Introduction toMatching Theory

E. Maskin

Jerusalem Summer School in Economic Theory

June 2014

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• much of economics is about markets– exchanges between buyers and sellers

• commonplace to suppose that sellers are heterogeneous– sell somewhat different goods– so buyers not indifferent between different sellers’

goods

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distinctive feature of matching markets: in addition to buyers caring about which seller they buy from,

sellers care which buyer they sell to • e.g., market for education: think of schools as sellers

and prospective students as buyers• not only do students have preferences over schools• but typically, schools have preferences over students

‒ view some students as more desirable than others

another (more technical) feature: indivisibilities (student will attend exactly one school - - or no school at all)

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Matching Theory

• which buyers are matched with which sellers in equilibrium?

• what are equilibrium prices?

– positive side

• which matchings between buyers and sellers have desirable properties ?e.g., stability or fairness

– normative side

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• how can we find such desirable matchings?– i.e., can we construct algorithms or mechanisms

that result in these matchings?– this is market design / implementation side

• we’ll look at all 3 sides in summer school– particular emphasis on second and third sides– but tomorrow, will look at first side (positive) in

order to study wage inequality

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Model• n sellers, each with 1 indivisible good

• m buyers, each wants to buy at most 1 good– one-to-one matching– in later lectures, will consider many-to-one matching (e.g. each student assigned to one

school, but each school assigned many students)

• buyer i (i=1,…,m) gets utility from obtaining seller j’s good (being matched with j)– could be positive or negative

• seller j ( j=1,…,n) gets utility from selling good to buyer i (being matched with i)

• each buyer and seller gets 0 utility from remaining unmatched: notationally,

(i matched with seller 0) (j matched with buyer 0)• two-sided matching (buyers and sellers are different populations)

– men matched with women to form marriages– violinists matched with pianists to form duos– will look at one-sided matching (single population) later in summer school

roommate problem and house assignment problem

( )iu j

( )iu j

( )jv i

(0) 0iu (0)jv

( ) could include cost of producing goodiv j

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For now, assume there exists perfectly transferable good (money)• let = price that buyer i pays for seller j’s good (could be negative)• buyer i’s payoff =• sellers j’s payoff =• let matching be matrix such that

ijp

( )i iju j p( )ij jp v i

0 0

1 for all 1,..., , 1,..., n m

ij i jj i

x x i m j n

1, if sells to ( 0 if unmatched; 0 if unmatched)

0, if doesn't sell to ij

j i j i i jx

j i

ijx

+

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• competitive equilibrium is such that

• claim: competitive equilibrium exists and is (essentially) unique– despite nonconvexity created by indivisibilities

*0 0 , with 0ij ij i jx p p p

( ) max ( ) , if 1i ij i ik ijk

u j p u k p x

( ) max ( ) , if 1ij j kj j ijk

p v i p v k x

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• to convexify,– let each buyer randomize over seller he buys from– let each seller randomize over buyer he sells to

• then (random) demand and supply correspondences satisfy standard convex-valuedness and upper hemicontinuity properties

• so equilibrium exists– with probability 1, no randomization in equilibrium

(because equilibrium matching maximizes sum of utilities, and so is generically unique - - see below)

– but even if there is randomization, can convert matching into no-randomization equilibrium

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e.g.,

can be converted to

• each buyer and seller indifferent between randomized equilibrium and deterministic equilibrium

i j

ji2

3

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13

13

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• for any matching , can obtain (by monetary transfers) any payoffs

for buyers and for sellers such that

(1)

• hence, from first welfare theorem (equilibrium is Pareto optimal),

equilibrium matching solves

(2)

• generically, unique solution to (2) (and no random solutions)• so, generically, unique equilibrium matching

– can be multiple prices supporting

ijx

1( , , )m

( ( ) ( ))i j i j iji j i j

u j v i x

ijx

ijx

arg max ( ( ) ( ))ij

ij i j ijx i j

x u j v i x

ijx

1( , , )m

12

ˆ there exist no coalitions and , matching

ˆ ˆ and transfers , such that

b sij

i j

C C x

t t

matching together with monetary transfers , is in ifij i jx t t core

1 1

0 m n

i ji j

t t

ˆ ˆ- 0 b s

i ji C j C

t t

ˆ- for all , if 1, then biji C x

0 0

ˆ ˆ- 1 and 1 for all , s b

b sij ij

j C i C

x x i C j C

is b sC C blocking coalition

ˆ ( ) ( ) if 1 i i i i iju j t u j t x (3)

(4) ˆ ( ) ( ) if 1 j j j j i jv i t v i t x

core matchings are stable

- analogously for all sj C

and if sj C

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claim: competitive equilibrium in core, where * ij ijx p

* *, for 1 i ij ijt p x

*, for 1j ij ijt p x

for all 1,..., i m

and for all 1,..., j n

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(5)

(6)

(7)

suppose to contrary there exist and b sC C

*that block , ij ijx p ˆ ˆˆand , , ij i jx t t

ˆ- then for any , if 1 biji C x

*ˆ ( ) ( ) 0, if 1 i i i ij iju j t u j p x

so, can assume , and hence sj C* *ˆ ( ) ( ) if 1 j j j i j i jv i t v i p x

- and so blocking coalition can do better by leaving out i

ˆ if 0 ( unmatched in blocking coalition) then > 0 ij i t

ˆ ˆ if 0, then rest of blocking coalition does even better by leaving out , i jt t i j

* ( ) ( ) ( ) + ( ) i j i ij j i ju j v i u j p v i p

ˆ ˆ so can assume 0, and thus adding (5) and (6), we have i jt t

but from definition of equilibrium * * * ( ) ( ) ( ) and ( ) ( ) , i ij i ij j ij j i ju j p i u j p v i p v i p

which, when added together, give * ( ) ( ) ( ) + ( ) , i j i ij j i ju j v i u j p v i p

contradicting (7)

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claim: any point , , in core is competitive equilibriumij i jx t t

if 0, then beacuse sum of all transfers sum is 0, there

exist , with <0, a contradiction of above

i j

i j

t t

i j t t

(8) ( ) ( ) ( ) + ( ) , where =1 i j i ij j i j i ju j v i u j p v i p x

from (8), we can choose such thatijp

if 1, then 0 ij i jx t t if 0, then sum of ' and ' payoffs ( )+ ( ) i j i jt t i s j s u j v i

but and can form blocking coalition and get ( )+ ( ) i ji j u j v i

if 1, let (so seller recieves and buyer pays ) ij ij j ij ijx p t p p

for such that 0, then being in the core implies ijj x

(9) ( ) ( ) i ij i iju j p u j p

(9) and (10) imply that , , is a C.E.ij i jx t t

and

(10) ( ) ( ) j ij j i jv i p v i p

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Assortative Matching• for each i and j, let

• assume for all i, j

• think of index i as positively correlated with buyer’s “productivity” (contribution to )

and j as correlated with seller’s productivity– then (11) says that buyer’s marginal productivity is

increasing in seller’s productivity and vice versa

– e.g., would hold if where f and g increasing

( ) ( ) ij i jw u j v i

1 1 1 1(11) i j i j i j ijw w w w

ijw

( ) ( ), ijw f i g j

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claim: given (11), there will be positive assortative matching in competitive equilibrium, i.e., for equilibrium matching ijx

if 1 and , then ij i jx x i i j j

1 1 1 1(11) i j ij i j ijw w w w

i.e., more productive buyers will be matched with more productive

sellers

–

* *implying that matching 1 yields higher sum of payoffs ij i jx x

* * * * ij i j ij i jw w w w

suppose to contrary that j j but from 11

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Now drop money from model– for some markets (e.g., public schools) buying and selling

goods may be problematic

• can no longer define competitive equilibrium• but can still speak of core

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matching in core if there do not existijx

ˆcoalitions and and with b sijC C x

0 0

ˆ ˆ 1 s b

ij ijj C i C

x x

such that

ˆ for all , if 1biji C x

(12) ( ) ( ), for 1 i j iju j u i x

and if sj C(13) ( ) ( ), for 1 j j i jv i v i x

analogously for all sj C

above simplifies to: in core if, for all , , 1 impliesij ijx i j x ( ) 0 and ( ) 0 i ju j v i

and there does not exist satisfying (12) and (13)

j

matching is called ijx stable

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claim: stable matching exists (Gale-Shapley)

proof is constructive (algorithmic):• in each stage some buyer i, not currently matched, proposes match

to favorite seller j (highest ) among those who have not

previously rejected him

• if seller j prefers i to current match rejects

• algorithm terminates when each unmatched buyer has been rejected by all sellers giving him positive utility

• called deferred acceptance algorithm, because seller’s “acceptance” of i only provisional

( ) 0 iu j

( ( ) ( ))j ji v i v i and replaces him with ; otherwise rejects and sticks with i i i i

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and

finiteness ensures that algorithm terminates

results in matching that is ijx stable

(12) ( ) ( ) i iu j u j

(13) ( ) ( ), where 1 j j i jv i v i x

if not, then for some , with =1 there exists with iji j x j

but must have proposed to previously i j

because from 12 he prefers to j j

and from 13 would have rejected and replaced him with j i i

so can't wind up with only ( ) (seller's utility can

only rise over time), a contradiction

jj v i

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• have looked at stable matching when buyers make proposals

• could do same for proposals by sellers• may get different matching– differs from transferable utility case, where stable matching

generically unique

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• for example: two buyers , two sellers

– if buyers propose, get – if sellers propose, get

• henceforth, focus on strict preferences

0 00 0

i i j j

j j i i

j j i i

1ij i jx x 1ij i jx x

,j j ,i i

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claim: • order of buyers doesn’t matter when buyers make proposals • every buyer (weakly) prefers outcome of buyer-proposal algorithm to

any other stable matching

* *

* * call seller for buyer if there exists stable matching   in which =1 ij i jj possible i x x

fix order of buyers and assume that before stage , no buyer is rejected by a seller

who is possible for him

s

* must have received proposal from for whomj i

* *

*(14) ( ) ( ) 0 j j

v i v i

by assumption about s*(15) ( ) ( ) for all sellers who are possible for i iu j u j j i

in (any) buyer-proposal algorithm, each buyer matched with his favorite

possible seller, i.e. order doesn't matter

* ˆ hence, (14) and (15) imply that ( , ) can block , a contradictioniji j x

thus, no buyer ever rejected by a seller who is possible for him

* *

* ˆ ˆ suppose rejected by at stage but =1 for some stable matching iji ji j s x x

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• be• symmetrically, each seller weakly prefers outcome of

seller-proposal algorithm to any other stable matching

sijx

let buyer-proposal stable matching bijx

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*

*

• suppose, to contrary, that some buyer can submit false

preferences and thereby induce , which he prefers to bij ij

i

x x

: in buyer-proposal algorithm, no buyer gains from

misrepresenting his preferences

claim

* * • by Gale-Shapley is stable wrt false preferences for

ijx i

* *• but will show that can be by agents other than ijx blocked i

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* * buyers who prefer to bij ijB x x

* * * sellers matched with buyers in ijS B x

* = sellers matched with * buyers b bijS B x

* * *

*

in every buyer in matched with some seller,

since each one strictly prefers

ij

ij

x B S

x

* * bcase I S S* choose such that bj S j S

* assume matched with in ijj i x thus prefers to his match in b

iji j x

(16) ( ) ( ) 0, where 1 bj j i jv i v i x

* * now, (because ), and so bi B j S *(17) ( ) ( ) where 1 i i i ju j u j x

* hence, (16) and (17) imply ( , ) block iji j x

• let• let

• in

- to prevent () from blocking

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* * buyers who prefer to bij ijB x x

* * * sellers matched with buyers in ijS B x

* sellers matched with * buyers in b bijS B x

* * bcase II S S

*

*

• in buyer-optimal algorithm (with true preferences), each buyer in  

is rejected by seller he’s matched with in ij

B

x* *

* *

• so because 1-to-1 correspondence between  and , each seller 

in must reject some buyer in in algorithm

b

b

B S

S B

** * * **• let be last seller in to get proposal from buyer in say, bj S B i** **• can’t be rejected by i j

* ultimately matched with seller from , so must make another proposal bS

**ˆmatched that he rejects when proposes, so i i

** * **• since rejected some other buyer in , must have had j B j

**ˆ ˆ ˆ ̂

ˆ(18) ( ) ( ), where 1 b

i i i ju j u j x

** contradicts choice of j

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* *

**

ˆ ˆ • because otherwise makes later proposal to seller in ,

contradicting choice of

bi B i S

j

** *ˆ • but 19 and 21   imply ( , ) block iji j x

**ˆ • because last buyer rejected by i j

** **

**ˆ ˆ21   ( ) ( )j j

v i v i

** ** **ˆ ˆ • so must propose to before i j j

** ** ** **

** **ˆ ˆ ˆ ˆ

ˆ20   ( ) ( ), where 1b

i i i ju j u j x

** ** *ˆ • let   be buyer matched with  in iji j x

**ˆ ˆ

ˆ19   ( ) ( )i i

u j u j

**ˆ ˆ ˆ ̂

ˆ ˆ • hence, ( ) ( ), where 1 and so from (18) i i i j

u j u j x

** *ˆ because i B• then

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summary of case II:

* * *

*

• buyers who gain from matched with same set of sellers

in as in

ij

bij ij

B x S

x x

** *

**

• consider last seller to get proposal from any buyer

in ** let  be buyer in algorithm with true preferences

j S

B i

* ** **ˆ • there must be buyer who proposes to just before i B j i

** *ˆ • so prefers to seller he's matched with in and hence in bij iji j x x

** *ˆ • prefers to buyer matched with in , who is rejected earlierijj i x

** *ˆ • so ( , ) blocks iji j x

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• dominant strategy for buyers to be truthful in buyer-proposal algorithm

• but sellers may not gain from true revealation of preferences • consider earlier example

0 00 0

i i j j

j j i i

j j i i

but if submits ranking 0 instead i

ji

if use buyer-proposal algorithm with true preferences, get 1ij i jx x

then algorithm gives 1 , which prefers ij i jx x j

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same example shows there is no algorithm guaranteeing stable matchings for which all players always have dominant strategies• suppose to contrary there is such a mechanism• consider preferences of example

• if

0 00 0

i i j j

j j i i

j j i i

dominant strategies lead to matching with 1ij i jx x

then has incentive to act as though preferences

are 0

j

i

i

then has incentive to play as though prefernces are 0 j

ij

only stable matching is 1ij i jx x • if dominant strategies lead to matching with 1ij i jx x