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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


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/Angelina


E,ening: reecte% boy &rite o girl

/

Billy Bob


+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



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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

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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 !

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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


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

L07 Dis.math

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