Download - Introduction to Matching Theory E. Maskin Jerusalem Summer School in Economic Theory June 2014
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
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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