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Matching
1 2 3 4 5
A B C D E
Boy
!irl
Today’s goal: to "#atch$ the boy an% the girl in a "goo%$ &ay'
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Stable Matching
Boy !irl
1: CBEAD A : 35214
2 : ABECD B : 52143
3 : DCBAE C : 43512
4 : ACDBE D : 12345
5 : ABDEC E : 23415
(hat i a stable #atching)
Coni%er the ollo&ing #atching' t i unstable &hy)
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Stable Matching
Boy !irl
1: CBEAD A : 35214
2 : ABECD B : 52143
3 : DCBAE C : 43512
4 : ACDBE D : 12345
5 : ABDEC E : 23415
* Boy 4 .reer girl C #ore than girl B 6hi current .artner'
* !irl C .reer boy 4 #ore than boy 1 6her current .artner'
So they ha,e the incenti,e to lea,e their current .artner
an% &itch to each other &e call uch a .air an unstable pair'
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Stable Matching
Boy !irl
1: CBEAD A : 35214
2 : ABECD B : 52143
3 : DCBAE C : 43512
4 : ACDBE D : 12345
5 : ABDEC E : 23415
A table #atching i a #atching &ith no untable .air an% e,ery one i #arrie%'
(hat i astable
#atching)
Can you in% a table #atching in thi cae)
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Stable Matching
Boy !irl
1: CBEAD A : 35214
2 : ABECD B : 52143
3 : DCBAE C : 43512
4 : ACDBE D : 12345
5 : ABDEC E : 23415
Can you in% a table #atching in thi cae)
Doe a table #atching al&ay e0it) 8ot clear/
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Stable Roommate
The Stable Roommate Problem:
*+here are 2n .eo.le'
*+here are n roo# each can acco##o%ate 2 .eo.le'
*Each .eron ha a .reerence lit o 2n91 .eo.le'
*in% a table #atching 6#atch e,eryone an% no untable .air'
Doe a table #atching al&ay e0it) 8ot clear/
(hen i it %iicult to in% a table #atching)
%ea: triangle relationship
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Stable Roommate
%ea: triangle relationship
* a .reer b #ore than c
* b .reer c #ore than a
* c .reer a #ore than b
* no one li-e %
So let; ay a i #atche% to b an% c i #atche% to %'
+hen b .reer c #ore than a an% c .reer b #ore than %'
8o table #atching e0it
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Stable Matching
Can you no& contruct an e0a#.le &here there i no table #atching)
!aleSha.ley 2?:
+here i al&ay a table #atching in the table #atching .roble#'
notclear/
+hi i #ore than a olution to a .ule:
*College A%#iion 6original !ale @ Sha.ley .a.er 1=>2
*Matching o.ital @ ei%ent'
Sha.ley recei,e% 8obel rie 212 in Econo#ic or it
+he .roo i bae% on a #arriage .roce%ure/
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Boy !irl
1: CBEAD A : 35214
2 : ABECD B : 52143
3 : DCBAE C : 43512
4 : ACDBE D : 12345
5 : ABDEC E : 23415
Stable Matching
(hy table #atching i eaier than table roo##ate)
Intuition: t i enough i &e only atiy one i%e
+hi intuition lea% u to a ,ery natural a..roach'
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The Marrying Procedure
Billy Bob
Bra%
Angelina
Morning: boy .ro.oe to their a,ourite girl
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Morning: boy .ro.oe to their a,ourite girlAternoon: girl reect all but a,ourite
Billy Bob
Bra%
Angelina
The Marrying Procedure
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/Angelina
Morning: boy .ro.oe to their a,ourite girlAternoon: girl reect all but a,ourite
E,ening: reecte% boy &rite o girl
/
Billy Bob
The Marrying Procedure
+hi .roce%ure i then re.eate% until all boy .ro.oe to a %ierent girl
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Boy !irl
1: CBEAD A : 35214
2 : ABECD B : 52143
3 : DCBAE C : 43512
4 : ACDBE D : 12345
5 : ABDEC E : 23415
Day 1
Morning: boy .ro.oe to their a,ourite girlAternoon: girl reect all but a,ourite
E,ening: reecte% boy &rite o girl
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Boy !irl
1: CBEAD A : 35214
2 : ABECD B : 52143
3 : DCBAE C : 43512
4 : ACDBE D : 12345
5 : ABDEC E : 23415
Morning: boy .ro.oe to their a,ourite girlAternoon: girl reect all but a,ourite
E,ening: reecte% boy &rite o girl
Day
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Boy !irl
1: CBEAD A : 35214
2 : ABECD B : 52143
3 : DCBAE C : 43512
4 : ACDBE D : 12345
5 : ABDEC E : 23415
Morning: boy .ro.oe to their a,ourite girlAternoon: girl reect all but a,ourite
E,ening: reecte% boy &rite o girl
Day !
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Boy !irl
1: CBEAD A : 35214
2 : ABECD B : 52143
3 : DCBAE C : 43512
4 : ACDBE D : 12345
5 : ABDEC E : 23415
Morning: boy .ro.oe to their a,ourite girlAternoon: girl reect all but a,ourite
E,ening: reecte% boy &rite o girlOFAG #arriage %ay
Day "
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!aleSha.ley 2?:
+hi .roce%ure al&ay in% a table #atching in the table #arriage .roble#'
Proo# o# $ale%Shapley Theorem
(hat %o &e nee% to chec-)
1' +he .roce%ure &ill ter#inate'
2' E,eryone i #arrie%'
3' 8o untable .air'
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Step 1 o# the Proo#
&laim 1' +he .roce%ure &ill ter#inate in at #ot n2
%ay'
1' e,ery girl i #atche% to e0actly one boy
then the .roce%ure &ill ter#inate'
2' Other&ie ince there are n boy an% n girl
there #ut be a girl recei,ing #ore than one .ro.oal'
3' She &ill reect at leat one boy in thi cae
an% thoe boy &ill &rite o that girl ro# their lit
an% .ro.oe to their ne0t a,ourite girl'
4' Since there are n boy an% each lit ha at #ot n girl
the .roce%ure &ill lat or at #ot n2
%ay'
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&laim !' +here i no untable .air'
Step ! o# the Proo#
(act' a girl ! reect a boy B
then ! &ill be #arrie% to a boy 6he li-e better than B'
Cae 1' ! i on B; lit then B i #arrie% to be the bet one on hi lit'
So B ha no incenti,e to lea,e'
Cae 2' ! i not on B; lit then ! i #arrie% to a boy he li-e better'
So ! ha no incenti,e to lea,e'
Coni%er any .air 6B!'
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!aleSha.ley 2?:
+here i al&ay a table #atching in the table #arriage .roble#'
Proo# o# $ale%Shapley Theorem
&laim 1' +he .roce%ure &ill ter#inate in at #ot n2 %ay'
&laim ' E,ery one i #arrie% &hen the .roce%ure to.'
&laim !' +here i no untable .air'
So the theore# ollo&'
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More )uestions *+ptional,
thi #arrying .roce%ure better or boy or or girl))
* All boy get the bet .artner i#ultaneouly
* All girl get the &ort .artner i#ultaneouly
(hy)
Can a boy %o better by lying)
Can a girl %o better by lying)
+hat i a#ong all .oible table #atching
boy get the bet .oible .artner i#ultaneouly'
8O
GES
Intuition: t i enough i &e only atiy one i%e
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