In this paper, we show that the one-to-one matching model of Mumcu and Saglam (2008) studying stability under interdependent preferences is refutable. We also give a sufficient characterization of the set of matchings that are rationalizable inside the core.One-to-one matching Stability Externalities Rationalizability

AyÃ¾e Mumcu

Ismail Saglam

English

Research Papers in Economics

Rationalizability of one-to-one matchings with externalities 
Ayþe Mumcu Ismail Saglam
Bogazici University TOBB University of Economics and Technology
Abstract
In this paper, we show that the one-to-one matching model of Mumcu and Saglam (2008)
studying stability under interdependent preferences is refutable. We also give a sufficient
characterization of the set of matchings that are rationalizable inside the core.
An early draft of this paper was written while the second author was affiliated with Bogazici University and the authors were
respectively visiting the University of Pennsylvania and Massachusetts Institute of Technology, to which they are grateful for
their hospitality. The second author also acknowledges the support of the Turkish Academy of Sciences, in the framework of
Distinguished Young Scientist Award Program (TUBA-GEBIP). The usual disclaimer applies.
Citation: Mumcu, Ayþe and Ismail Saglam, (2008) "Rationalizability of one-to-one matchings with externalities." Economics
Bulletin, Vol. 4, No. 32 pp. 1-8
Submitted: September 1, 2008.  Accepted: November 18, 2008.
URL: http://economicsbulletin.vanderbilt.edu/2008/volume4/EB-08D70031A.pdf1 Introduction
The last 45 years following the appearance of the seminal work by Gale
and Shapley (1962) have witnessed a rapidly growing literature in matching
theory studying the microfoundations of equilibrium in marriage and labor
markets, in college admissions and school choice problems, and recently in
organ exchange. Undoubtedly, stability, as the relevant notion of economic
eﬃciency, has invariably been one of the main concerns of researchers and
market designers in evaluating possible matching rules andprocedures. While
many eﬀorts have in this literature been spent on characterizing the set of
stable matchings in a given market or for a given problem, an existential
question as to the validity of matching models with regard to the used sta-
bility concepts was delayed until it was very recently posed by Echenique
(2008): Can there be any set of matchings for a given society or a market
that is incompatible with the predictions of the matching model at hand with
respect to the employed stability notions? As Echenique (2008) points out,
the answer to this question is important when the preferences of individuals
are unknown as it allows one to know whether a matching theory at hand
has testable implications.
In this paper, we extend the inquiry of Echenique (2008) that he answers
in a marriage model under independent preferences to the marriage model
of Mumcu and Saglam (2008) that characterizes stable one-to-one matchings
under interdependent preferences.1
Following Echenique (2008), we say that a set of matchings H in a given
marriage market is rationalizable inside the stable set if there exists a prefer-
ence proﬁle such that the corresponding stable set contains H. Similarly, we
say that the set H is rationalizable inside the core if there exists a preference
proﬁle such that the corresponding core contains H.
We show that Mumcu and Saglam’s (2008) marriage model with external-
1Matching under externalities was also studied for marriage markets by Sasaki and
Toda (1996), Roy Chowdhury (2004), and Hafalir (2008f), and for a housing market by
Mumcu and Saglam (2007).
1ities is refutable, since for any society facing at least two diﬀerent matchings
there exists at least one collection of matchings, e.g. the set of all conceivable
matchings, that is not rationalizable inside the stable set or inside the core.
We also give a suﬃcient characterization of the set of matchings that are
rationalizable inside the core.
The organization of the rest of the paper is as follows: In Section 2,
we introduce our model that borrows from Mumcu and Saglam (2008). We
present our results in Section 3. Finally, Section 4 concludes.
2 The Model
We consider a marriage market involving a set of men, M and a set of women,
W. We assume that M and W are nonempty, ﬁnite and disjoint, and satisfy
|M||W| ≥ 2, i.e. there exist at least three agents in the society and at least
one member from each gender. We denote a generic agent by i, a generic man
by m, and a generic woman by w. We denote the society by N = M ∪ W.
A matching is a one-to-one function, µ, from N to itself, such that for
each m ∈ M and for each w ∈ W we have µ(m) = w if and only if µ(w) = m.
Moreover, either µ(m) ∈ W or µ(m) = m, and similarly either µ(w) ∈ M
or µ(w) = w. If µ(m) = w, then m and w are matched to each other. If
µ(i) = i, then i is single. Let MN denote the set of all matchings in society
N.
Given any matching µ, let µm,w denote the matching at which (i) m and w
are a couple, i.e., µm,w(m) = w, (ii) their mates under µ, if they exist, become
single, i.e., µm,w(µ(m)) = µ(m) if µ(m) / ∈ {m,w} and µm,w(µ(w)) = µ(w) if
µ(w) / ∈ {w,m}, and (iii) the marital status and the mates of all other agents
are preserved, i.e., µm,w(i) = µ(i) for all i / ∈ {m,w,µ(m),µ(w)}.
Each agent has a complete, transitive, and strict preference relation over
the matchings in MN. Pi represents the preference relation of agent i, while
P = (Pi)i∈N denotes the preference proﬁle of the society. We respectively
write µ >i µ￿ and µ ≥i µ￿ to mean i strictly and weakly prefers µ to µ￿. A
2marriage market is a triple (M,W,P).
For any proﬁle P and any l ∈ {1,2,...,|MN|}, let Pi[l] denote the lth-
ranked matching from top in the ordering Pi of agent i.
We say that agent i individually blocks matching µ (via µi,i) if µi,i >i
µ. A matching is individually rational if it is not individually blocked by
any agent. For a given matching µ, (m,w) is a blocking pair if µ(m) ￿= w,
µm,w >m µ and µm,w >w µ. A matching is stable if it is individually rational
and if there are no blocking pairs. We denote the set of stable matchings
(the stable set) for the marriage market (M,W,P) by S(M,W,P).
A matching ˆ µ dominates another matching µ via a blocking coalition
ˆ M ∪ ˆ W of men and women such that ˆ µ( ˆ M ∪ ˆ W) = ˆ M ∪ ˆ W, ˆ µ(µ(ˆ m)) = µ(ˆ m)
for any ˆ m ∈ ˆ M if µ(ˆ m) / ∈ ˆ W ∪ {ˆ m}, ˆ µ(µ( ˆ w)) = µ( ˆ w) for any ˆ w ∈ ˆ W if
µ( ˆ w) / ∈ ˆ M ∪ { ˆ w}, ˆ µ(i) = µ(i) for any i / ∈ ˆ M ∪ ˆ W ∪ µ( ˆ M ∪ ˆ W), and ˆ µ >i µ
for all i ∈ ˆ M ∪ ˆ W. In the above deﬁnition, members of the blocking coalition
can only be matched within the coalition. In addition, the previous mate, if
exists, of any agent in the blocking coalition becomes single under the new
matching unless he or she is inside the blocking coalition, too. Moreover, the
mates and marital status of all other agents are unchanged.
The set of all matchings dominated by no other matching is called the
core and denoted by C(M,W,P).
For a given society N, let H ⊂ MN be a subset of available matchings.
We say that H is rationalizable inside the stable set if there exists a pref-
erence proﬁle P such that H ⊂ S(M,W,P). Similarly, we say that H is
rationalizable inside the core if there exists a preference proﬁle P such that
H ⊂ C(M,W,P).
We simply note that a set H ⊂ MN is rationalizable inside the core
only if it is rationalizable inside the stable set. Echenique (2008) shows that
under independent preferences MN is not rationalizable inside the stable set
(equalling the core) if the number of men and the number of women are the
same and at least three. We extend this result in our ﬁrst proposition.
33 Results
Proposition 1. For any society N satisfying |M||W| ≥ 2 and having strict
and interdependent preferences, MN is not rationalizable inside the stable
set (hence not rationalizable inside the core).
Proof. Suppose, MN is rationalizable inside the stable set by some pref-
erence proﬁle P; i.e., MN ⊂ S(M,W,P). Let µs denote the matching at
which every agent is single. Pick any (m,w) ∈ M × W. Denote by µs
m,w the
matching at which (m,w) is the unique married couple. Then, µs
m,w >m µs
and µs
m,w >w µs by the assumed stability of µs
m,w. This implies that µs can-
not be in S(M,W,P), a contradiction.
Proposition 1 establishes that the whole set of matchings cannot be ratio-
nalizable, hence our matching model is testable. As the proof of the propo-
sition clearly shows, what drives this impossibility result is the presence of
the matching, called µs, where every individual is single inside the set of all
matchings, MN. In fact, the following example shows that one cannot argue
that MN\{µs} is not rationalizable.
Example 1. Consider M = {m1,m2} and W = {w1}. The three possible
matchings are denoted by µ1, µ2, and µ3. At µ1 and µ2, w1 is matched to m1
and m2 respectively, while at µ3 every agent is single. Let the preferences be
Pm1 = µ2µ1µ3, P m2 = µ1µ2µ3, and Pw1 = µ1µ2µ3. It is easy to check that
S(M,W,P) = C(M,W,P) = {µ1,µ2}. So, H = {µ1,µ2} is rationalizable
inside the core (hence inside the stable set).
It is then natural to ask here which proper subsets of MN can be ratio-
nalizable. When the preferences are independent, Echenique (2008) is able
to show that any set of matchings in which no agent is matched with the
same partner under diﬀerent matchings is rationalizable. Below, we will es-
tablish a similar result under interdependent preferences. But, we have to
4ﬁrst introduce the following deﬁnitions.
For any society N with the preference proﬁle P, we call a proper subset
V of matchings MN top-matching collection if V is nonempty, and for all
i ∈ N we have Pi[k] ∈ V for all k ∈ {1,2,...,|V|}. Given a society N and
an agent i ∈ N, two matchings µ,µ￿ ∈ MN are called connected by agent i
if µ(i) = µ￿(i) and unconnected by agent i otherwise. Given a society N and
a coalition T of agents in N, a matching µ￿ is reachable by T from another
matching µ if the set of all individuals that connect µ to µ￿ is N\T. Let
R(µ,µ￿) denote the unique coalition by which µ￿ is reachable from µ.
Our main result, Proposition 2 below, shows that any collection of match-
ings, any two members of which are unconnected by each member of the
society, is rationalizable inside the core. To give the intuition underlying this
result, we will simply sketch the proof here before its formal introduction.
When the elements of a collection of matchings are pairwise unconnected,
the unique blocking coalition that is capable to reach from one matching to
another one always involves the smallest of the set of men and the set of
women, say without loss of genarality the set of men. Then it suﬃces for us
to ﬁnd a preference proﬁle under which (i) a given collection of matchings will
be a top-matching collection and (ii) for any two matchings inside the given
collection, there will always exist a man as a member of the corresponding
blocking coalition who will be in conﬂict with the rest of the coalition as to
the comparison of the two matchings. In the proof of Proposition 2, we use
the well-known ‘Condorcet cycle’ over the collection of matchings commonly
top-ranked by all agents in the society so as to construct a preference proﬁle
that rationalizes a given collection of pairwise-unconnected matchings inside
the core.
Proposition 2. For any society N satisfying |M||W| ≥ 2 and having
strict and interdependent preferences, consider H ⊂ MN such that |H| ≤
min{|M|,|W|} and no pair of matchings µk,µl ∈ H are connected by any
agent in N. Then H is rationalizable inside the core (hence inside the stable
5set).
Proof. Consider any society N satisfying |M||W| ≥ 2 and having strict and
interdependent preferences. Assume without loss of generality that |M| ≤
|W|. Pick any H = {µ1,µ2,...,µH} ⊂ MN such that no pair of matchings
µk,µl ∈ H are connected by any agent in N.
Now, enumerate agents from 1 to |N|, and let M = {1,2,...,|N|−|W|},
i.e. individuals enumerated with the smallest |M| numbers are all men. Let
P be a preference proﬁle such that P i[k] = µl with l = (k + i − 2)mod H
for all i ∈ {1,...,H} and k ∈ {1,2,...,H}. Explicitly writing the ﬁrst H
components of P as
P 1 = µ1 µ2 ... µH−1 µH ...
P 2 = µ2 µ3 ... µH µ1 ...
... ... ... ... ... ... ... ...
P H = µH µ1 ... µH−2 µH−1 ...
we cannotice the (well-known Condorcet) cycle overthe top-H ranked match-
ings. Also, let Pi = P 1 for all i ∈ {H + 1,...,|N|}.
Obviously, H is a top-matching collection under the constructed prefer-
ence proﬁle. So, no coalition of individuals can block any matching in H via
any other matching in MN\H. It is obvious that C(M,W,P) = H if |H| = 1.
Now, suppose |H| ≥ 2. Pick any two matchings µ,µ￿ ∈ H such that µ ￿= µ￿.
We have R(µ,µ￿) ⊃ M ⊃ {1,2,...,H}, since H ≤ |M| = min{|M|,|W|} by
assumption. Then, by the construction of ￿P1,...,PH￿, there always exists
an agent (a man) in R(µ,µ￿) who prefers µ to µ￿ and withstands the coalition
R(µ,µ￿). Since µ and µ￿ were arbitrary, we have H ⊂ C(M,W,P).
Example 2. Let M = {m1,m2} and W = {w1}, and consider the three pos-
sible matchings µ1, µ2, and µ3 as deﬁned in Example 1. The sets H1 = {µ1}
and H2 = {µ2} satisfy the connectedness hypothesis in Proposition 2. More-
over, |H1| = |H2| = 1 = min{|M|,|W|}. Therefore H1 and H2 are both
6rationalizable by Proposition 2. Indeed, the preference proﬁles that rational-
ize H1 and H2 respectively place µ1 and µ2 at the top position for each agent.
We should ﬁnally remark that the studied refutable matching model is
not exactly identiﬁable, as similarly to Echenique (2008) there may exist
many diﬀerent preference proﬁles that rationalize some sets of matchings.
4 Concluding Remarks
In this paper, we have showed that Mumcu and Saglam’s (2008) marriage
model with externalities is refutable, and hence it has testable implications
(Proposition 1). We have also established that if a collection of matchings is
not rationalizable inside the core, then some agents must have the same mate
under more than one matching (Proposition 2). We should here emphasize
that our second result simply characterizes collections of matchings that are
not rationalizable. However, a suﬃciency result such as Proposition 2 is still
valuable, as already remarked by Echenique (2008) in his framework of inde-
pendent preferences, since it has an important implication for empirical tests
of matching theory at hand, requiring some pairs of agents to be identiﬁed
under more than one matching in the available data.
References
Echenique F (2008) What matchings can be stable? The testable implications
of matching theory. Mathematics of Operations Research 33(3), 757-768.
Gale D, Shapley LS (1962) College admissions and the stability of marriage.
Am Math Mon 69, 9-15.
Hafalir IE (2008f) Stability of marriage with externalities. International
Journal of Game Theory, forthcoming.
Mumcu A, Saglam I (2007) The core of a housing market with external-
ities. Economics Bulletin 3(55), 1-5.
Mumcu A, Saglam I (2008) Characterizing stable one-to-one matchings under
7interdependent preferences. Working Papers 0806, TOBB University of
Economics and Technology, Department of Economics.
Roy Chowdhury P (2004) Marriage markets with externalities. Indian Sta-
tistical Institute, Planning Unit, New Delhi Discussion Papers 04-11.
Sasaki H, Toda M (1996) Two-sided matching problems with externalities.
Journal of Economic Theory 70(1), 93-108.
8

Rationalizability of one-to-one matchings with externalities

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics