Introduction to
Matching Theory
E. Maskin
Jerusalem Summer School in Economic Theory
June 2014
• 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
2
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)
3
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
4
• 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
5
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 ui ( j ) from obtaining seller j’s good (being
matched with j)
– ui ( j ) could be positive or negative
• seller j ( j=1,…,n) gets utility v j (i ) from selling good to buyer i (being
matched with i)
 vi ( j ) could include cost of producing good
• each buyer and seller gets 0 utility from remaining unmatched: notationally,
ui (0)  0 (i matched with seller 0) v j (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
6
For now, assume there exists perfectly transferable good (money)
• let pij = price that buyer i pays for seller j’s good (could be negative)
• buyer i’s payoff = ui ( j )  pij
+ v j (i )
• sellers j’s payoff = pij 
• let matching be matrix  xij  such that
1, if j sells to i (j  0 if i unmatched; i  0 if j unmatched)
 xij  
0, if j doesn't sell to i

n
x
j  0
ij 
m
  xij  1 for all i  1,..., m, j  1,..., n
i  0
7
• competitive equilibrium is xij* ,  pij  with pi0  p0 j  0 such that
ui ( j )  pij  max ui ( k )  pik  , if xij  1
k
pij  v j (i )  max  pkj  v j ( k )  , if xij  1
k
• claim: competitive equilibrium exists and is (essentially) unique
– despite nonconvexity created by indivisibilities
8
• 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 convexvaluedness 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
9
e.g.,
i
2
j
3
1
1
i
2
3
3
3
j
can be converted to
• each buyer and seller indifferent between randomized
equilibrium and deterministic equilibrium
10
• for any matching xij  , can obtain (by monetary transfers) any payoffs
(1 , ,  m ) for buyers and ( 1 , ,  m ) for sellers such that
(1)
 i    j   (ui ( j )  v j (i )) xij
i
j
i
j
• hence, from first welfare theorem (equilibrium is Pareto optimal),
equilibrium matching xij  solves
(2)
 (u ( j )  v (i )) x
x   arg max
 

ij
xij
i
j
i
j
ij
• generically, unique solution to (2) (and no random solutions)
• so, generically, unique equilibrium matching xij 
– can be multiple prices supporting xij 
11
matching xij  together with monetary transfers ti , t j  is in core if

m
n
t  t
i 1
i
j 1
j
0
 there exist no coalitions C b and C s , matching xˆij 
tˆ ,tˆ  such that
and transfers
-
j
j
0
 tˆ   tˆ
i
iC
-
i
jC
b

jC s
0
s
xˆij  1 and

iC b
0
xˆij  1 for all i  C b , j  C s
- for all i  C b , if xˆij  1, then
(3)
ui ( j)  tˆi  ui ( j )  ti if xij  1
and if j   C s
(4)
v j (i )  tˆj  v j (i)  t j if xij  1
- analogously for all j  C s
 C b C s is blocking coalition
 core matchings are stable
12
claim: competitive equilibrium xij*   pij  in core, where
for all i  1,..., m
ti   pij* , for xij*  1
and for all j  1,..., n
t j  pij , for xij*  1
13
 suppose to contrary there exist C b and C s
and xˆij , tˆi , tˆj  that block xij* ,  pij 
- then for any i  C b ,if xˆij  1
ui ( j)  tˆi  ui ( j )  pij  0, if xij*  1
if j  0 (i unmatched in blocking coalition) then tˆi > 0
- and so blocking coalition can do better by leaving out i
so, can assume j  C s , and hence
(6)
v j (i )  tˆj  v j (i )  pi*j if xi*j  1
if tˆi  tˆj  0, then rest of blocking coalition does even better by leaving out i, j
so can assume tˆi  tˆj  0, and thus adding (5) and (6), we have
(5)
(7)
ui ( j)  v j (i )  ui ( j )  pij + v j (i )  pi*j
but from definition of equilibrium
ui ( j)  pij* (i )  ui ( j )  pij and v j (i )  pij*  v j (i )  pi*j ,
which, when added together, give
ui ( j)  v j (i )  ui ( j )  pij + v j (i )  pi*j ,
contradicting (7)
14
 

claim: any point xij , ti , t j in core is competitive equilibrium
 if xij  1, then ti  t j  0
 if ti  t j  0, then sum of i ' s and j ' s payoffs  ui ( j )+ v j (i )
but i and j can form blocking coalition and get ui ( j )+ v j (i )
 if ti  t j  0, then beacuse sum of all transfers sum is 0, there
exist i, j with ti  t j <0, a contradiction of above
 if xij  1, let pij  t j (so seller recieves pij and buyer pays pij )
 for j such that xij  0, then being in the core implies
(8)
ui ( j)  v j (i )  ui ( j )  pij + v j (i )  pij , where xij =1
 from (8), we can choose pij such that
(9)
ui ( j)  pij  ui ( j )  pij
and
(10)
v j (i )  pij  v j (i)  pij
 (9) and (10) imply that xij , ti , t j  is a C.E.
15
Assortative Matching
• for each i and j, let
wij  ui ( j )  v j (i )
• assume for all i, j
(11)
wi 1 j 1  wi j 1  wi 1 j  wij
• think of index i as positively correlated with buyer’s
“productivity” (contribution to wij )
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 wij  f (i ) g ( j ), where f and g
increasing
16
(11)
wi 1 j 1  wij 1  wi 1 j  wij
claim: given (11), there will be positive assortative matching in
competitive equilibrium, i.e., for equilibrium matching xij 
 if xij  xij  1 and i  i , then j  j 
– i.e., more productive buyers will be matched with more productive
sellers
 suppose to contrary that j  j
 but from 11
wij*  wi*j  wij*  wi*j
implying that matching xij*  xi*j  1 yields higher sum of payoffs
17
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
18
matching xij in core if there do not exist
coalitions C b and C s and xˆij  with
 xˆij   xˆij  1
jC s
0
iC b
0
such that
(12)
 for all i  C b , if xˆij  1
ui ( j)  u j (i ), for xij  1
and if j  C s
(13)
v j (i )  v j (i), for xij  1
 analogously for all j  C s
 above simplifies to: xij in core if, for all i, j, xij  1 implies
ui ( j )  0 and v j (i )  0
and there does not exist j satisfying (12) and (13)
 matching xij  is called stable
19
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 ui ( j)  0) among those who have not
previously rejected him
• if seller j prefers i to current match i ( v j (i )  v j (i)) rejects
i and replaces him with i; otherwise rejects i and sticks with i
• 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
20
 finiteness ensures that algorithm terminates
 results in matching xij  that is stable
 if not, then for some i, j with xij =1 there exists j with
(12)
ui ( j)  ui ( j )
and
(13)
v j (i )  v j (i), where xij  1
 but i must have proposed to j previously
because from 12  he prefers j to j
 and from 13 j would have rejected i and replaced him with i
 so j can't wind up with only v j (i) (seller's utility can
only rise over time), a contradiction
21
• 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
22
• for example: two buyersi, i, two sellers  j, j
i
j
j
0
i
j
j
0
j j
i i
i i
0 0
– if buyers propose, get xij  xij  1
– if sellers propose, get xij  xij  1
• henceforth, focus on strict preferences
23
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 j * possible for buyer i * if there exists stable matching xij  in which xi* j* =1
 fix order of buyers and assume that before stage s, no buyer is rejected by a seller
who is possible for him
 suppose i  rejected by j * at stage s but xˆi* j* =1 for some stable matching xˆij 
 j * must have received proposal from i for whom
(14)
v j* (i)  v j* (i * )  0
 by assumption about s
(15)
ui ( j * )  ui ( j) for all sellers j who are possible for i
 hence, (14) and (15) imply that (i, j * ) can block xˆij , a contradiction
 thus, no buyer ever rejected by a seller who is possible for him
 in (any) buyer-proposal algorithm, each buyer matched with his favorite
possible seller, i.e. order doesn't matter
24
b
let
buyer-proposal
stable
matching
x
•
be  ij 
s
x
• symmetrically, each seller weakly prefers outcome  ij  of
seller-proposal algorithm to any other stable matching
25
claim : in buyer-proposal algorithm, no buyer gains from
misrepresenting his preferences
• suppose, to contrary, that some buyer i * can submit false
preferences and thereby induce xij* , which he prefers to xijb 
• by Gale-Shapley xij*  is stable wrt false preferences for i *
• but will show that xij*  can be blocked  by agents other than i * 
26
• let B*  buyers who prefer xij*  to xijb 
*
*
*
• let S  sellers matched with B buyers in xij 
 in xij*  every buyer in B*matched with some S * seller,
•
S *b
since each one strictly prefers xij* 
= sellers matched with B * buyers in xijb 
case I



S *  S *b
choose j  S such that j  S *b
assume j  matched with i in  xij* 
thus i prefers j to his match in xijb 
- to prevent (𝑖 ′ , 𝑗 ′ ) from blocking
𝑏
𝑥𝑖𝑗
(16) v j (i)  v j (i)  0, where xibj  1
 now, i  B* (because j  S *b ), and so
(17) ui ( j)  ui ( j) where xi*j  1
 hence, (16) and (17) imply ( i, j) block xij* 
27
B*  buyers who prefer xij*  to xijb 
S *  sellers matched with B* buyers in xij* 
S *b  sellers matched with B * buyers in xijb 
case II S *  S *b
• in buyer-optimal algorithm (with true preferences), each buyer in B*
is rejected by seller he’s matched with in xij* 
• so because 1-to-1correspondence between B* and S *b , each seller
in S *b must reject some buyer in B* in algorithm
• let j ** be last seller in S *b to get proposal from buyer in B*  say, i ** 
• i ** can’t be rejected by j **
 ultimately matched with seller from S *b , so must make another proposal
 contradicts choice of j **
• since j ** rejected some other buyer in B* , j ** must have had
matched iˆ that he rejects when i ** proposes, so
(18) uiˆ ( j ** )  uiˆ ( ˆj ), where xiˆb ˆj  1
28
• iˆ  B* because otherwise iˆ makes later proposal to seller in S *b ,
contradicting choice of j **
• hence, uiˆ ( ˆj )  uiˆ ( ˆj), where xi**
ˆ ˆj   1 and so from (18)
19 
uiˆ ( j ** )  uiˆ ( ˆj)
• let iˆ** be buyer matched with j ** in xij* 
• then because iˆ**  B*
**
**
b
ˆ
20
u
(
j
)

u
(
j
),
where
x
1
  iˆ**
iˆ**
iˆ** ˆj**
• so iˆ** must propose to j ** before ˆj **
• because iˆ last buyer rejected by j **
 21 v j (iˆ)  v j (iˆ** )
but 19  and  21 imply (iˆ, j ** ) block xij* 
**
•
**
29
summary of case II:
• buyers B* who gain from xij*  matched with same set of sellers S *
in xij*  as in xijb 
• consider last seller j **  S * to get proposal from any buyer
in B **  let i ** be buyer  in algorithm with true preferences
• there must be buyer iˆ  B* who proposes to j ** just before i **
• so iˆ prefers j ** to seller he's matched with in xijb  and hence in xij* 
• j ** prefers iˆ to buyer matched with in xij* , who is rejected earlier
• so (iˆ, j ** ) blocks xij* 
30
• dominant strategy for buyers to be truthful in buyer-proposal
algorithm
• but sellers may not gain from true revealation of preferences
• consider earlier example
i
j
j
0
i
j
j
0
j j
i i
i i
0 0
 if use buyer-proposal algorithm with true preferences, get xij  xij  1
i
 but if j submits ranking 0 instead
i
then algorithm gives xij  xij  1 , which j prefers
31
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
i
j
j
0
i
j
j
0
j j
i i
i i
0 0
• if dominant strategies lead to matching with xij  xij  1
 then j has incentive to act as though preferences
i
are 0
i
 only stable matching is xij  xij  1
• if dominant strategies lead to matching with xij  xij  1
j
 then i has incentive to play as though prefernces are 0
j
32