In this paper we prove the impossibility of stability rules that satisfy a concept weaker than nonbossiness. Stability and nonbossiness are essential to matching theory. However, Kojima Fuhito(2010) shows that a matching mechanism that is both stable and nonbossy dose not exist. We define a new concept that is weaker than nonbossiness and consider whether or not stability and the new concept are compatible. Unfortunately, we show that these properties are incompatible.matching, stability, non-damaging bossy, impossibility theorem.

Taisuke Matsubae

English

Research Papers in Economics

     
 
 
  
  
Volume 30, Issue 3 
  
Impossibility of Stable and Non-damaging bossy Matching Mechanism 
  
 
 
Taisuke Matsubae  
Competition Policy Research Center, Japan Fair Trade Commission 
Abstract 
In this paper we prove the impossibility of stability rules that satisfy a concept weaker than nonbossiness. Stability and 
nonbossiness are essential to matching theory. However, Kojima Fuhito(2010) shows that a matching mechanism that 
is both stable and nonbossy dose not exist. We define a new concept that is weaker than nonbossiness and consider 
whether or not stability and the new concept are compatible. Unfortunately, we show that these properties are 
incompatible. 
I am grateful to Fuhito Kojima for insightful discussion. I am thanks to Francis Bloch and an anonymous referee. 
Citation: Taisuke Matsubae, (2010) ''Impossibility of Stable and Non-damaging bossy Matching Mechanism'', Economics Bulletin, Vol. 30 
no.3 pp. 2092-2096. 
Submitted: Apr 26 2010.   Published: August 12, 2010. 
 
     1. Introduction
Matching is one of the important functions of markets. Decisions
regarding the person to be selected for a job, the applicant to be ad-
mitted to a school, the person someone chooses to marry, etc. are
problems that are very important for people who are involved in them.
This is because the outcomes of these problems have a great eﬀect on
their lives and careers. The theory of two-sided matching, introduced
by Gale and Shapley (1962), analyzes such matching problems between
two types of agents, such as between workers and ﬁrms, applicants and
universities, and men and women.
One of the most attractive applications of the matching theory is
the designing of mechanisms that determine a matching in any mar-
ket. Empirical studies have shown that mechanisms satisfying some
properties often succeed and those not satisfying the properties often
fail in real world applications. See Roth (2002) for evidence.
When designing mechanisms in matching markets, stability plays a
central role in the theory. A mechanism is stable if no agent and no
pair of agents have the incentive to deviate from the outcome produced
by the mechanism. If some agents deviate from the outcome, the other
agents may not participate in the market; thus, it is important to
consider this property at the time of desgining mechanisms.
The concept of nonbossiness was introduced by Satterthwaite and
Sonnenschin (1981). This property is important in many allocation
problems. A mechanism is nonbossy if an agent cannot change allo-
cation of other agents without changing her own allocation. Kojima
Fuhito (2010)insists that nonbossiness requires an aspect of fairness,
because it may be unfair for an agent to be aﬀected by changes of
reported preferences of someone else even though the change has no
consequence on the allocation of the latter. Further, if a mechanism is
not nonbossy, it may invite strategic manipulation.
1
Although stability and nonbossiness are important properties when
a designer considers a mechanism, Kojima Fuhito (2010) shows that
these properties are incompatible in matching markets.
We study the possibility of a stable mechanism that an agent makes
the allocation of other agents better oﬀ even if she inﬂuences the allo-
cation of other agents without changing her own allocation. To do so,
we deﬁne a weaker concept than nonbossy called non-damaging bossy.
Thus, nonbossiness implies non-damaging bossy. A mechanism is non-
damaging bossy if an agent cannot make the allocation of other agents
worse oﬀ unless doing so also changes her own allocation.
1See Kojima Fuhito (2010) with regard to this discussion.
12
Unfortunately, we show that there does not exist such a mechanism.
Thus, the stable mechanism cannot avoid the situation where an agent
makes the allocation of some agent worse oﬀ without changing her own
allocation.
2. Model
We consider a (one to one) matching problem with tuple (S,C,P).
Let S and C be ﬁnite and disjoint sets of students and colleges, re-
spectively. Each student s 2 S has a strict, transitive, and complete
preference Ps over C [ fsg; each college c 2 C has a strict, transitive,
and complete preference Pc over S [ fcg. For each s 2 S, let Ps be
a set of all possible preferences over C [ fsg and similarly, for each
c 2 C, let Pc be a set of all possible preferences over S [fcg. We write
P 2 P =
∏
i∈S∪C Pi. A matching µ is a mapping from the set S [ C
onto itself and satisfying (i) for all s 2 S, µ(s) 2 C [ fsg; (ii) for all
c 2 C, µ(c) 2 S [ fcg; and (iii) for all i 2 S [ C, µ(µ(i)) = i.
A matching µ is individually rational at preference proﬁle P if µ(i)Pii
or µ(i) = i for all i 2 S [ C. A blocking pair of µ at preference proﬁle
P is a pair fs,cg 2 S  C such that cPsµ(s) and sPcµ(c). A matching
µ is stable at preference proﬁle P if it is individually rational and there
exists no blocking pair of µ.
Let M be the set of all possible matchings on S[C. A mechanism is a
procedure used to determine a matching for each matching market, that
is, a mechanism on P is a mapping ϕ from P to M. A stable mechanism
is a mechanism that selects a matching that is stable with respect to
the submitted preference proﬁle. Gale and Shapley (1962) show the
existence of a stable mechanism. They propose deferred acceptance
algorithms, which ﬁnd stable matchings for all preference proﬁles.
3. Result
Kojima Fuhito (2010) proves that there does not exist a mechanism
that is stable and nonbossy. The concept of nonbossoiness was in-
troduced by Satterthwaite and Sonnenschin (1981). A mechanism is
nonbossy if an agent cannot change the allocation of other agents with-
out changing her own allocation. This concept is formally deﬁned as
follows.
Denition 3.1. A mechanism ϕ is nonbossy if, for any P and P ′
i,
ϕi(P ′
i,P−i) = ϕi(P) implies ϕ(P ′
i,P−i) = ϕ(P).
Remark 3.2 (Theorem 1. (Kojima Fuhito (2010))). There dose not
exist a mechanism that is stable and nonbossy.3
We study the possibility of a mechanism that is stable and a concept
that is weaker than nonbossiness. The weaker concept is deﬁned as
follows.
Denition 3.3. A mechanism ϕ is non-damaging bossy if, for any P
and P ′
i, ϕi(P ′
i,P−i) = ϕi(P) implies ϕ(P ′
i,P−i)Rϕ(P).
A mechanism is non-damaging bossy if an agent does not make the
allocation of other agents worse oﬀ without changing her own alloca-
tion.
Remark 3.4. Nonbossy implies a Non-damaging bossiness.
Unfortunately, we show that there does not exist a mechanism that
satisﬁes stable and non-damaging bossy. We prove the statement using
a example borrowed from Kojima Fuhito (2010).
Theorem 3.5. There does not exist a mechanism that is stable and
non-damaging bossy.
Proof. Consider a market with three students and colleges with prefer-
ences P given by
Ps1 : c2,c3,c1,s1;
Ps2 : c2,c3,c1,s2;
Ps3 : c1,c2,c3,s3;
Pc1 : s1,s2,s3,c1;
Pc2 : s3,s2,s1,c2;
Pc3 : c3;
where Ps1 indicates that her ﬁrst choice is to be matched to college c2,
her second choice is to be matched to college c1, her third choice is to
be matched to college c3, and her forth choice is to remain unmatched,
for example. In this market, there exists unique stable matching ϕ(P)
given by
ϕ(P) =
(
c1 c2 c3  
s1 s3   s2
)
which means that c1 is matched to s1, c2 is matched to s3, and c2 and
s2 are unmatched. Consider P ′
s2 given by P ′
s2 : s2. Then, there are two
stable matching, µ and µ′, given by
µ =
(
c1 c2 c3  
s3 s1   s2
)4
and
µ
′ =
(
c1 c2 c3  
s1 s3   s2
)
respectively. Therefore, any stable mechanism must choose one of µ or
µ′ when the preference P ′ = (P ′
s2,P−s2) is stated.
Suppose the mechanism chooses µ. Thus, ϕ(P ′
s2,P−s2) = µ. In this
case, we have ϕs2(P ′
s2,P−s2) = ϕs2(P). However, for example, agent c1
prefers ϕc1(P) to ϕc1(P ′
s2,P−s2). Thus, ϕ is non-damaging bossy.
On the other hand, suppose the mechanism chooses µ′. Thus, ϕ(P ′
s2,P−s2) =
µ′. Now, consider P
′′
c3 : s1,s3,c3. Then, a stable mechanism ϕ(P ′
s2,P
′′
c3,P−s2;c3)
is induced by
ϕ(P
′
s2,P
′′
c3,P−s2;c3) =
(
c1 c2 c3  
s3 s1   s2
)
Therefore, we have that ϕc3(P ′
s2,P
′′
c3,P−s2;c3) = ϕc3(P ′
s2,P−s2). How-
ever, an agent c1 prefers ϕc1(P ′
s2,P−s2) to ϕc1(P ′
s2,P
′′
c2,P−s2;c2). Thus, ϕ
is not non-damaging bossy. 
Kojima Fuhito (2010) proved the impossibility of mechanism that is
stable and nonbossy. Thus, he showed that stable mechanisms cannot
avoid the situation where an agent inﬂuences allocation of other agents
without changing her own allocation. Moreover, we prove the impossi-
bility of mechanism that is stable and non-damaging bossy. Thus, sta-
ble mechanism cannot avoid the even situation where an agent makes
allocation of some agent worse oﬀ without changing her own alloca-
tion. We interpret that this result is more negative result than the
one derived by Kojima Fuhito (2010) because there does exist a stable
mechanism where an agent makes allocation of other agents worse oﬀ
without changing her own allocation.
References
Gale, D., Shapley, L. S. (1962) “College admissions and the stability
of marriage” American Mathematical Monthly 69: 9–15
Kojima, F.(2010) “Impossibility of stable and nonbossy matching
mechanism” Economic Letters 107: 69–70.
Roth, A. E. (2002) “The economist as engineer: Game theory, experi-
mentation, and computation as tools for design economics” Econo-
metrica 70: 1341–1378.
Satterthwaite, M.and Sonnenschein, H. (1981) “Strategy-Proof Alloca-
tion Mechanisms at Diﬀerentiable Points” Review of Economic Stud-
ies 48: 587–597.

College admissions and the stability of marriage”

Strategy-Proof Allocation Mechanisms at Diﬀerentiable Points”

The economist as engineer: Game theory, experimentation, and computation as tools for design economics”

Impossibility of Stable and Non-damaging bossy Matching Mechanism

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Impossibility of Stable and Non-damaging bossy Matching Mechanism

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