i4
i1
i1
i2
i2
i5
i5
i6
i7
i4
i3
i4
i3
In this example candidate i1 is a pseudo star. He is ranked by d1 and d2 , and his rank is
always weakly higher than the rank of the hired candidate. Candidate i2 , on the contrary,
is not a pseudo star, because for d4 he is ranked below the hired candidate.
Since we know that the final matching is stable, pseudo star are similar to stars in that
the department that hire them is necessarily the top choice among all the other positions
that rank that candidate. So in our case we know that i1 necessarily prefers d2 to d1 . Note,
however, that this deduction is only valid for a given pre-matching profile. In our predicting
algorithm it could be that at a previous step i1 was ranked by, say, d4 , but below the hired
candidate, i4 . In this case we obviously cannot infer the relative ranking of d4 and d1 in i1 ’s
preferences, and thus at that previous step i1 was not a pseudo star.
The structure of our code works as follows when considering stars (and pseudo stars).
The algorithm is similar to the algorithm presented in the paper, except that we also search
for pseudo stars. That search is only triggered when there are no market stars.
15
Predicting Algorithm
Input: A pair (P, µ), where µ is the output of a regular matching algorithm with input
2 ⇥(P ).
Step 1 Identify all impossible match pairs in P and construct P 1 := R(P ).
Step h for h
2
h.1 Identify the market stars in P h .
If there are no market stars, identify the pseudo stars in P h .
If there are not stars and no pseudo stars, then STOP.
h.2 If there are stars (or pseudo stars), elimimate them from the rankings of the
positions that do not hire them. Let Peh be the pre-matching profile obtained.
h.3 Construct P h+1 := R(Peh ) and go to Step h + 1.
It is routine to check that with this version of the Predicting Algorithm the results of
Section 4 continue to hold. The proof of Proposition 9, which is the unique result dealing
with market stars, simply needs some minor modifications in the phrasing (paragraphs 2
and 3 in the proof).
Table 7 reports the number of market stars and pseudo stars used in the Predicting
Algorithm for every year. Note that the total number of stars and pseudo stats outnumber
the number of positions extracted. Indeed, very few pseudo stars (at most three per year)
identified at step h.2 of the above algorithm are not mapped to a position by the algorithm.
In the example given in Table 6, candidate i1 is a pseudo star but the algorithm cannot
clear any of the four positions (even after iterations). In that example, there is no position
extracted and one pseudo star.
C.4
Simulations
Table 8 reports the detailed results of the simulations.
16
year
Positions extracted
Number Stars
Number Pseudo Stars
1999
28
21
11
2000
14
9
6
2001
22
20
2
2002
14
13
2
2003
17
15
3
2004
18
18
0
2005
16
14
5
2006
29
22
9
2007
25
22
4
2008
16
13
6
2009
18
16
4
2010
14
9
6
2011
28
24
7
2012
19
15
5
2013
13
10
4
19.4
16
4.7
Average
Table 7: Extractions
References
[1] Echenique, Federico and Juan S. Pereyra 2016. “Strategic complementarities and
unraveling in matching markets.” Theoretical Economics, 11: 1–39.
[2] Halaburda, Hanna 2010. “Unravelling in two-sided matching markets and similarity
of preferences.” Games and Economic Behavior , 69(2): 365–393.
17
year
Original
Random
Reshu✏e
1999
32.2%
31%
33.1%
2000
49.4%
40.8%
43.8%
2001
44.2%
41.1%
47.1%
2002
52.9%
42.6%
48.3%
2003
44.3%
40.7%
38%
2004
50.7%
44.1%
47.8%
2005
44.9%
38%
46.3%
2006
39.1%
27.8%
40.3%
2007
47%
36.7%
45.3%
2008
36%
28.1%
34.4%
2009
33.9%
29.3%
30.2%
2010
34.2%
27.3%
36%
2011
38.8%
28.5%
35.3%
2012
60.2%
53.3%
51%
2013
58.9%
48.3%
52.2%
average
44.4%
37.2%
41.9%
Table 8: % of predicted positions
18