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 schools I matching workers to firms in labor markets I matching patients to compatible organ donors I allocating space, positions, tasks I 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 of concrete 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?
Mihai Manea (MIT) Matching Theory June 27, 2016 4 / 53 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. . .
Mihai Manea (MIT) Matching Theory June 27, 2016 5 / 53 Kidney Exchange
I Some patients who need a kidney find a willing donor. The patient may be incompatible with the donor, in which case a direct transplant is not feasible.
Figure : Blood type compatibility
I A kidney exchange matches two (or more) incompatible donor-patient pairs and swaps donors. I How to design efficient kidney exchange mechanisms? Incentive and fairness requirements?
Mihai Manea (MIT) Matching Theory June 27, 2016 6 / 53 One-to-One Matching
Mihai Manea (MIT) Matching Theory June 27, 2016 7 / 53 The Marriage Problem
A one-to-one matching or marriage problem (Gale and Shapley 1962) is a triple (M, W, R), where
I M = {m1, ..., mp} is a set of men
I W = {w1, ..., wq} is a set of women I 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 can be matched to multiple men, many-to-one matching.
Mihai Manea (MIT) Matching Theory June 27, 2016 8 / 53 Preferences
Consider a man m 0 0 I wPmw : man m prefers woman w to woman w
I wPmm: man m prefers woman w to being single
I mPmw: woman w is unacceptable for man m Similar interpretation for women. Assumption: All preferences are strict.
Mihai Manea (MIT) Matching Theory June 27, 2016 9 / 53 Matchings
The outcome of a marriage problem is a matching. A matching is a function µ : M ∪ W → M ∪ W such that I µ(m) ∈ W ∪ {m}, ∀m ∈ M I µ(w) ∈ M ∪ {w}, ∀w ∈ W I µ (m) = w ⇐⇒ µ(w) = m, ∀m ∈ M, w ∈ W.
Assumption: There are no externalities. Agent i ∈ M ∪ W prefers a matching µ to a matching ν iff µ(i)Piν(i).
Mihai Manea (MIT) Matching Theory June 27, 2016 10 / 53 Stability
A matching µ is blocked by an agent i ∈ M ∪ W if iPiµ(i). A matching is individually 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 m and w prefer each other to their partners under µ, i.e.,
wPmµ(m)& mPw µ(w).
A matching is stable if it is not blocked by any agent or pair of agents.
Mihai Manea (MIT) Matching Theory June 27, 2016 11 / 53 Stability and the Core
Proposition 1 The set of stable matchings coincides with the core of the associated cooperative game.
Proof. A matching µ is in the core if there exists no matching ν and coalition S ⊂ M ∪ W such that ν(i)Piµ(i) and ν(S) ⊂ S...
Mihai Manea (MIT) Matching Theory June 27, 2016 12 / 53 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 a stable matching. Step 1. Each man proposes to his first choice (if acceptable). Each woman 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 highest choice (if any). Each woman tentatively accepts her most preferred acceptable 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) at the last step. Any woman who has never tentatively accepted someone or any man who has been rejected by all acceptable women remains single.
Mihai Manea (MIT) Matching Theory June 27, 2016 13 / 53 Example
Pm1 : w2 w1 w3 m1
Pm2 : w1 w2 w3 m2
Pm : w1 w2 w3 m3 | 3 {z } Men’s Preferences
Pw1 : m1 m3 m2 w1
Pw2 : m2 m1 m3 w2
Pw : m2 m1 m3 w3 | 3 {z } Women’s Preferences
The resulting matching is ! m m m µ = 1 2 3 . w1 w2 w3
Mihai Manea (MIT) Matching Theory June 27, 2016 14 / 53 I Women get weakly better off and men get weakly worse off as the algorithm 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 propose to unacceptable women and women immediately reject unacceptable men. I Suppose the pair (m, w) blocks µ. Then wPmµ(m) implies that m proposed to w in the algorithm and, as they are not matched with each other, w rejected m in favor of someone better. But w gets weakly better throughout the algorithm, hence µ(w)Pw m, which contradicts the assumption that (m, w) blocks µ.
Mihai Manea (MIT) Matching Theory June 27, 2016 15 / 53 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 stable matching. I Roth (1991) studied the British medical match, where various regions use different matching mechanisms. Stable mechanisms outlast unstable ones.
Mihai Manea (MIT) Matching Theory June 27, 2016 16 / 53 Evidence from the Medical Match
Market Stable Still in use NRMP yes yes (new design 98-) Edinburgh (’69) yes yes Cardiff yes yes Birmingham no no Edinburgh (’67) no no Newcastle no no Sheffield no no Cambridge no yes London Hospital no yes
Mihai Manea (MIT) Matching Theory June 27, 2016 17 / 53 Men-Optimal Stable Matching Theorem 2 (Gale and Shapley 1962) There exists a men-optimal stable matching that every man weakly prefers to any other stable matching. Furthermore, the men-proposing deferred acceptance algorithm delivers the men-optimal stable matching.
Proof. We say that w is achievable for m if there is some stable matching µ with µ(m) = w. For a contradiction, suppose a man is rejected by an achievable woman at some stage of the deferred acceptance algorithm. Consider the first step of the algorithm in which a man m is rejected by an achievable woman w. Let µ be a stable matching where µ(m) = w. Then 0 0 w tentatively accepted some other man m at this step, so (i) m Pw m. Since this is the first time a man is rejected by an achievable woman, (ii) 0 wPm0 µ(m ). By (i) and (ii), (m0, w) blocks µ, contradicting the stability of µ.
Mihai Manea (MIT) Matching Theory June 27, 2016 18 / 53 The Opposing Interests of Men and Women
Analagous to the men-optimal stable matching, there is a women-optimal stable matching (obtained by a version of the deferred acceptance algorithm where women propose).
I µM: men-optimal stable matching I µW : women-optimal stable matching
Theorem 3 (Knuth 1976) µW is the worst stable matching for each man. Similarly, µM is the worst stable matching for each woman.
Example with 2 men, 2 women, with “reversed” preferences
Mihai Manea (MIT) Matching Theory June 27, 2016 19 / 53 Proof of Opposing Interests
Suppose there is a man m and stable matching µ such that W µ (m)Pmµ(m). Then m is not single under µW . Let w = µW (m). Clearly, w , µ(m), so m , µ(w). W W By the definition of µ , m = µ (w)Pw µ(w). But then (m, w) blocks µ, yielding the desired contradiction.
Mihai Manea (MIT) Matching Theory June 27, 2016 20 / 53 Relevance of Opposing Interests
The result shows that different stable matchings may benefit different market participants. In particular, each version of the deferred acceptance algorithm 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 previous NRMP algorithm had hospitals proposing. Medical students argued that the system favors hospitals over doctors and called for the doctor-proposing version of the mechanism.
Mihai Manea (MIT) Matching Theory June 27, 2016 21 / 53 Agents Matched at Stable Matchings µ˜(W) := µ(W) ∩ M: set of men who are matched under µ µ˜(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 |µ˜M(W)| ≥ |µ˜(W)| ≥ |µ˜W (W)|, since any man matched under µ (µW ) is also matched under µM (µ) I similarly, |µ˜W (M)| ≥ |µ˜(M)| ≥ |µ˜M(M)| I obviously, |µ˜M(W)| = |µ˜M(M)| & |µ˜W (W)| = |µ˜W (M)|, hence all inequalities hold with equality I in particular, |µ˜M(W)| = |µ˜(W)|; since any man matched under µ is also matched under µM, we get µ˜M(W) =µ ˜(W) I analogous argument for women
Mihai Manea (MIT) Matching Theory June 27, 2016 22 / 53 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 filling their positions. It has been argued that the matching mechanism should be adjusted so that more doctors go to rural areas. The theorem shows that this is not feasible if stable matchings are implemented. Also, if some students were matched at some stable matchings and not others, they would find it unfair if one of the matchings that do not include them is selected.
Mihai Manea (MIT) Matching Theory June 27, 2016 23 / 53 Join and Meet
Definition 1 For any matchings µ and µ0, the function µ ∨M µ0 : M ∪ W → M ∪ W (join of µ and µ0) assigns each man the more preferred of his two assignments under µ and µ0 and each woman the less preferred.
( 0 M 0 µ(m) if µ(m)Rmµ (m) µ ∨ µ (m) = 0 0 µ ()m if µ ()m Pmµ() m ( 0 M 0 µ(w) if µ (w)Rw µ(w) µ ∨ µ (w) = 0 0 µ (w) if µ(w)Pw µ (w)
µ ∧M µ0 : M ∪ W → M ∪ W (meet of µ and µ0) is defined analogously, by reversing preferences.
Mihai Manea (MIT) Matching Theory June 27, 2016 24 / 53 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 ! m m m µ = 1 2 3 w1 w 2 w 3 ! m m m µ0 = 1 2 3 w3 w1 w2
The join and meet of µ and µ0 are ! m m m w w w µ ∨M µ0 = 1 2 3 1 2 3 w1 w2 w2 m1 m2 m3 ! m m m w w w µ ∧M µ0 = 1 2 3 1 2 3 w3 w1 w3 m2 m3 m1
Neither is a matching!
Mihai Manea (MIT) Matching Theory June 27, 2016 25 / 53 The Lattice Structure of the Set of Stable Matchings
Theorem 5 (Conway) If µ and µ0 are stable matchings, then µ ∨M µ0 and µ ∧M µ0 are 1 matchings 2 stable.
We prove the result for the join. The proof for the meet is similar.
Mihai Manea (MIT) Matching Theory June 27, 2016 26 / 53 Proof of Part 1. µ ∨M µ0 is a matching I The sets of single agents under µ and µ0 are identical (Theorem 4), hence also identical under µ ∨M µ0. I If a man-woman pair are matched to each other under both µ and µ0, this also holds under µ ∨M µ0. I Consider a man m with different mates under µ and µ0. W.l.o.g., 0 M 0 assume w := µ(m)Pmµ (m). Then µ ∨ µ (m) = w. M 0 0 I We need to show that µ ∨ µ (w) = m. Else, m = µ(w)Pw µ (w) and 0 hence (m, w) blocks µ , contradicting its stability.
Mihai Manea (MIT) Matching Theory June 27, 2016 27 / 53 Proof of Part 2. µ ∨M µ0 is stable
I For a contradiction, suppose that (m, w) blocks µ ∨M µ0. W.l.o.g., assume µ ∨M µ0(w) = µ(w). I Then
M 0 M 0 mPw [µ ∨ µ (w)] = µ(w) and wPm[µ ∨ µ (m)]Rmµ(m),
so (m, w) blocks µ, contradicting its stability. Discuss median stable matchings.
Mihai Manea (MIT) Matching Theory June 27, 2016 28 / 53 Strategic Behavior
I We know many desirable properties of stable matchings, given information about the preferences of market participants. I But in reality, preferences are private information, so the clearinghouse needs to rely on the participants’ reports. I Do participants have incentives to state their preferences truthfully?
Mihai Manea (MIT) Matching Theory June 27, 2016 29 / 53 Direct Mechanisms
Fix M and W, so that each preference profile R defines a marriage problem.
Ri: set of all preference relations for agent i Q R = i∈M∪W Ri: set of all preference profiles R−i: set of all preferences for all agents except i M: set of all matchings A mechanism is a systematic procedure which determines a matching for every marriage problem. Formally, a mechanism is a function ϕ : R → M.
Mihai Manea (MIT) Matching Theory June 27, 2016 30 / 53 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 stable matching for each problem
Mihai Manea (MIT) Matching Theory June 27, 2016 31 / 53 Preference Revelation Games Each mechanism ϕ induces a preference revelation game for every preference profile R where I the set of players is M ∪ W
I the strategy space for player i is Ri I the outcome is determined by the mechanism—if agents report R0, the outcome is ϕ(R0)
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 every player in the induced preference revelation game. Formally, a mechanism ϕ is strategy-proof if
0 ϕ(R−i, Ri)(i) Ri ϕ(R−i, Ri )(i), ∀ ∈ ∪ ∀ 0 ∈ R ∀ ∈ R i M W, Ri, Ri i, R−i −i.
Mihai Manea (MIT) Matching Theory June 27, 2016 32 / 53 ϕM (ϕW ) is Not Strategy-Proof
I 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 produces M ϕ (R) = {(m1, w1), (m2, w2)}.
I If w1 instead reports
P0 : m , w , m w1 2 1 1 M M 0 then ϕ produces ϕ (R ) = {(m1, w2), (m2, w1)}, which w1 prefers to ϕM(R).
I Hence w1 has incentives to misreport her preferences and the deferred acceptance mechanism is not strategy-proof.
Mihai Manea (MIT) Matching Theory June 27, 2016 33 / 53 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 Rw1 : m2 m1 w1
Rm2 : w2 w1 m2 Rw2 : m1 m2 w2 In this problem there are only two stable matchings, ! ! m m m m µM = 1 2 and µW = 1 2 . w1 w2 w2 w1
Let ϕ be any stable mechanism. Then ϕ(R) = µM or ϕ(R) = µW . next slide. . .
Mihai Manea (MIT) Matching Theory June 27, 2016 34 / 53 Proof (Continuation). M Suppose that ϕ(R) = µ . If w1 misrepresents her preferences to be
R0 : m , w , m w1 2 1 1 then µW is the unique stable matching for the manipulated economy (R , R0 ). Since ϕ is stable, it must be that ϕ(R , R0 ) = µW . But then −w1 w1 −w1 w1 W M ϕ is not strategy-proof, as µ Pw1 µ .
W If, on the other hand, ϕ(R) = µ then m1 can report false preferences
R0 : w , m , w m1 1 1 2 and ensure that his favorite stable matching µM is selected by ϕ, since it is the only stable matching for the manipulated economy (R , R0 ). −m1 m1
Mihai Manea (MIT) Matching Theory June 27, 2016 35 / 53 A Stronger Negative Result Every stable matching is Pareto efficient (proof?) and individually rational. Theorem 7 (Alcalde and Barbera 1994) There exists no mechanism that is Pareto efficient, individually rational and strategy-proof.
Proof. For R from the previous proof, any efficient and individually rational mechanism ϕ satisfies ϕ(R) = µM or ϕ(R) = µW . Suppose ϕ(R) = µM. Moreover, ϕ(R , R0 ) ∈ {µW , µ}, where µ = {(m , w )}. −w1 w1 2 2 If ϕ(R , R0 ) = µW , we obtain a contradiction as before. −w1 w1 Suppose ϕ(R , R0 ) = µ. Consider R0 : m , w , m . The only efficient −w1 w1 w2 1 2 2 0 W 0 W and individually rational matching at (RM, RW ) is µ , so ϕ(RM, RW ) = µ . But then m = ϕ (R , R0 ) R ϕ (R , R0 ) = m , and w has 1 w2 M W w2 w2 −w1 w1 2 2 incentives to report R0 at (R , R0 ). w2 −w1 w1
Mihai Manea (MIT) Matching Theory June 27, 2016 36 / 53 Incentives of Men Under ϕM
Theorem 8 (Dubins and Freedman 1981, Roth 1982) Truth-telling is a weakly dominant strategy for all men under the men-optimal stable mechanism.
Similarly, truth-telling is a weakly dominant strategy for all women under the women-optimal stable mechanism.
Mihai Manea (MIT) Matching Theory June 27, 2016 37 / 53 Many-to-One Matching
Mihai Manea (MIT) Matching Theory June 27, 2016 38 / 53 College Admissions Problems
A college admissions problem (Gale and Shapley 1962) is a 4-tuple (C, S, q, R) where
I C = {c1, ..., cm} is a set of colleges
I S = {s1, ..., sn} is a set of students I q = (qc1 ,..., qcm ) is a vector of college capacities I R = (Rc1 ,..., Rcm , Rs1 ,..., Rsn ) is a list of preferences.
Rs: preference relation over colleges and being unassigned, i.e., C∪ {} s S Rc : preference relation over sets of students, i.e., 2 Pc (Ps): strict preferences derived from Rc (Rs)
Mihai Manea (MIT) Matching Theory June 27, 2016 39 / 53 College Preferences
Suppose colleges have rankings over individual students. How should they compare between sets of students? If T is a set consisting of c’s 2nd & 4th choices and T 0 consists of its rd th 0 00 st th 00 3 & 4 choices, then TPc T . If T contains c’s 1 & 5 , then T ?c T . Multiple Pc ’s are consistent with the same ranking of singletons, but this is not essential for the definition of stability.
Rc is responsive (Roth 1985) if I whether a student is acceptable for c I the relative desirability of two students do not depend on other students in the class.
Mihai Manea (MIT) Matching Theory June 27, 2016 40 / 53 Responsiveness
Formally, Rc is responsive if
1 for any T ⊂ S with |T| < qc and s ∈ S \ T,
(T ∪ {s}) Pc T ⇐⇒ {s} Pc ∅
0 2 for any T ⊂ S with |T| < qc and s, s ∈ S \ T,
0 0 (T ∪ {s}) Pc (T ∪ {s }) ⇐⇒ {s} Pc {s }.
Mihai Manea (MIT) Matching Theory June 27, 2016 41 / 53 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, and 3 s ∈ µ(c) ⇐⇒ µ(s) = {c} for all c ∈ C and s ∈ S.
Mihai Manea (MIT) Matching Theory June 27, 2016 42 / 53 Stability
A matching µ is blocked by a college c ∈ C if there exists s ∈ µ(c) such that ∅ 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) and 2 either 0 0 1 there exists s ∈ µ(c) such that {s} Pc {s } or 2 |µ(c)| < qc and {s} Pc ∅. A matching is stable if it is not blocked by any agent or pair.
Mihai Manea (MIT) Matching Theory June 27, 2016 43 / 53 Stable Matchings and the Core
(c, s) blocks a matching µ if c Ps µ(s) and either 0 0 I there exists s ∈ µ(c) such that {s} Pc {s }, which means that the coalition {c, s} ∪ µ(c) \{s0} can weakly block µ in the associated cooperative game, or
I |µ(c)| < qc and {s} Pc ∅, hence the coalition {c, s} ∪ µ(c) can weakly block µ in the cooperative game. A coalition weakly blocks an outcome of a cooperative game if it has a feasible action that makes every member weakly better off, with at least one strict preference → core defined by weak domination (contained in the standard core). This is the right concept of stability in many-to-one settings, as colleges may block a matching by admitting new students while holding on to some old ones. Proposition 2 (Roth 1985) The weak domination core coincides with the set of stable matchings.
Mihai Manea (MIT) Matching Theory June 27, 2016 44 / 53 The Deferred Acceptance Algorithm
The following student applying deferred acceptance algorithm yields a stable matching. Step 1. Each student “applies” to her first choice college. Each college tentatively accepts the most preferred acceptable applicants up to its quota and rejects all others. Step k ≥ 2. Any student rejected at step k − 1 applies to his next highest choice (if any). Each college considers both the new applicants and the students held at step k − 1 and tentatively accepts the most preferred acceptable applicants from the combined pool up to its quota; the other students are rejected. The algorithm terminates when there are no new applications, in finite time.
Mihai Manea (MIT) Matching Theory June 27, 2016 45 / 53 The Correspondence between Many-to-One and One-to-One Matchings
Many (but not all) results for the marriage problem extend to the college admissions problem. The following trick is useful in proofs. For any college admissions problem (C, S, q, R), construct the related marriage problem as follows.
1 qc I “Divide” each college c into qc distinct “seats” c ,..., c . Each seat has unit capacity and ranks students according to c’s preferences over singletons. (This is feasible when Rc is responsive, and hence consistent with a unique ranking of students. . . but not for more general preferences.) C∗ denotes the resulting set of college seats. I For any student s, extend her preferences to C∗ by replacing each 1 qc college c in her original preferences Rs with the block c ,..., c , in this order.
Mihai Manea (MIT) Matching Theory June 27, 2016 46 / 53 Example
The college admissions problem defined by
C = {c1, c2}, qc1 = 2, qc2 = 1, S = {s1, s2} and
R R R R c1 c2 s1 s2 ! {s , s }{s } c c c c 1 2 1 1 2 µ = 1 2 {s2}{s2} c2 c1 s2 s1 {s1}
1 2 is transformed into the marriage problem with M = {c1 , c1 , c2}, W = S and
R 1 R 2 Rc2 Rs Rs c1 c1 1 2 1 1 2 ! s2 s2 s1 c1 c2 ∗ c1 c1 c2 2 1 µ = 2 . s1 s1 s2 c1 c1 s2 c1 s1 2 c2 c1
Mihai Manea (MIT) Matching Theory June 27, 2016 47 / 53 Stability Lemma
In the related marriage problem I 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 respective colleges, whereas seats at the same college are ranked according to their index. Given a matching for a college admissions problem, it is straightforward to define the corresponding matching for its related marriage problem: for any college c, assign the students matched to c in the original problem to seats 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 the corresponding matching for the related marriage problem is stable.
Mihai Manea (MIT) Matching Theory June 27, 2016 48 / 53 Further Results
The Stability Lemma can be used to extend many results from marriage problems 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 is assigned precisely the same set of students at every stable matching.
Mihai Manea (MIT) Matching Theory June 27, 2016 49 / 53 Preferences over Stable Matchings Any two classes to which a college can be stably matched are ranked in the following strong sense. Theorem 10 (Roth and Sotomayor 1989)
If µ and ν are stable matchings such that µ(c) Pc ν(c), then
0 0 {s} Pc {s }, ∀s ∈ µ(c), s ∈ ν(c) \ µ(c).
The set of stable matchings depends only on colleges’ ranking of individual students. The same is true about preferences over stable matchings. Corollary 1 0 Suppose the preferences Pc and Pc are consistent with the same ranking of individuals and P−c is a preference profile for C ∪ S \{c}. Let Σ denote 0 the common set of stable matchings for (Pc , P−c ) and (Pc , P−c ). Then
0 µ(c) Pc ν(c) =⇒ µ(c) Pc ν(c), ∀µ, ν ∈ Σ.
Mihai Manea (MIT) Matching Theory June 27, 2016 50 / 53 Some Properties Do Not Extend to Many-to-One Settings
Not all properties carry over to many-to-one matchings, especially those concerning incentives. I No stable mechanism is strategy-proof for colleges (Roth 1985). In particular, even the college-proposing deferred acceptance rule is not strategy-proof for colleges. Intuition: a college is like a coalition of players in terms of strategies. I On the contrary, student-proposing deferred acceptance is still strategy-proof for students. Why? I Colleges may benefit simply by misreporting capacities. Sonmez (1997) shows that no stable mechanism is immune to misreporting capacities.
Mihai Manea (MIT) Matching Theory June 27, 2016 51 / 53 Incentives for Colleges under Stable Mechanisms
Consider the college admissions problem with
C = {c1, c2}, qc1 = 2, qc2 = 1, S = {s1, s2} and the following preferences
R R0 R R R c1 c1 c2 s1 s2 {s , s }{s }{s } c c 1 2 2 1 1 2 . {s2} ∅ {s2} c2 c1 {s1}
Each of the problems (R , R ) and (R0 , R ) has a unique stable c1 −c1 c1 −c1 matching, ! ! c c c c 1 2 and, respectively, 1 2 . s1 s2 s2 s1 Hence college c benefits from reporting R0 instead of R under any 1 c1 c1 stable mechanism (including the college-optimal stable one).
Mihai Manea (MIT) Matching Theory June 27, 2016 52 / 53 Manipulation of ϕC via Capacities
Consider the college admissions problem with
C = {c1, c2}, qc1 = 2, qc2 = 1, S = {s1, s2} and the following preferences
Rc1 Rc2 Rs1 Rs2 {s , s }{s } c c 1 2 1 1 2 . {s2}{s2} c2 c1 {s1}
Let q0 = 1 be a potential capacity manipulation by college c . We have c1 1 ! ! c c c c ϕC (R, q) = 1 2 and ϕC (R, q0 , q ) = 1 2 . c1 c2 s1 s2 s2 s1
C Hence c1 benefits under ϕ by underreporting its number of seats.
Mihai Manea (MIT) Matching Theory June 27, 2016 53 / 53 MIT OpenCourseWare https://ocw.mit.edu
14.16 Strategy and Information Spring 2016
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