Economics Letters 150 (2017) 135–137
Contents lists available at ScienceDirect
Economics Letters
journal homepage: www.elsevier.com/locate/ecolet
Incompatibility between stability and consistency
Mustafa Oğuz Afacan a, *, Umut Mert Dur b
a
b
Faculty of Arts and Social Sciences, Sabanci University, Orhanli, Tuzla, 34956, Istanbul, Turkey
Poole School of Management, North Carolina State University, 27695, Raleigh, NC, USA
highlights
• Consistency is a desirabe robustness property.
• Stability is the core notion in matching problems.
• In the one-sided matching context, we show that no stable mechanism is consistent.
article
info
Article history:
Received 8 September 2016
Received in revised form 8 November 2016
Accepted 18 November 2016
Available online 25 November 2016
JEL classification:
C78
I28
a b s t r a c t
Stability is a main concern in the school choice problem. However, it does not come for free. The literature
shows that stability is incompatible with Pareto efficiency. Nevertheless, it has been ranked over Pareto
efficiency by many school districts, and thereof, they are using stable mechanisms. In this note, we reveal
another important cost of stability: ‘‘consistency’’, which is a robustness property that requires from a
mechanism that whenever some students leave the problem along with their assignments, the remaining
students’ assignments do not change after running the mechanism in the smaller problem. Consequently,
we show that no stable mechanism is consistent.
© 2016 Elsevier B.V. All rights reserved.
Keywords:
Matching theory
Market design
Consistency
Stability
Incompatibility
School choice
1. Introduction
Balinski and Sönmez (1999) interpret stability as a desirable
fairness requirement in the school choice context. Since then, it
has been a major desideratum in both practical and theoretical
one-sided matching problems, such as school choice, dorm assignment, and house allocation. However, stability comes at the
expense of efficiency. The literature shows both theoretically and
empirically that its efficiency cost can be severe (see Kesten (2010)
and Abdulkadiroğlu et al. (2009)). However, many school districts
have been using stable mechanisms (e.g., Boston, New York City,
Chicago, and Wake County (see Abdulkadiroğlu et al. (2005b, a);
Pathak and Sönmez (2013), and Dur et al. (2016)).
In this note, we reveal another important cost of stability.
‘‘Consistency’’ is a well-known robustness property, which re-
*
Corresponding author.
E-mail addresses: [email protected] (M.O. Afacan), [email protected]
(U.M. Dur).
http://dx.doi.org/10.1016/j.econlet.2016.11.022
0165-1765/© 2016 Elsevier B.V. All rights reserved.
quires from a mechanism to assign the same schools to the remaining students after some students leave the problem with
their assignments.1 As already discussed in literature, consistency
is a highly desirable property because its violation incentivizes
students to manipulate the centralized mechanism through precommitments.2 This type of strategic behavior may, in turn, lead
sincere students to appeal the outcome. Similarly, for schoolchoice systems with a secondary round such as appeals, under
a consistent mechanism, students do not have the incentive to
strategically file an appeal after the main round (Kojima and
Ünver, 2014). Moreover, the importance of consistency for stable
1 See Thomson (forthcoming) for a detailed survey on consistency. Sönmez and
Ünver (2010); Velez (2014); Kojima and Ünver (2014); Dur (2013), and Klaus and
Klijn (2013) provide characterizations of consistent mechanisms.
2 Specifically, under a non-consistent mechanism, a group of students may come
together, and for the sake of others in the group, some of them can pre-arrange with
the school that he would already be matched with under the mechanism outcome.
136
M.O. Afacan, U.M. Dur / Economics Letters 150 (2017) 135–137
mechanisms is exacerbated because of the pervasive welfare effects of pre-commitments on students under stable mechanisms
(see Afacan (2013)).3
It is well-known that the celebrated student-proposing deferred acceptance (DA) mechanism (Gale and Shapley, 1962) is not
consistent. However, this does not imply a general incompatibility
between stability and consistency. In this note, we show that they
are incompatible in that there is no stable mechanism that is consistent. Hence, this result reveals another cost of stability, as well
as its inefficiency. On the other hand, serial dictatorship is Pareto
efficient and consistent (Ergin, 2000), that is, Pareto efficiency does
not have a consistency cost, unlike stability.
Our paper is in the same spirit as that of Kojima (2010), which
shows the incompatibility of non-bossiness and stability in twosided matching markets. However, Kojima (2010)’s impossibility
result does not hold in one-sided matching markets. For instance,
school-proposing DA is non-bossy in the school choice framework (Afacan and Dur, 2016). Hence, although consistency implies
non-bossiness (Dogan and Klaus, 2016), our result is not implied
by Kojima (2010).
In the following section, we present the school choice model. In
Section 3, we formally define consistency and provide our impossibility result.
2. Model
A school choice problem is a tuple (I , S , q, P , ≻) where
• I = {i1 , i2 , . . . , in } is the finite set of students,
• S = {s1 , s2 , . . . , sm } is the finite set of schools,
• q = (qs )s∈S is the quota vector where qs is the number of
available seats in school s,
• P = (Pi )i∈I is the preference profile where Pi is the strict preference of student i over the schools and being unassigned
option, which is denoted by ∅,
• ≻= (≻s )s∈S is the priority profile where ≻s is the strict
priority relation of school s over I.
Let P be the set of strict preference relations and q∅ = ∞. In the
remainder of the paper, we fix S and ≻, and represent the problem
with (I , q, P). For every student i, we write Ri for the at-least-asgood-as relation associated with Pi .
A matching µ : I → S ∪ {∅} is a function such that |µ(i)| = 1
and |µ−1 (s)| ≤ qs for all i ∈ I and s ∈ S. Let M denote the set
of matchings. With a slight abuse of notation, we use µi instead of
µ(i) henceforth.
A matching µ is individually-rational if, for any student i,
µi Ri ∅. A matching µ is non-wasteful if there does not exist a
student–school pair (i, s) such that s Pi µi and |µ−1 (s)| < qs . A
matching µ is fair if there does not exist a student–school pair (i, s)
such that s Pi µi and i≻s j for some j ∈ µ−1 (s). A matching µ is
stable if it is individually-rational, non-wasteful, and fair.
A mechanism Φ is a systematic way that selects a matching
for each problem (I , q, P). We write Φ (I , q, P) and Φi (I , q, P) to
denote the matching selected by Φ , and the assignment of student
i, respectively. A mechanism Φ is stable if, for any problem (I , q, P),
Φ (I , q, P) is stable.
3. Result
A mechanism is consistent if, whenever some students are
removed with their assignments, the mechanism selects the same
assignments for the remaining students. We formally define it
below.
3 Consistency is important for problems outside matching theory as well. For
instance, Serrano (1995) and Krishna and Serrano (1996) reveal its importance
for the allocative performance of mechanisms in various strategic settings.
Definition 1. A mechanism Φ is consistent if, for any problem
(I , q, P), Ī ⊂ I, and i ∈ Ī, Φi (I , q, P) = Φi (Ī , q̄, (Pi )i∈Ī ) where, for
any school s, q̄s = qs − |j ∈ I \ Ī : Φj (I , q, P) = s|.4
Now we are ready to present our impossibility result.
Theorem 1. There does not exist a mechanism that is stable and
consistent.
Proof. On the contrary, suppose Φ is a stable and consistent mechanism. Consider a problem where S = {s1 , s2 , s3 , s4 },
I = {i1 , i2 , i3 , i4 , i5 }. Each school has unit capacity, that is, q =
(1, 1, 1, 1). Consider the following preference profile P and priorities ≻:
Pi1
Pi2
Pi3
Pi4
Pi5
≻s 1
≻s 2
≻s 3
≻s 4
s2
s3
s1
s1
s2
s3
s4
s4
∅
:
∅
:
i2
i4
i1
i3
i1
i3
i4
i2
i5
∅
:
i1
i2
i4
i3
:
:
:
:
:
:
In this problem, there exists a unique stable matching µ such
that µi1 = s2 , µi2 = s1 , µi3 = s3 , µi4 = ∅, and µi5 = s4 . Since
Φ is stable, Φ (I , q, P) = µ. Let us remove students i3 , i4 , and i5
with their assignments. By consistency, Φ assigns i1 to s2 and i2
to s1 when the remaining students and seats are considered, i.e.,
Φi1 (Ī , q̄, (Pi )i∈Ī ) = µi1 = s2 , and Φi2 (Ī , q̄, (Pi )i∈Ī ) = µi2 = s1 where
Ī = {i1 , i2 } and q̄ = (1, 1, 0, 0).
Now consider the following preference profile P ′ :
Pi′
Pi′
Pi′
Pi′
Pi′
s2
s3
s1
s1
s2
s4
s2
s4
∅
:
∅
:
∅
:
1
2
:
3
4
5
:
Let Pi = Pi1 , Pi = Pi2 , and q′ = (1, 1, 0, 1). At problem
2
1
(I , q′ , P ′ ), there exists a unique stable matching ν such that νi1 = s1 ,
νi2 = s2 , νi3 = ∅, νi4 = ∅, and νi5 = s4 . Since Φ is stable,
Φ (I , q′ , P ′ ) = ν . Let us remove students i3 , i4 , and i5 with
their assignments. By consistency, Φ assigns i1 to s1 and i2 to
s2 when the remaining students and seats are considered, i.e.,
Φi1 (Ī , q̄′ , (Pi′ )i∈Ī ) = νi1 = s1 , and Φi2 (Ī , q̄′ , (Pi′ )i∈Ī ) = νi2 = s2 where
Ī = {i1 , i2 } and q̄′ = (1, 1, 0, 0).
Note that (Ī , q̄′ , (Pi′ )i∈Ī ) = (Ī , q̄, (Pi )i∈Ī ) but Φ (Ī , q̄′ , (Pi′ )i∈Ī ) ̸ =
Φ (Ī , q̄, (Pi )i∈Ī ), which is a contradiction. ■
′
′
Acknowledgments
We are grateful to the associate editor and anonymous referee
for their comments and suggestions.
References
Abdulkadiroğlu, A., Pathak, P.A., Roth, A.E., 2005a. The new york city high school
match. Amer. Econ. Rev. Pap. Proc. 95, 364–367.
Abdulkadiroğlu, A., Pathak, P.A., Roth, A.E., 2009. Strategy-proofness versus efficiency in matching with indifferences: redesigning the NYC high school match.
Amer. Econ. Rev. 99, 1954–1978.
Abdulkadiroğlu, A., Pathak, P.A., Roth, A.E., Sönmez, T., 2005b. The boston public
school match. American Economic Review Papers and Proceedings 95, 368–
372.
Afacan, M.O., 2013. The welfare effects of pre-arrangements in matching markets.
Econom. Theory 53(1), 139–151.
Afacan, M.O., Dur, U., 2016. When preference misreporting is harm[less]ful? Mimeo.
Balinski, M., Sönmez, T., 1999. A tale of two mechanisms: student placement.
J. Econom. Theory 84, 73–94.
4 Alternatively, we can define q̄ = |j ∈ Ī : Φ (I , q, P) = s|. Our result holds
s
j
under this definition too.
M.O. Afacan, U.M. Dur / Economics Letters 150 (2017) 135–137
Dogan, B., Klaus, B., 2016. Object allocation via immediate acceptance: characterizations and an affirmative action application. Mimeo.
Dur, U., 2013. A characterization of the top trading cycles mechanism in the school
choice problem. Mimeo.
Dur, U., Hammond, R. and Morrill, T., 2016. Identifying the harm of manipulable
school-choice mechanisms. Mimeo.
Ergin, H., 2000. Consistency in house allocation problems. J. Math. Econom. 34,
77–97.
Gale, D., Shapley, L.S., 1962. College admissions and the stability of marriage. Amer.
Math. Monthly 69, 9–15.
Kesten, O., 2010. School choice with consent. Q. J. Econ. 125, 1297–1348.
Klaus, B., Klijn, F., 2013. Local and global consistency properties for student placement. J. Math. Econom. 49, 222–229.
Kojima, F., 2010. Impossibility of stable and nonbossy matching mechanisms.
Econom. Lett. 107, 123–135.
137
Kojima, F., Ünver, M.U., 2014. The boston school-choice mechanism: an axiomatic
approach. Econom. Theory 55, 515–544.
Krishna, V., Serrano, R., 1996. Multilateral bargaining. Rev. Econom. Stud. 63, 61–80.
Pathak, P.A., Sönmez, T., 2013. School admissions reform in Chicago and England:
Comparing Mechanisms by their vulnerability to manipulation. Amer. Econ.
Rev. 103, 80–106.
Serrano, R., 1995. A market to implement the core. J. Econom. Theory 67, 285–294.
Sönmez, T., Ünver, M.U., 2010. House allocation with existing tenants: a characterization. Games Econom. Behav. 69, 425–445.
Thomson, W.L., 2008. Fair allocation rules, in: K. Arrow, A. Sen, K. Suzumura
(Eds.), (forthcoming), In Handbook of Social Choice and Welfare, North-Holland,
Amsterdam, New York.
Velez, R., 2014. Consistent strategy-proof assignment by hierarchical exchange.
Econom. Theory 56, 125–156.