Week 4
Matching
GM Ch1 and H Ch 9,10,11
Marriage market
Y MUN mi mn w wa NOTATION
Preferences tobestrict
assumed PmW w mew
M W the
Hiew y overMuli tobesingleto Pm
Y
married
equivalent
MatchingMMuW Mowsuchthat
AmenMIMEWUM
twewMuleMow
tiensuchthatMil i MMilliiiethe of s pouse myself
ismyself
Stability
Mlw inn
mandwarenotmatched
ApairImw blocksMit m
m wMulmpreferswtomaten
hmmm wprefersmtomatch
Misindividuallyrationaliftimlits i
Mis pairwisestableif i r F blockingpairs
Deferred acceptance algorithm
• Men (rejected in the previous round) make proposals to their favourite women who have not
rejected them yet (but may have accepted others)
• Women reject all but their favourite proposal
• This is repeated until no rejections are made within a round
MEN WOMEN ROUND ROUND
2
kg bc d a w
men
a
w xw w b
w x
a ab
yz I b x
p ropose
the
Beingproposergivesyou
x wz y a b ba preferences d
y yy z c c ed
best
ofallstablematchings
z z z z d d d c
if bothmatchingsare
thesamethestable
MEN WOMEN ROUND ROUND
2 isunique
matching
I kq a bc d w Y
a w a w a w anagent
unmatched inon
x b x b x besoin
will
I
stablematching
women
p ropose W WW b a c
all
x w z y a b b a preference t q d z
y y y z c c ed
z z z z dd dc
NOTATION
Mla x Mlb w
or
at x bow
, Proof of stability
• The outcome is individually rational because
◦ No man proposes to a women that he nds unacceptable
◦ No woman accepts an o er from a man she nds unacceptable
• The outcome cannot have a blocking pair, proof by contraction:
◦ Suppose A and B are a blocking pair
◦ Because B proposed to A before proposing to nal match, if he had to make subsequent
proposals he must have been rejected by A
◦ A only rejects B if alternative proposal is preferred, so prefers nal match to B
Optimality
Mm bestablematchingwhenmenpropose
AstableM t meM MmlmlemMlmI
t wew Mmw Mw
Sw 2
1 roof Imw is achievable if IstableMMw m
byw ifImw is
achievable
Claim misneverrejected
letwife
retina hitotitisattheendofthe be
theywould p aired
process
w willnotrejecthimby contradiction
mis rejectedatroundkbywa
roundk
wrecievedproposalmamat
Let7stableMm wandMmi w
mnotrejectedbyw infirstkrounds
wemw
mam and wemw means mw is ablockingpair
acontradiction
2 Mw withblockingpairlmw
Bycontr FweW FMMmw contradiction
menMlw wmMm whichis a
others tablematchingtowomanoptimalmatching
PmMw Mw Mm'sMM Mprefersmanoptimalmatchingtoanyotherstablematchingandany
PwMm Mm Mw t MwWpreferswomanoptimalmatchingtoanyotherstablematchingandanyotherstablematchingto manoptimalmatching
DA + report
So far it was assumed that the algorithm knows true preferences
Suppose:
W Mam W
We Memewe Mm m hi meow
M Wewzem
We w m
egwelies Wa Miwema thenMm m wz matsw
es
i Wihasincentiveto lie MmandMiifMw
if
There is no mechanism that for any matching problem ensures
• The matching is stable with respect to submitted preferences
fi
ff fi
Matching
GM Ch1 and H Ch 9,10,11
Marriage market
Y MUN mi mn w wa NOTATION
Preferences tobestrict
assumed PmW w mew
M W the
Hiew y overMuli tobesingleto Pm
Y
married
equivalent
MatchingMMuW Mowsuchthat
AmenMIMEWUM
twewMuleMow
tiensuchthatMil i MMilliiiethe of s pouse myself
ismyself
Stability
Mlw inn
mandwarenotmatched
ApairImw blocksMit m
m wMulmpreferswtomaten
hmmm wprefersmtomatch
Misindividuallyrationaliftimlits i
Mis pairwisestableif i r F blockingpairs
Deferred acceptance algorithm
• Men (rejected in the previous round) make proposals to their favourite women who have not
rejected them yet (but may have accepted others)
• Women reject all but their favourite proposal
• This is repeated until no rejections are made within a round
MEN WOMEN ROUND ROUND
2
kg bc d a w
men
a
w xw w b
w x
a ab
yz I b x
p ropose
the
Beingproposergivesyou
x wz y a b ba preferences d
y yy z c c ed
best
ofallstablematchings
z z z z d d d c
if bothmatchingsare
thesamethestable
MEN WOMEN ROUND ROUND
2 isunique
matching
I kq a bc d w Y
a w a w a w anagent
unmatched inon
x b x b x besoin
will
I
stablematching
women
p ropose W WW b a c
all
x w z y a b b a preference t q d z
y y y z c c ed
z z z z dd dc
NOTATION
Mla x Mlb w
or
at x bow
, Proof of stability
• The outcome is individually rational because
◦ No man proposes to a women that he nds unacceptable
◦ No woman accepts an o er from a man she nds unacceptable
• The outcome cannot have a blocking pair, proof by contraction:
◦ Suppose A and B are a blocking pair
◦ Because B proposed to A before proposing to nal match, if he had to make subsequent
proposals he must have been rejected by A
◦ A only rejects B if alternative proposal is preferred, so prefers nal match to B
Optimality
Mm bestablematchingwhenmenpropose
AstableM t meM MmlmlemMlmI
t wew Mmw Mw
Sw 2
1 roof Imw is achievable if IstableMMw m
byw ifImw is
achievable
Claim misneverrejected
letwife
retina hitotitisattheendofthe be
theywould p aired
process
w willnotrejecthimby contradiction
mis rejectedatroundkbywa
roundk
wrecievedproposalmamat
Let7stableMm wandMmi w
mnotrejectedbyw infirstkrounds
wemw
mam and wemw means mw is ablockingpair
acontradiction
2 Mw withblockingpairlmw
Bycontr FweW FMMmw contradiction
menMlw wmMm whichis a
others tablematchingtowomanoptimalmatching
PmMw Mw Mm'sMM Mprefersmanoptimalmatchingtoanyotherstablematchingandany
PwMm Mm Mw t MwWpreferswomanoptimalmatchingtoanyotherstablematchingandanyotherstablematchingto manoptimalmatching
DA + report
So far it was assumed that the algorithm knows true preferences
Suppose:
W Mam W
We Memewe Mm m hi meow
M Wewzem
We w m
egwelies Wa Miwema thenMm m wz matsw
es
i Wihasincentiveto lie MmandMiifMw
if
There is no mechanism that for any matching problem ensures
• The matching is stable with respect to submitted preferences
fi
ff fi
