mit 14.16 s16 matching theory lecture slides · pdf filecooperative game. proof. ... roth...

Matching Theory

Mihai Manea

MIT

Based on slides by Fuhito Kojima.

Market Design

I Traditional economics focuses mostly on decentralized markets.I Recently, economists are helping to design economic institutions for

centralized markets.I placing students in schoolsI matching workers to firms in labor marketsI matching patients to compatible organ donorsI allocating space, positions, tasksI auctioning electromagnetic spectrum, landing slots at aiports

I The economics of market design analyzes and develops institutions.Practical solutions require attention to the details and objectives ofconcrete markets.

Mihai Manea (MIT) Matching Theory June 27, 2016 2 / 53

Hospitals and Residents

I Graduating medical students are hired as residents at hospitals.I In the US more than 20,000 doctors and 4,000 hospitals are matched

through a clearinghouse, the National Resident Matching Program.I Doctors and hospitals submit preference rankings and the

clearinghouse uses an algorithm to assign positions.I Some centralized markets succeed, while others fail. What makes a

good matching mechanism?


School Choice

I School districts use centralized student placement mechanisms.I School districts take into account the preferences of students and

decide the priorities each school assigns to students.I What is a desirable student placement mechanism? Walking

distance, siblings, affirmative action, test scores. . .


Kidney ExchangeI Some patients who need a kidney find a willing donor. The patient

may be incompatible with the donor, in which case a direct transplantis not feasible.

Figure : Blood type compatibility

I A kidney exchange matches two (or more) incompatible donor-patientpairs and swaps donors.

I How to design efficient kidney exchange mechanisms? Incentive andfairness requirements?


One-to-One Matching


The Marriage Problem

A one-to-one matching or marriage problem (Gale and Shapley 1962)is a triple (M,W ,R), whereI M = {m1, ...,mp} is a set of menI W = {w1, ...,wq} is a set of womenI R = (Rm1 , . . . ,Rmp ,Rw1 , . . . ,Rwq ) is a preference profile.

For m ∈ M, Rm is a preference relation over W ∪ {m}.For w ∈ W , Rw is a preference relation over M ∪ {w}.

Pm,Pw denote the strict preferences derived from Rm,Rw .

In applications men and women correspond to students and schools,doctors and hospitals, etc. Extend theory to the case where a woman canbe matched to multiple men, many-to-one matching.


Preferences

Consider a man mI wPmw′: man m prefers woman w to woman w′

I wPmm: man m prefers woman w to being singleI mPmw: woman w is unacceptable for man m

Similar interpretation for women.

Assumption: All preferences are strict.


Matchings

The outcome of a marriage problem is a matching. A matching is afunction µ : M ∪W → M ∪W such thatI µ(m) ∈ W ∪ {m},∀m ∈ MI µ(w) ∈ M ∪ {w},∀w ∈ WI µ (m) = w ⇐⇒ µ(w) = m,∀m ∈ M,w ∈ W .

Assumption: There are no externalities. Agent i ∈ M ∪W prefers amatching µ to a matching ν iff µ(i)Piν(i).


Stability

A matching µ is blocked by an agent i ∈ M ∪W if iPiµ(i). A matching isindividually rational if it is not blocked by any agent.

A matching µ is blocked by a man-woman pair (m,w) ∈ M ×W if both mand w prefer each other to their partners under µ, i.e.,

wPmµ(m) & mPwµ(w).

A matching is stable if it is not blocked by any agent or pair of agents.


Stability and the Core

Proposition 1The set of stable matchings coincides with the core of the associatedcooperative game.

Proof.A matching µ is in the core if there exists no matching ν and coalitionS ⊂ M ∪W such that ν(i)Piµ(i) and ν(S) ⊂ S. . . �


The Deferred Acceptance (DA) Algorithm

Theorem 1 (Gale and Shapley 1962)Every marriage problem has a stable matching.

The following men-proposing deferred acceptance algorithm yields astable matching.Step 1. Each man proposes to his first choice (if acceptable). Eachwoman tentatively accepts her most preferred acceptable proposal (if any)and rejects all others.Step k ≥ 2. Any man rejected at step k − 1 proposes to his next highestchoice (if any). Each woman tentatively accepts her most preferredacceptable proposal to date and rejects the rest.

The algorithm terminates when there are no new proposals, in finite time.

Each woman is matched with the man whose proposal she holds (if any) atthe last step. Any woman who has never tentatively accepted someone orany man who has been rejected by all acceptable women remains single.


Example

Pm1 : w2 � w1 � w3 � m1

Pm2 : w1 � w2 � w3 � m2

Pm3 : w1 � w2 � w3 � m3︸︷︷︸Men’s Preferences

Pw1 : m1 � m3 � m2 � w1

Pw2 : m2 � m1 � m3 � w2

Pw3 : m2 � m1 � m3 � w3︸︷︷︸Women’s Preferences

The resulting matching is

µ =

(m1 m2 m3

w1 w2 w3

).


I Women get weakly better off and men get weakly worse off as thealgorithm proceeds.

I The algorithm eventually stops, producing a matching µ.I µ is stable

I µ cannot be blocked by any individual agent, since men never proposeto unacceptable women and women immediately reject unacceptablemen.

I Suppose the pair (m,w) blocks µ. Then wPmµ(m) implies that mproposed to w in the algorithm and, as they are not matched with eachother, w rejected m in favor of someone better. But w gets weaklybetter throughout the algorithm, hence µ(w)Pwm, which contradicts theassumption that (m,w) blocks µ.


Stable Mechanisms in Real Markets

I Stability is theoretically appealing, but is it relevant in applications?I Roth (1984) showed that the NRMP algorithm is equivalent to a

(hospital-proposing) DA algorithm, so NRMP produces a stablematching.

I Roth (1991) studied the British medical match, where various regionsuse different matching mechanisms. Stable mechanisms outlastunstable ones.


Evidence from the Medical Match

Market Stable Still in useNRMP yes yes (new design 98-)Edinburgh (’69) yes yesCardiff yes yesBirmingham no noEdinburgh (’67) no noNewcastle no noSheffield no noCambridge no yesLondon Hospital no yes


Men-Optimal Stable MatchingTheorem 2 (Gale and Shapley 1962)There exists a men-optimal stable matching that every man weaklyprefers to any other stable matching. Furthermore, the men-proposingdeferred acceptance algorithm delivers the men-optimal stable matching.

Proof.We say that w is achievable for m if there is some stable matching µ withµ(m) = w. For a contradiction, suppose a man is rejected by anachievable woman at some stage of the deferred acceptance algorithm.

Consider the first step of the algorithm in which a man m is rejected by anachievable woman w. Let µ be a stable matching where µ(m) = w. Thenw tentatively accepted some other man m′ at this step, so (i) m′Pwm.Since this is the first time a man is rejected by an achievable woman, (ii)wPm′µ(m′).

By (i) and (ii), (m′,w) blocks µ, contradicting the stability of µ. �


The Opposing Interests of Men and Women

Analagous to the men-optimal stable matching, there is a women-optimalstable matching (obtained by a version of the deferred acceptancealgorithm where women propose).

I µM : men-optimal stable matchingI µW : women-optimal stable matching

Theorem 3 (Knuth 1976)

µW is the worst stable matching for each man. Similarly, µM is the worststable matching for each woman.

Example with 2 men, 2 women, with “reversed” preferences


Proof of Opposing Interests

Suppose there is a man m and stable matching µ such thatµW (m)Pmµ(m).

Then m is not single under µW . Let w = µW (m). Clearly, w , µ(m), som , µ(w).

By the definition of µW , m = µW (w)Pwµ(w). But then (m,w) blocks µ,yielding the desired contradiction.


Relevance of Opposing Interests

The result shows that different stable matchings may benefit differentmarket participants. In particular, each version of the deferred acceptancealgorithm favors one side of the market at the expense of the other.

This point was part of a policy debate in NRMP in the 90s. The previousNRMP algorithm had hospitals proposing. Medical students argued thatthe system favors hospitals over doctors and called for thedoctor-proposing version of the mechanism.


Agents Matched at Stable Matchingsµ̃(W) := µ(W) ∩M: set of men who are matched under µµ̃(M) := µ(M) ∩W : set of women who are matched under µ

Theorem 4 (McVitie and Wilson 1970)The set of matched agents is identical at every stable matching.

Proof.Let µ be an arbitrary stable matching.I |µ̃M(W)| ≥ |µ̃(W)| ≥ |µ̃W (W)|, since any man matched under µ (µW ) is

also matched under µM (µ)I similarly, |µ̃W (M)| ≥ |µ̃(M)| ≥ |µ̃M(M)|

I obviously, |µ̃M(W)| = |µ̃M(M)| & |µ̃W (W)| = |µ̃W (M)|, hence allinequalities hold with equality

I in particular, |µ̃M(W)| = |µ̃(W)|; since any man matched under µ isalso matched under µM , we get µ̃M(W) = µ̃(W)

I analogous argument for women �


Relevance of the Result

One motivation is the allocation of residents to hospitals in rural areas.Rural hospitals are not attractive to residents and have difficulties fillingtheir positions. It has been argued that the matching mechanism shouldbe adjusted so that more doctors go to rural areas. The theorem showsthat this is not feasible if stable matchings are implemented.

Also, if some students were matched at some stable matchings and notothers, they would find it unfair if one of the matchings that do not includethem is selected.


Join and Meet

Definition 1For any matchings µ and µ′, the function µ ∨M µ′ : M ∪W → M ∪W (joinof µ and µ′) assigns each man the more preferred of his two assignmentsunder µ and µ′ and each woman the less preferred.

M{µ(m) if µ(m)R

µ ∨ µ′(m mµ′(m)

) = ′ m if ′ m P m

µ ∨M µ′(w) =

{ µ ( ) µ ( ) mµ( )

µ(w) if µ′(w)Rwµ(w)µ′(w) if µ(w)Pwµ

′(w)

µ ∧M µ′ : M ∪W → M ∪W (meet of µ and µ′) is defined analogously, byreversing preferences.


Example

Pm1 : w1 w2 w3 m1 Pw1 : m2 m3 m1 w1

Pm2 : w2 w3 w1 m2 Pw2 : m3 m1 m2 w2

Pm3 : w2 w1 w3 m3 Pw3 : m1 m2 m3 w3

µ =

(m1 m2 m3

w w w

)µ′ =

( 1 2 3

m1 m2 m3

w3 w1 w2

)The join and meet of µ and µ′ are

µ ∨M µ′ =

(m1 m2 m3 w1 w2 w3

w1 w( 2 w2 m1 m2 m3

M m1 m2 m3 w w

)µ ∧ µ′ 1 2 w3= w3 w1 w3 m2 m3 m1

)Neither is a matching!


The Lattice Structure of the Set of Stable Matchings

Theorem 5 (Conway)

If µ and µ′ are stable matchings, then µ ∨M µ′ and µ ∧M µ′ are1 matchings2 stable.

We prove the result for the join. The proof for the meet is similar.


Proof of Part 1.µ ∨M µ′ is a matchingI The sets of single agents under µ and µ′ are identical (Theorem 4),

hence also identical under µ ∨M µ′.I If a man-woman pair are matched to each other under both µ and µ′,

this also holds under µ ∨M µ′.I Consider a man m with different mates under µ and µ′. W.l.o.g.,

assume w := µ(m)Pmµ′(m). Then µ ∨M µ′(m) = w.

I We need to show that µ ∨M µ′(w) = m. Else, m = µ(w)Pwµ′(w) and

hence (m,w) blocks µ′, contradicting its stability. �


Proof of Part 2.µ ∨M µ′ is stable

I For a contradiction, suppose that (m,w) blocks µ ∨M µ′. W.l.o.g.,assume µ ∨M µ′(w) = µ(w).

I Then

mPw [µ ∨M µ′(w)] = µ(w) and wPm[µ ∨M µ′(m)]Rmµ(m),

so (m,w) blocks µ, contradicting its stability. �

Discuss median stable matchings.


Strategic Behavior

I We know many desirable properties of stable matchings, giveninformation about the preferences of market participants.

I But in reality, preferences are private information, so theclearinghouse needs to rely on the participants’ reports.

I Do participants have incentives to state their preferences truthfully?


Direct Mechanisms

Fix M and W , so that each preference profile R defines a marriageproblem.

Ri : set of all preference relations for agent iR =

∏i∈M∪W Ri : set of all preference profiles

R−i : set of all preferences for all agents except iM: set of all matchings

A mechanism is a systematic procedure which determines a matching forevery marriage problem. Formally, a mechanism is a function ϕ : R →M.


Stable Mechanisms

A mechanism ϕ is stable if ϕ(R) is stable for each R ∈ R.

ϕM (ϕW ) : the mechanism that selects the men-(women-)optimal stablematching for each problem


Preference Revelation GamesEach mechanism ϕ induces a preference revelation game for everypreference profile R whereI the set of players is M ∪WI the strategy space for player i is Ri

I the outcome is determined by the mechanism—if agents report R ′,the outcome is ϕ(R ′)

I i’s preferences over outcomes are given by his true preference Ri .

A mechanism ϕ is strategy-proof if, for every (true) preference profile R,truthful preference revelation is a (weakly) dominant strategy for everyplayer in the induced preference revelation game.

Formally, a mechanism ϕ is strategy-proof if

ϕ(R−i ,Ri)(i) Ri ϕ(R−i ,Ri′)(i),

∀i ∈ M ∪W , ∀Ri ,R ′ .i ∈ Ri ,∀R−i ∈ R−i


ϕM (ϕW ) is Not Strategy-ProofI Let M = {m1,m2},W = {w1,w2} and

Pm1 : w1,w2,m1

Pm2 : w2,w1,m2

Pw1 : m2,m1,w1

Pw2 : m1,m2,w2.

I When each agent reports his true preferences, ϕM producesϕM(R) = {(m1,w1), (m2,w2)}.

I If w1 instead reports

P′w1: m2,w1,m1

then ϕM produces ϕM(R ′) = {(m1,w2), (m2,w1)}, which w1 prefers toϕM(R).

I Hence w1 has incentives to misreport her preferences and thedeferred acceptance mechanism is not strategy-proof.


Incompatibility of Stability and Strategy-Proofness

Theorem 6 (Roth 1982)There exists no mechanism that is both stable and strategy-proof.

Proof.Consider the following 2 men, 2 women problem

Rm1 : w1 w2 m1

Rm2 : w2 w1 m2

Rw1 : m2 m1 w1

Rw2 : m1 m2 w2

In this problem there are only two stable matchings,

µM =

(m1 m2

w1 w2

)and µW =

(m1 m2

w2 w1

).

Let ϕ be any stable mechanism. Then ϕ(R) = µM or ϕ(R) = µW .next slide. . .


Proof (Continuation).

Suppose that ϕ(R) = µM . If w1 misrepresents her preferences to be

R ′w1: m2,w1,m1

then µW is the unique stable matching for the manipulated economy(R−w1 ,R

′w1

). Since ϕ is stable, it must be that ϕ(R−w1 ,R′w1

) = µW . But thenϕ is not strategy-proof, as µW Pw1 µ

M .

If, on the other hand, ϕ(R) = µW then m1 can report false preferences

R ′m1: w1,m1,w2

and ensure that his favorite stable matching µM is selected by ϕ, since it isthe only stable matching for the manipulated economy (R−m1 ,R

′m1

). �


A Stronger Negative ResultEvery stable matching is Pareto efficient (proof?) and individually rational.

Theorem 7 (Alcalde and Barbera 1994)There exists no mechanism that is Pareto efficient, individually rationaland strategy-proof.

Proof.For R from the previous proof, any efficient and individually rationalmechanism ϕ satisfies ϕ(R) = µM or ϕ(R) = µW . Suppose ϕ(R) = µM .Moreover, ϕ(R−w1 ,R

′w1

) ∈ {µW , µ}, where µ = {(m2,w2)}.

If ϕ(R−w1 ,R′w1

) = µW , we obtain a contradiction as before.

Suppose ϕ(R−w1 ,R′w1

) = µ. Consider R ′w2: m1,w2,m2. The only efficient

and individually rational matching at (RM ,R ′W ) is µW , so ϕ(RM ,R ′W ) = µW .But then m1 = ϕw2(RM ,R ′W ) Rw2 ϕw2(R−w1 ,R

′w1

) = m2, and w2 hasincentives to report R ′w2

at (R−w1 ,R′w1

). �


Incentives of Men Under ϕM

Theorem 8 (Dubins and Freedman 1981, Roth 1982)Truth-telling is a weakly dominant strategy for all men under themen-optimal stable mechanism.

Similarly, truth-telling is a weakly dominant strategy for all women underthe women-optimal stable mechanism.


Many-to-One Matching


College Admissions Problems

A college admissions problem (Gale and Shapley 1962) is a 4-tuple(C ,S, q,R) whereI C = {c1, ..., cm} is a set of collegesI S = {s1, ..., sn} is a set of studentsI q = (qc1 , . . . , qcm ) is a vector of college capacitiesI R = (Rc1 , . . . ,Rcm ,Rs1 , . . . ,Rsn ) is a list of preferences.

Rs : preference relation over colleges and being unassigned, i.e., C sR : preference relation over sets of students, i.e., 2S

∪ { }

c

Pc(Ps): strict preferences derived from Rc(Rs)


College Preferences

Suppose colleges have rankings over individual students. How should theycompare between sets of students?

If T is a set consisting of c ’s 2nd & 4th choices and T ′ consists of its3rd & 4th choices, then T PcT ′. If T ′′ contains c ’s 1st & 5th , then T ?cT ′′.Multiple Pc ’s are consistent with the same ranking of singletons, but this isnot essential for the definition of stability.

Rc is responsive (Roth 1985) ifI whether a student is acceptable for cI the relative desirability of two students

do not depend on other students in the class.


Responsiveness

Formally, Rc is responsive if1 for any T ⊂ S with |T | < qc and s ∈ S \ T ,

(T ∪ {s}) Pc T ⇐⇒ {s}Pc ∅

2 for any T ⊂ S with |T | < qc and s, s′ ∈ S \ T ,

(T ∪ {s}) Pc (T ∪ {s′}) ⇐⇒ {s}Pc {s′}.


Matchings

The outcome of a college admissions problem is a matching.Formally, a matching is a correspondence µ : C ∪ S ⇒ C ∪ S such that

1 µ(c) ⊆ S with |µ(c)| ≤ qc for all c ∈ C (we allow µ(c) = ∅),2 µ(s) ⊆ C ∪ {s} with |µ(s)| = 1 for all s ∈ S, and3 s ∈ µ(c) ⇐⇒ µ(s) = {c} for all c ∈ C and s ∈ S.


Stability

A matching µ is blocked by a college c ∈ C if there exists s ∈ µ(c) suchthat ∅Pc {s}.A matching µ is blocked by a student s ∈ S if s Ps µ(s).A matching µ is blocked by a pair (c, s) ∈ C × S if

1 c Ps µ(s) and2 either

1 there exists s′ ∈ µ(c) such that {s}Pc {s′} or2 |µ(c)| < qc and {s}Pc ∅.

A matching is stable if it is not blocked by any agent or pair.


Stable Matchings and the Core

(c, s) blocks a matching µ if c Ps µ(s) and eitherI there exists s′ ∈ µ(c) such that {s}Pc {s′}, which means that the

coalition {c, s} ∪ µ(c) \ {s′} can weakly block µ in the associatedcooperative game, or

I |µ(c)| < qc and {s}Pc ∅, hence the coalition {c, s} ∪ µ(c) can weaklyblock µ in the cooperative game.

A coalition weakly blocks an outcome of a cooperative game if it has afeasible action that makes every member weakly better off, with at leastone strict preference→ core defined by weak domination (contained inthe standard core). This is the right concept of stability in many-to-onesettings, as colleges may block a matching by admitting new studentswhile holding on to some old ones.

Proposition 2 (Roth 1985)The weak domination core coincides with the set of stable matchings.


The Deferred Acceptance Algorithm

The following student applying deferred acceptance algorithm yields astable matching.Step 1. Each student “applies” to her first choice college. Each collegetentatively accepts the most preferred acceptable applicants up to itsquota and rejects all others.Step k ≥ 2. Any student rejected at step k − 1 applies to his next highestchoice (if any). Each college considers both the new applicants and thestudents held at step k − 1 and tentatively accepts the most preferredacceptable applicants from the combined pool up to its quota; the otherstudents are rejected.

The algorithm terminates when there are no new applications, in finitetime.


The Correspondence between Many-to-One andOne-to-One Matchings

Many (but not all) results for the marriage problem extend to the collegeadmissions problem. The following trick is useful in proofs.For any college admissions problem (C ,S, q,R), construct the relatedmarriage problem as follows.I “Divide” each college c into qc distinct “seats” c1, . . . , cqc . Each seat

has unit capacity and ranks students according to c ’s preferencesover singletons. (This is feasible when Rc is responsive, and henceconsistent with a unique ranking of students. . . but not for moregeneral preferences.) C∗ denotes the resulting set of college seats.

I For any student s, extend her preferences to C∗ by replacing eachcollege c in her original preferences Rs with the block c1, . . . , cqc , inthis order.


Example

The college admissions problem defined byC = {c1, c2}, qc1 = 2, qc2 = 1,S = {s1, s2} and

Rc1 Rc2

{s1, s2} {s1}

{s2} {s2}

{s1}

Rs1 Rs2

c1 c2

c2 c1µ =

(c1 c2

s2 s1

)

is transformed into the marriage problem with M = {c11 , c

21 , c2},W = S and

Rc11

Rc21

Rc2

s2 s2 s1

s1 s1 s2

Rs1 Rs2

c11 c2

c21 c1

1c2 c2

1

µ∗ =

(c1

1 c21 c2

s2 c21 s1

).


Stability Lemma

In the related marriage problemI each seat at a college c is an individual unit that has preferences

consistent with Pc

I students rank seats at different colleges as they rank the respectivecolleges, whereas seats at the same college are ranked according totheir index.

Given a matching for a college admissions problem, it is straightforward todefine the corresponding matching for its related marriage problem: forany college c, assign the students matched to c in the original problem toseats at c, such that students ranked higher by Pc get lower indexed seats.

Lemma 1 (Roth 1985)A matching in a college admissions problem is stable if and only if thecorresponding matching for the related marriage problem is stable.


Further Results

The Stability Lemma can be used to extend many results from marriageproblems to college admissions.I The student (college) proposing deferred acceptance algorithm

produces the student-(college-)optimal stable matching.I Opposing interests, lattice structure.I The rural hospital theorem also extends. The following stronger

version holds.

Theorem 9 (Roth 1986)Any college that does not fill all its positions at some stable matching isassigned precisely the same set of students at every stable matching.


Preferences over Stable MatchingsAny two classes to which a college can be stably matched are ranked inthe following strong sense.

Theorem 10 (Roth and Sotomayor 1989)If µ and ν are stable matchings such that µ(c) Pc ν(c), then

{s} Pc {s′},∀s ∈ µ(c), s′ ∈ ν(c) \ µ(c).

The set of stable matchings depends only on colleges’ ranking of individualstudents. The same is true about preferences over stable matchings.

Corollary 1Suppose the preferences Pc and P′c are consistent with the same rankingof individuals and P−c is a preference profile for C ∪ S \ {c}. Let Σ denotethe common set of stable matchings for (Pc ,P−c) and (P′c ,P−c). Then

µ(c) Pc ν(c) =⇒ µ(c) P′c ν(c),∀µ, ν ∈ Σ.


Some Properties Do Not Extend to Many-to-One Settings

Not all properties carry over to many-to-one matchings, especially thoseconcerning incentives.I No stable mechanism is strategy-proof for colleges (Roth 1985). In

particular, even the college-proposing deferred acceptance rule is notstrategy-proof for colleges. Intuition: a college is like a coalition ofplayers in terms of strategies.

I On the contrary, student-proposing deferred acceptance is stillstrategy-proof for students. Why?

I Colleges may benefit simply by misreporting capacities. Sonmez(1997) shows that no stable mechanism is immune to misreportingcapacities.


Incentives for Colleges under Stable Mechanisms

Consider the college admissions problem withC = {c1, c2}, qc1 = 2, qc2 = 1,S = {s1, s2} and the following preferences

Rc1 R ′c1Rc2

{s1, s2} {s2} {s1}

{s2} ∅ {s2}

{s1}

Rs1 Rs2

c1 c2

c2 c1.

Each of the problems (Rc1 ,R−c1) and (R ′c1,R−c1) has a unique stable

matching, (c1 c2

s1 s2

)and, respectively,

(c1 c2

s2 s1

).

Hence college c1 benefits from reporting R ′c1instead of Rc1 under any

stable mechanism (including the college-optimal stable one).


Manipulation of ϕC via Capacities

Consider the college admissions problem withC = {c1, c2}, qc1 = 2, qc2 = 1,S = {s1, s2} and the following preferences

Rc1 Rc2

{s1, s2} {s1}

{s2} {s2}

{s1}

Rs1 Rs2

c1 c2

c2 c1.

Let q′c1= 1 be a potential capacity manipulation by college c1. We have

ϕC(R , q) =

(c1 c2

s1 s2

)and ϕC(R , q′c1

, qc2) =

(c1 c2

s2 s1

).

Hence c1 benefits under ϕC by underreporting its number of seats.


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