Discussion Paper No.321
AN AFFIRMATIVE ACTION POLICY AS STRATEGY ON
TWO-SIDED MATCHING MARKET

Taisuke Matsubae
Chuo University

Faculty of Economics
Chuo University
October 2019

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, JapanAN AFFIRMATIVE ACTION POLICY AS STRATEGY ON TWO-SIDED MATCHING
MARKETTAISUKE MATSUBAE
Faculty of Economics, Chuo University
Email: matsubae@tamacc.chuo-u.ac.jp
A BSTRACT. This paper investigates a market situation in which firms freely can choose whether they implement an affirmative action policy or not. We show that affirmative action policies can be abused by firms.
Specifically, we demonstrate that there are market situations where a firm’s implementation of affirmative action policies can employ more desirable workers, but is detrimental to the minority workers in the firm.
JEL Classification number: C70, D61.
Keywords: Matching, Stability, Affirmative Action, Stable Mechanism, Abuse-Proof.

1. I NTRODUCTION
Matching markets arise in many economic environments. Matching markets are two-sided of the market,
called men and women, workers and firms, students and colleges, and interns and hospitals and one sides
want to be matched other sides. Many researchers have begun to investigate the theory of a two-sided
matching market because of its theoretical appeal and the relevance of its design to real-world institutions.
This paper investigates a environment in which firms can strategically choose an affirmative action policy inthe context of a two sided matching market.
The two sided matchign market (also called by many to one matching problem) is pioneered by Gale andShapley (1962). The worker-proposing deferred acceptance algorithm proposed by them has many appealing properties. A matching induced by the worker-proposing deferred acceptance algorithm is stable andPareto dominates any other stable matching. Moreover, the woker-proposing deferred acceptance algorithm
has a desirable property with regard to wokers’ strategically action. Thus, it makes the truthful reporting ofpreferences a dominant strategy for all wokers. 1
On the other hand, there are many studies focused on firms’ strategically action. Any stable mechanism(including, in this case, the worker-proposing deferred acceptance mechanism) does not make the truthfulreporting of preferences a dominant strategy for every firms. Thus, a stable mechanism can be manipulated
via preference in the case where a firm will gain in terms of true preference by submitting a false preference
instead of its true preference. Roth (1982) showed that any stable mechanism can be manipulable via
preferences for firms. Firm’s capacities are also private information; a mechanism can be manipulable interms of capacities if a firm can gain in terms of true preference by underreporting its true capacities. Ehlers(2010) shows that manipulation via capacities can be equivalently described by two types of manipulation
via capacities. He shows that no mechanism exists that is stable and excludes the possibility of non-TypeII-manipulation via capacities.2 3 He also shows if mechanism is manipulable via capacities, then themechanism is manipulable via preferences.
In many cases (i.e. college admission or job search), affirmative action policies have been playing animportant role in the pursuit of racial desegregation. Affirmative action is the attempt to promote equal
opportunities for all groups of society. It is often instituted in government and educational settings to ensure
I am thank Fuhito Kojima for discussion in early stages of this research.
1
See Roth and Sotomayor (1990) for more detailed discussion.
2
Non-Type II-manipulation means that firms with no vacant positions are unable to manipulate the situation by dropping someof its filled positions.
3
Sönmez (1997) showed that when there are at least three workers and two firms, there exists no mechanism that is stable andnon-manipulable via capacities.
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that minority groups within a society are included in all programs. There are many studies that have examined affirmative action. (e.g. Abdulkadiroğlu (2005) and Kojima (2012)) Abdulkadiroğlu (2005) showed
that, in the college preference domain, including in the context of the college admission problem, the student(worker)-proposing deferred acceptance mechanism makes the truthful revelation of preferences a dominant
strategy for every student.
In Kojima (2012)’s investigation of the welfare effects of affirmative action policies in school choice,he found that there are certain market situations in which an affirmative action policy inevitably hurt every
minority students. 4
In this paper, we conduct an investigation, against this background, to examine whether firms can adapt
their affirmative action policies to promote equal opportunity for minority workers. To examine this, we
assess whether firms implement affirmative action policies to employ more desirable workers or not. Thus,
we consider the case in which a firm can strategically implement affirmative action policies. We define that afirm abuses its affirmative action policies when the implementation of the affirmative action policies benefits
the firm but is to the detriment of the minority workers in the firm. We define a mechanism as abuse-proof if
the firm cannot benefit from the policy and minority workers are not put to a disadvantage, can benefit fromthe policy and minority workerss are not put a disadvantage, or can benefit from the policy and minority
workers are put an advantage. The analytical approach of this paper follows in the tradition of impossibility
studies from the existing literature. For example, see Roth (1982), Sönmez (1997), Sönmez (1999), andKojima (2012).
The paper is organized as follows: Section 2 introduces the two-sided matching market-stability. Section
3 describes various affirmative action policies and defines what we mean by misuse-proof; we also present
our results in terms of the stable mechanism examined. In Section 4, we provide concluding remarks.
2. M ODEL
We consider the following college admission problem with type-specific capacities. We focus on thesimple situation in which there are only two types of majority students and minority students. 5
College admission problem with type-specific capacities, or simply a problem consists of the following terms:(1) a finite set of students S = {s1 , s2 , . . . , sn },(2) a finite set of colleges C = {c1 , c2 , . . . , cm },(3) a students’ preference profile RS = (Rs1 , Rs2 , . . . , Rsn ), where Rs is a preference relation over C andbeing unmatched (being unmatched is denoted by ∅). We assume that preferences are strict. We
write c′ Ps c if and only if c′ Rs c but not cRs c′ . If c ≻s ∅, then c is said to be acceptable to s. The
set of students are partitioned to two subsets; the set SM of majority students and Sm of minority
students;(4) a colleges’ preference profile ⪰C = (⪰c1 , ⪰c2 , . . . , ⪰cm ), where ⪰c is the preference relation of college c ∈ C over S. We assume that the preference relations are strict and responsive with quota. 6 7
We write s′ ≻c s if and only if s′ ⪰c s but not s ≻c s′ . If S′ ≻c ∅, then set of students S′ ⊂ S is said
acceptable for c ;(5) for each c ∈ C, qc = (qc , qMc ) is the capacity of c: The first component qc represents the total capacityof college c, while the second component qM
c represents the type-specific capacity for the majority
students.
Let a problem be G = (S,C, (Rs )s∈S , (⪰)c∈C , (qc )c∈C ).
4

Hafalir, Bumin, Muhammed and Yildirim (2013) considered welfare effect of minority by anothe affimative action poicy. They
showed that there is minority student who admits to better college by their affimative action policy.
5Note that our results do not depend on the assumption that there are more majority students than minority students.
6The preferences ⪰ of college c are responsive with quata q if (a) for all s, s′ ∈ S, if {s} ≻ {s′ }, then for any S′ ⊆ S \ {s, s′ },ccc′S ∪ {s} ≻c S′ ∪ {s′ }, (b) for all s, s′ ∈ S if {s} ≻c ∅, then for any S′ ⊆ S such that |S′ | < qc , S′ ∪ {s} ≻c S′ , and (c) ∅ ≻c S′ for any
S′ ⊆ S with |S′ | > qc .
7
Note that while our results are showed under responsive preference, showing an impossibility with respect to the requirement
immediately implies an impossibility result for a wider requirement.

AN AFFIRMATIVE ACTION POLICY AS STRATEGY ON TWO-SIDED MATCHING MARKET

3

We define the notation of matching introduced by Kojima (2012). A matching µ is a function from theset S ∪C to the set of all subsets of S ∪C such that(1) |µ (s)| = 1 for every students s, and µ (s) = ∅ if s ∈/ µ (c) for all c ∈ C; 8 9(2) For all s ∈ S and c ∈ C, µ (s) = c if and only if s ∈ µ (c);(3) |µ (c)| ≤ qc and µ (c) ⊆ S for any college c;(4) |µ (c) ∩ SM | ≤ qM
c for all c ∈ C.
This matching requires (4) in addition to the standard requirements (1)-(3). (4) means that the number ofmajority students matched to each college c represents its type-specific capacity qMc or fewer. Let M be theset of matching.
We use the definition of stable matching introduced by Kojima (2012). A matching µ is stable if(1) µ (s)Rs ∅ for all s ∈ S, and for all c ∈ C and all s ∈ µ (c), s ⪰ ∅, and(2) if there exists a pair (s, c) ∈ S ×C such that cPs µ G (s), then we have either(a) |µ (c)| = qc and s′ ≻c s for all s′ ∈ µ (c), or′′M(b) s ∈ SM , |µ (c) ∩ SM | = qM
c , and s ≻c s for all s ∈ ( µ (c) ∩ S ).
This definition is standard except for (2b), but later part of (1) is different from Kojima (2012) . Condition(2b) means that when student s is a majority student, even if she prefers college c to the college matched toher and the slots of the college are left, if the slots of c for majority students are filled by students who havehigher preference than her, then she cannot block the matching.
A mechanism is a function ϕ that, for each problem G, associates a matching M . Note that ϕ (G) ∈ M
for any problem G. A mechanism ϕ is stable if ϕ (G) is a stable matching for any given G. We denote college
to which student s match at a problem G as ϕs (G), and the set of students to which college c is matched at a
problem G as ϕc (G).
We denote the stable mechanism as ϕ stable (·). 10
3. A BUSE - PROOF FOR AFFIRMATIVE ACTION POLICIES
We consider a market situation in which a college can freely implements an affirmative action policy.
However, we assume that any college cannot change quota or preference except implementing an affirmative
action policies. For example, by law or rule, the preferences orderings of college is decided.
Let G = (S,C, (Rs )s∈S , (⪰)c∈C , (qc , q−c )) be a problem. We say that another problem G̃ = (S,C, (Rs )s∈S , (⪰M
)c∈C , (q̃c , q−c )) is obtained via college c’s implementation of an affirmative action policy if qc = q̃c , qM
c > q̃cM
and for all c′ ∈ (C \ {c}), qc′ = q̃c′ and qM
c′ = q̃c′ .
Definition 1. Given a mechanism ϕ , a college c abuses the affirmative action policy in problem G if there
exist a problem G̃ and minority student s ∈ (ϕc (G̃) \ SM ) such that ϕc (G̃) ≻c ϕc (G), ϕs (G)Ps c, and G̃ is aproblem in which the college c implements affirmative action policy.
This defines a market situation in which, when a college implements an affirmative action policy, thedecision benefits the college, but is to the detriment of the new minority students who is matched to thecollege in the problem G̃ and is not matched to the college in the problem G. Clearly, the conditions of thismarket situation are entirely contrary to those intended by the architects of affirmative action. Hence, we
consider whether there might be a mechanism that can eliminate such a market situation.
In a situation where, for any problem, no colleges abuse the affirmative action policy under a mechanism,
then we can say the mechanism is abuse-proof for the affirmative action policy. We hope that a stablemechanism is abuse-proof for an affirmative action policy. Unfortunately, the next result shows any stablemechanism is not abuse-proof for affirmative action policy.
Theorem 1. There exists no stable mechanism which is abuse-proof for the affirmative action policy.
8µ (c) = {s ∈ S | µ (s) = c}

Because each student is matched to exactly one school or no school, we will omit set brackets and write µ (s) = c instead ofµ (s) = {c} and µ (s) = s instead of µ (s) = {s}.
10We call a mechanism a stable mechanism if for any problem, the mechanism induce stable matching.
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Proof. The proof is provided via a counter example. Hence, we consider a problem
G = (S,C, (Rs )s∈S , (⪰)c∈C , (qc )c∈C ).
The problem G can be expressed as follows: Let S = {s1 , s2 , s3 , s4 }, C = {c1 , c2 , c3 }, SM = {s1 , s2 , s4 },Sm = {s3 }. Student preferences are given by
Rs1
Rs2
Rs3
Rs4

c1 , c2 , c3 ,
c1 , c2 , c3 ,
c2 , c1 , c3 ,
c1 , c2 , c3 ,

:
:
:
:

where the notational convention is that colleges are listed in order of preferences and colleges not on thepreference list are unacceptable. College preferences and capacities are given by⪰c1: s3 , s2 , s1 , s4

qc1 = (qc1 , qMc1 ) = (2, 2),⪰c2: s1 , s2 , s3 , s4

qc2 = (qc2 , qMc2 ) = (1, 1),⪰c3: s1 , s2 , s3 , s4

qc3 = (qc3 , qMc3 ) = (1, 1),

where the notational convention here is that students are listed in their order of preference: At college c1 , forinstance, student s3 has the highest preference, s1 has the second highest preference, s2 has the third highest
preference, and s4 has the lowest preference.
There exists a unique stable matching ϕ stable (G) in this problem G , which is given by
(
)
c1
c2 c3
stable(G) =,ϕs1 , s2 s3 s4which means that c1 is matched to s1 and s2 , c2 is matched to s3 , and c3 is matched to s4 .
Consider the case in which college c1 implements the affirmative action policy, (q̃c )c∈C = (q̃c1 , qc2 ),where q̃c1 = (2, 1), qc2 = (1, 1), and qc3 = (1, 1). Thus, the problem
G̃ = (S,C, (Rs )s∈S , (⪰c )c∈C , (q̃c )c∈C ),is that college c1 implements the affirmative action policy. In the problem G̃, there is a unique stablematching ϕ stable (G̃) given by
(
)
c1
c2 c3
ϕ stable (G̃) =.
s2 , s3 s1 s4
The minority student s3 in college c1 is strictly worse off under G̃ than they are under G. Moreover,college c1 is strictly better off under G̃ than it would be under G. Therefore, it is clear that the college c1 has
incentive to abuse the affirmative action policy.
□
The result suggests that no stable mechanism can remove the incentive for a college to abuse the affirmative action policy. This result may sound that colleges always abuse affirmative action policy. However, thisintuition is incorrect. We present a matching market such that stable mechanism can help minority students
without allowing the college to abuse their affirmative action policy. To do show it, we borrow the exampleof Kojima (2012).
Example 1 (Example 1 of Kojima (2012)). Let G = (S,C, (Rs )s∈S , (⪰)c∈C , (qc )c∈C ). The problem G is asfollows. Let S = {s1 , s2 , s3 , s4 }, C = {c1 , c2 }, SM = {s1 , s2 , s3 }, and Sm = {s4 }. Student preferences are given
by
Rs1
Rs2
Rs3
Rs4

:
:
:
:

c1 ,
c1 ,
c1 , c2 ,
c2 , c1 .

AN AFFIRMATIVE ACTION POLICY AS STRATEGY ON TWO-SIDED MATCHING MARKET

5

College preferences and capacities are given by⪰c1: s1 , s4 , s2 , s3

qc1 = (qc1 , qMc1 ) = (2, 2),⪰c2: s3 , s4 , s2 , s1

qc2 = (qc2 , qcM2 ) = (1, 1).

The unique stable matching ϕ stable (G) in this problem G is given by
(
)
c1
c2 ∅
stableϕ(G) =s1 , s4 s3 s2
Let us consider the case in which college c1 implements an affirmative action policy, (q̃c )c∈C = (q̃c1 , qc2 ),where q̃c1 = (2, 1) and qc2 = (1, 1). The problem isG̃ = (S,C, (Rs )s∈S , (⪰c )c∈C , (q̃c )c∈C ).
In the problem G̃, there is a unique stable matching ϕ stable (G̃) which is given by
(
)
c1
c2 ∅
stable(G̃) =.
ϕs1 , s4 s3 s2
The minority student in c1 , s4 is not strictly worse off under G̃ than they are under G. In addition, c1 is notbetter off under G̃ than it would be under G. Therefore, c1 does not abuse the affirmative action policy.
Next, we consider the case in which college c2 implements an affirmative action policy, (q′c )c∈C =(qc1 , q′c2 ), where qc1 = (2, 2) and q′c2 = (1, 0). The problem isG′ = (S,C, (Rs )s∈S , (⪰c )c∈C , (q′c )c∈C ).
In the problem G′ , there is a unique stable assignment ϕ stable (G′ ), which is given by
(
)
c1
c2 ∅
stable′ϕ(G ) =.
s1 , s2 s4 s3
The minority student in c2 , s4 is strictly better off under G′ than they are under G. In contrast, college c2
is worse off under G′ than it would be under G. Therefore, c2 dose not abuse the affirmative action policy.
No colleges abuse the affirmative action policy for any stable mechanism. Thus, this example shows that astable mechanism is abuse-proof for an affirmative action policy.
So far, we have seen that colleges will abuse their affirmative action policy under any stable mechanism.
Kojima (2012) defines an alternative affirmative action policy, which he calls preference-based affirmative
˜ c∈C , (qc )c∈C ) is said to have a stronger preference-based
action policy. This situation: G̃ = (S,C, (Rs )s∈S , (⪰)
affirmative action policy than G = (S,C, (Rs )s∈S , (⪰)c∈C , (qc )c∈C ) if, for every c ∈ C and s, s′ ∈ S, s ⪰c s′ and˜ c s′ .
s ∈ Sm imply s⪰
We consider the situation in which a college can freely implement preference-based affirmative action
˜ c , ⪰−c ), (qc )c∈C ), where
policy under any stable mechanism in the following way: Let G̃ be (S,C, (Rs )s∈S , (⪰˜⪰c means that college c’s preference improve one minority student who is ranked lower than a majority
student, while the relative ranking of each student within her own group remains fixed. 11 We define abuse
for the preference-based affirmative action policy by a college as we did before.
Definition 2. A college c abuses the preference-based affirmative action policy in problem G if there
exist a problem G̃ and s ∈ (ϕcstable (G̃) \ SM ) such that ϕcstable (G̃) ≻c ϕcstable (G), ϕsstable (G)Ps c, and G̃ is aproblem in which the college c implements preference-based affirmative action policy.
If, for any problem, no colleges are found to abuse the preference-based affirmative action policy undera certain mechanism, then we say the mechanism is abuse-proof for the preference-based affirmative
action policy. Although the current definition considers whether there is a strategy-proof for college preferences, our requirements are different from those of the mechanism because our conditions also include theminority student’s welfare in the college. If a mechanism holds this definition, then the mechanism naturally
holds the strategy-proof for college preferences, but the inverse dose not hold.
11⪰−c = (⪰c′ )c′ ∈C\{c′ } .

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Theorem 2. There exists no stable mechanism which is abuse-proof for the preference-based affirmative
action policy.
Proof. We will prove this theorem with the example.
Let G = (S,C, (Rs )s∈S , (⪰)c∈C , (qc )c∈C ). The problem G is is expressed thus: Let S = {s1 , s2 , s3 , s4 , s5 },C = {c1 , c2 , c3 }, SM = {s1 , s3 }, Sm = {s2 , s4 , s5 }. Student preferences are given by
Rs1
Rs2
Rs3
Rs4
Rs5

:
:
:
:
:

c3 , c2 , c1 ,
c2 , c1 , c3 ,
c1 , c3 , c2 ,
c1 , c2 , c3 ,c1 .

College preferences are as follows.
⪰c1: s1 , s2 , s3 , s4 , s5 ,qc1 = (qc1 , qMc1 ) = (2, 2),⪰c2: s1 , s2 , s3 , s4 , s5 ,qc2 = (qc2 , qMc2 ) = (1, 1),⪰c3: s3 , s1 , s2 , s4 , s5 ,qc3 = (qc3 , qMc3 ) = (1, 1).

In this problem, there is a unique stable matching ϕ stable (G) given by()
c1
c2 c3 ∅
stable(G) =.
ϕs3 , s4 s2 s1 s5
In the case, college c1 implements the preference-based affirmative action policy as follows:′⪰c1: s1 , s2 , s4 , s5 , s3

qc1 = (qc1 , qMc1 ) = (2, 2).

In problem G̃, a unique stable matching ϕ stable (G̃) is given by()
c1
c2 c3 ∅
stable(G̃) =.
ϕs1 , s4 s2 s3 s5
College c1 is better off under G̃ it would be than under G, but the minority student s2 is worse off underG̃ they would be than under G. Thus, ϕ stable (G̃) is found not to be an abuse-proof for the preference-based
affirmative action policy.
□
Note that this result dose not imply that a college has incentive to abuse the preference-based affirmative
action policy under stable mechanism. The following example shows that there exists a stable mechanism
which is abuse-proof for the preference-based affirmative action policy.
Example 2. Let G = (S,C, (Rs )s∈S , (⪰)c∈C , qc )c∈C ). The problem, G is as follows. Let S = {s1 , s2 , s3 },C = {c1 , c2 }, SM = {s1 }, Sm = {s2 , s3 }. Student preferences are given by
Rs1
Rs2
Rs3

: c2 , c1 ,: c1 , c2 ,: c2 , c1 ,College preferences are as follows.
⪰c1: s1 , s2 , s3

qc1 = (qc1 , qMc1 ) = (1, 1),⪰c2: s1 , s3 , s2

qc2 = (qc2 , qMc2 ) = (1, 1).

In this problem, there is a unique stable matching ϕ stable (G)which is given by
(
)
c c ∅
ϕ stable (G) = 1 2
s2 s1 s3
The case in which a college c1 implements the preference-based affirmative action policy is expressed asfollows,′⪰c1: s2 , s1 , s3

qc1 = (qc1 , qMc1 ) = (1, 1).

AN AFFIRMATIVE ACTION POLICY AS STRATEGY ON TWO-SIDED MATCHING MARKET

7

Specifically, it is only preference treatment in which the college c1 implements the preference-based affirmative action policy.
In problem G̃, there is a unique stable matching ϕ stable (G̃) which is given by
(
)
c c ∅
ϕ stable (G̃) = 1 2
s2 s1 s3 .
The minority student in college c1 is not found to be strictly better off under G̃ than they would be underG.
Next, we consider the case in which college c2 implements the preference-based affirmative action policy
as follows,′⪰c2 : s3 , s1 , s2

qc2 = (qc2 , qMc2 ) = (1, 1).

Note that this is the only preference treatment in which c2 implements the preference-based affirmative
action policy.
′′
In problem G̃ , there is a unique stable matching ϕ stable (G̃ ) which is given by()
c1 c2 ∅′
stable(G̃ ) =
ϕs2 s3 s1′

The minority student s3 in college c2 is found to be strictly better off under ϕ stable (G̃ ) than they would beunder ϕ stable (G).
The minority student in a college is never worse off when their college implements the preference-based
affirmative action policy, regardless of the college. Thus, in this problem, the stable mechanism is found to
be abuse-proof for the preference-based affirmative action policy.
4. C ONCLUSION
This paper investigated whether affirmative action policies work to accomplish its intended purpose in
the context of a college admission problem. We attempted to identify a stable mechanism that would be an
abuse-proof for affirmative action policies. Unfortunately, our results suggest that this does not exist. Thisimplies that college affirmative action policies do not necessarily promote equal opportunities for minority
students. This finding correlated with the findings of Kojima (2012), who also found that affirmative action
policies were likely to be harmful to minority students. He identified certain environments in which theimplementation of an affirmative action policy was inevitably to the detriment of every minority student
under any stable mechanism. Our results imply that this may be an outcome of colleges’ abuse of affirmative
action policies.
We find that a market situation exists in which, once an affirmative action policy is implemented, a college
benefits but the minority students in the school are worse off. More specifically, we establish that this applies
under any stable mechanism. Moreover, this misfortune is found to be unavoidable regardless of whether
alternative affirmative action policies are implemented; for instance, this still applies in situations whereminority students are given preferential treatment. Indeed, we establish that, in situations where colleges
give minority students preferential treatment, the college benefits, while minority students in the college areworse off; this applies under any stable mechanism.
As both our results and those of Kojima (2012) suggest, caution should be exercised when employing
affirmative action policies. While affirmative action may be an attempt to promote equal opportunity for theminority, clearly, the policy can be abused by a college.
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