CakeCuttingAlgorithms

EricPacuit

January

7,2007

ILLC,University

ofAmsterdam

staff.science.uva.nl/∼epacuit

epacuit@science.uva.nl

PlanforToday

Discuss

somefairdivisionalgorithms

•Whatdoes

itmeanto

�fairly�dividegoods?

•IndivisibleGoods

•DivisibleGoods(C

uttingaCake)

�DivideandChoose

�SurplusProcedure

�Banach-K

naster

Last

Dim

inisher

�Dubins-SpanierMovingKnifeProcedure

Main

Question

How

dowecu

tacake

fairly?

Main

Question

How

dowecu

tacake

fairly?

•anydesirablesetofgood

s(orchoresormixtures)

•each

may

bedivisibleorindivisible

•theremay

berestrictions(such

asthenumber

ofgoodsaplayer

may

receive)

Main

Question

How

dowecu

tacake

fairly?

•discreteprocedure

•continuousmovingknifeprocedures

•compensationprocedures(usingmoney

asadivisiblemedium

forindivisibleobjects)

Main

Question

How

dowecu

tacake

fairly?

•Interested

notonly

intheexistence

ofa(fair)divisionbutalso

aconstructiveprocedure

(analgorithm)for�ndingit

Main

Question

How

dowecu

tacake

fairly?

•Di�erentresultsknow

nfor2,3,4,.

..cutters!

Main

Question

How

dowecu

tacake

fairly?

•Manywaysto

makethisprecise!

Fairness

Conditions

•Proportional:

(fortwoplayers)

each

player

receives

atleast

50%

oftheirvaluation.

•Envy-Free:nopartyiswillingto

giveupitsallocationin

exchangefortheother

player'sallocation,so

noplayersenvies

anyoneelse.

•Equitable:

each

player

values

itsallocationthesame

accordingto

itsownva

luationfunction.

•E�ciency:

thereisnoother

divisionbetterforeverybody,or

betterforsomeplayersandnotworsefortheothers

Fairness

Conditions

•Proportional:

(fortwoplayers)

each

player

receives

atleast

50%

oftheirvaluation.

•Envy-Free:nopartyiswillingto

giveupitsallocationin

exchangefortheother

player'sallocation,so

noplayersenvies

anyoneelse.

•Equitable:

each

player

values

itsallocationthesame

accordingto

itsownva

luationfunction.

•E�ciency:

thereisnoother

divisionbetterforeverybody,or

betterforsomeplayersandnotworsefortheothers

Fairness

Conditions

•Proportional:

(fortwoplayers)

each

player

receives

atleast

50%

oftheirvaluation.

•Envy-Free:nopartyiswillingto

giveupitsallocationin

exchangefortheother

player'sallocation,so

noplayersenvies

anyoneelse.

•Equitable:

each

player

values

itsallocationthesame

accordingto

itsownva

luationfunction.

•E�ciency:

thereisnoother

divisionbetterforeverybody,or

betterforsomeplayersandnotworsefortheothers

Fairness

Conditions

•Proportional:

(fortwoplayers)

each

player

receives

atleast

50%

oftheirvaluation.

•Envy-Free:nopartyiswillingto

giveupitsallocationin

exchangefortheother

player'sallocation,so

noplayersenvies

anyoneelse.

•Equitable:

each

player

values

itsallocationthesame

accordingto

itsownva

luationfunction.

•E�ciency:

thereisnoother

divisionbetterforeverybody,or

betterforsomeplayersandnotworsefortheothers

Truthfulness

Someproceduresask

playersto

representtheirpreferences.

This

representationneednotbe

�truthful�

Typically,itisassumed

thatagents

willfollow

amaxim

instrategy

(maxim

izethesetofitem

sthatare

guaranteed)

Main

References

S.BramsandA.Taylor.FairDivision:From

Cake-Cuttingto

Dispute

Reso-

lution.1996.

J.RobertsonandW.Webb.Cake-CuttingAlgorithms:

BeFair

IfYouCan.

1998.

J.Barbanel.TheGeometryofE�cientFairDivision.2005.

IndivisibleGoods

S.Brams,P.EdelmanandP.Fishburn.ParadoxesofFairDivision.

Journal

ofPhilosophy,98:6(2001).

IndivisibleGoods

S.Brams,P.EdelmanandP.Fishburn.ParadoxesofFairDivision.

Journal

ofPhilosophy,98:6(2001).

•Playerscannotcompensate

each

other

withsidepayments

•Allplayershavepositivevalues

foreveryitem

•LiftPreferencesto

Sets:

Aplayer

prefers

aset

Sto

aset

Tif

�Shasasmanyelem

ents

as

T

�foreveryitem

int∈

T−

Sthereisadistinct

item

s∈

S−

T

thattheplayer

prefers

tot.

IndivisibleGoods:

Envy-Freeness

andE�ciency

Auniqueen

vy-freedivisionmaybe

ine�

cien

t

A:

12

34

56

B:

43

21

56

C:

51

26

34

A:{

1,3}

B:{

2,4}

C:{

5,6}

IndivisibleGoods:

Envy-Freeness

andE�ciency

Auniqueen

vy-freedivisionmaybe

ine�

cien

t

A:

12

34

56

B:

43

21

56

C:

51

26

34

A:{

1,3}

B:{

2,4}

C:{

5,6}

Thisistheuniqueen

vy-freeoutcome.

IndivisibleGoods:

Envy-Freeness

andE�ciency

Auniqueen

vy-freedivisionmaybe

ine�

cien

t

A:

12

34

56

B:

43

21

56

C:

51

26

34

A:{

1,3}

B:{

2,4}

C:{

5,6}

Thedivision

(12,

34,5

6)pareto-dominatestheabovedivision

IndivisibleGoods:

Envy-Freeness

andE�ciency

Auniqueen

vy-freedivisionmaybe

ine�

cien

t

A:

12

34

56

B:

43

21

56

C:

51

26

34

A:{

1,3}

B:{

2,4}

C:{

5,6}

Thedivision

(12,

34,5

6)pareto-dominatestheabovedivision

IndivisibleGoods:

Envy-Freeness

andE�ciency

Auniqueen

vy-freedivisionmaybe

ine�

cien

t

A:

12

34

56

B:

43

21

56

C:

51

26

34

A:{

1,3}

B:{

2,4}

C:{

5,6}

Thedivision

(12,

34,5

6)pareto-dominatestheabovedivision

IndivisibleGoods:

Envy-Freeness

andE�ciency

Auniqueen

vy-freedivisionmaybe

ine�

cien

t

A:

12

34

56

B:

43

21

56

C:

51

26

34

A:{

1,3}

B:{

2,4}

C:{

5,6}

Thedivision

(12,

34,5

6)pareto-dominatestheabovedivision

IndivisibleGoods:

Envy-Freeness

andE�ciency

Auniqueen

vy-freedivisionmaybe

ine�

cien

t

A:

12

34

56

B:

43

21

56

C:

51

26

34

A:{

1,3}

B:{

2,4}

C:{

5,6}

How

ever,(1

2,34

,56)

isnot(necessarily)envy-free

IndivisibleGoods:

Envy-Freeness

andE�ciency

Auniqueen

vy-freedivisionmaybe

ine�

cien

t

A:

12

34

56

B:

43

21

56

C:

51

26

34

A:{

1,3}

B:{

2,4}

C:{

5,6}

Thereisnoother

division,includingane�

cientone,that

guarantees

envy-freeness.

IndivisibleGoods:

Envy-Freeness

andE�ciency

Theremaybe

noen

vy-freedivision,even

when

allplayers

have

di�eren

tpreference

rankings

IndivisibleGoods:

Envy-Freeness

andE�ciency

Theremaybe

noen

vy-freedivision,even

when

allplayers

have

di�eren

tpreference

rankings

Trivialifallplayershavethesamepreference.

IndivisibleGoods:

Envy-Freeness

andE�ciency

Theremaybe

noen

vy-freedivision,even

when

allplayers

have

di�eren

tpreference

rankings A

:1

23

B:

13

2

C:

21

3

Threedivisionsare

e�cient:

(1,3

,2),

(2,1

,3)and

(3,1

,2).

How

ever,noneofthem

are

envy-free.

IndivisibleGoods:

Envy-Freeness

andE�ciency

Theremaybe

noen

vy-freedivision,even

when

allplayers

have

di�eren

tpreference

rankings A

:1

23

B:

13

2

C:

21

3

Threedivisionsare

e�cient:

(1,3

,2),

(2,1

,3)and

(3,1

,2).

How

ever,noneofthem

are

envy-free.

Infact,thereisnoenvy-freedivision.

2Players,1Cake

Twoplayers

Aand

B

Thecakeistheunitinterval[0

,1]

Only

parallel,verticalcuts,perpendicularto

thehorizontalx-axis

are

made

2Players,1Cake

Each

player

hasacontinuousvaluemeasure

v A(x

)and

v B(x

)on

[0,1

]such

that

•v A

(x)≥

0and

v B(x

)≥

0for

x∈

[0,1

]

•v A

and

v Bare

�nitelyadditive,non-atomic,absolutely

continuousmeasures

•theareasunder

v Aand

v Bon

[0,1

]is1(probabilitydensity

function)

2Players,1Cake

Each

player

hasacontinuousvaluemeasure

v A(x

)and

v B(x

)on

[0,1

]such

that

•v A

(x)≥

0and

v B(x

)≥

0for

x∈

[0,1

]

•v A

and

v Bare

�nitelyadditive,non-atomic,absolutely

continuousmeasures

•theareasunder

v Aand

v Bon

[0,1

]is1(probabilitydensity

function)

valueof�nitenumberofdisjointpiecesequals

theva

lueoftheir

union(hen

ce,nosubp

ieceshave

greaterva

luethanthelarger

piece

containingthem

).

2Players,1Cake

Each

player

hasacontinuousvaluemeasure

v A(x

)and

v B(x

)on

[0,1

]such

that

•v A

(x)≥

0and

v B(x

)≥

0for

x∈

[0,1

]

•v A

and

v Bare

�nitelyadditive,non-atomic,absolutely

continuousmeasures

•theareasunder

v Aand

v Bon

[0,1

]is1(probabilitydensity

function)

asinglecu

t(w

hichde�

nes

theborder

ofapiece)hasnoareaandso

hasnova

lue.

2Players,1Cake

Each

player

hasacontinuousvaluemeasure

v A(x

)and

v B(x

)on

[0,1

]such

that

•v A

(x)≥

0and

v B(x

)≥

0for

x∈

[0,1

]

•v A

and

v Bare

�nitelyadditive,non-atomic,absolutely

continuousmeasures

•theareasunder

v Aand

v Bon

[0,1

]is1(probabilitydensity

function)

nopo

rtionofcake

isofpo

sitive

measure

foroneplayerandzero

measure

foranother

player.

CuttingaCake:DivideandChoose

Procedure:oneplayer

cuts

thecakeinto

twoportionsandthe

other

player

choosesone.

Suppose

Aisthecutter.

IfAhasnoinform

ationabouttheother

player'spreferences,then

A

should

cutthecakeatsomepoint

xso

thatthevalueoftheportion

totheleftofxisequalto

thevalueoftheportionto

theright.

Thisstrategycreatesanenvy-freeande�cientallocation,butit

isnotnecessarilyequitable.

CuttingaCake:DivideandChoose

Suppose

Avalues

thevanilla

halftw

iceasmuch

asthechocolate

half.Hence,

v A(x

)=

4/3

x∈

[0,1

/2]

2/3

x∈

(1/2,

1]

v B(x

)=

1/2

x∈

[0,1

/2]

1/2

x∈

(1/2,

1]

Ashould

cutthecakeat

x=

3/8:

(4/3

)(x−

0)=

4/3(

1/2−

x)+

2/3(

1−

1/2)

Note

thattheportionsare

notequitable(B

receive

5/8according

tohisvaluation)

SurplusProcedure

SurplusProcedure

1.Independently,

Aand

Breport

theirvaluefunctions

f Aand

f B

over

[0,1

]to

areferee.

Theseneednotbethesameas

v Aand

v B.

SurplusProcedure

1.Independently,

Aand

Breport

theirvaluefunctions

f Aand

f B

over

[0,1

]to

areferee.

Theseneednotbethesameas

v Aand

v B.

2.Therefereedetermines

the50-50points

aand

bofA

and

B

accordingto

f Aand

f B,respectively.

SurplusProcedure

1.Independently,

Aand

Breport

theirvaluefunctions

f Aand

f B

over

[0,1

]to

areferee.

Theseneednotbethesameas

v Aand

v B.

2.Therefereedetermines

the50-50points

aand

bofA

and

B

accordingto

f Aand

f B,respectively.

3.If

aand

bcoincide,thecakeiscutat

a=

b.Oneplayer

is

randomly

assigned

thepiece

totheleftandtheother

tothe

right.

Theprocedure

ends.

SurplusProcedure

1.Independently,

Aand

Breport

theirvaluefunctions

f Aand

f B

over

[0,1

]to

areferee.

Theseneednotbethesameas

v Aand

v B.

2.Therefereedetermines

the50-50points

aand

bofA

and

B

accordingto

f Aand

f B,respectively.

3.If

aand

bcoincide,thecakeiscutat

a=

b.Oneplayer

is

randomly

assigned

thepiece

totheleftandtheother

tothe

right.

Theprocedure

ends.

4.Suppose

aisto

theleftofb(T

hen

Areceives

[0,a

]andB

receives

[b,1

]).Cutthecakeapoint

cin

[a,b

]atwhichthe

playersreceivethesamepropo

rtion

pofthecakein

this

interval.

SurplusProcedure

Aprocedure

isstrategy-proofifmaxim

inplayersalwayshavean

incentiveto

let

f A=

v Aand

f B=

v B.

Let

cbethecut-pointthatguarantees

proportionalequitabilityand

ethecut-pointthatguarantees

equitabilityofthesurplus.

Theorem

TheSurplusProcedure

isstrategy-proof,whereasany

procedure

thatmakes

ethecut-pointisstrategy-vulnerable.

3Players,2Cuts

FactIt

isnotalwayspossibleto

divideacakeamongthreeplayers

intoenvy-freeandequitable

portionsusing2cuts.

More

than2Players

Adivisionissuper-envyfreeifeveryplayer

feelsallother

players

received

strictly

less

that

1/nofthetotalvalueofthecake.

Theorem

(Barbenel)

Asuper

envy-freedivisionexists

ifand

only

iftheplayer

measuresare

linearlyindependent.

(infact,there

are

in�nitelymanysuch

divisions)

J.Barbanel.Superenvy-freecakedivisionandindependence

ofmeasures.

J.

Math.Anal.Appl.(1996).

Banach-K

nasterLast

Dim

inisherProcedure

Suppose

thereare

ndi�erentagents:

p1,.

..,p

n.

Procedure:

•The�rstperson(p

1)cuts

outapiece

whichheclaim

sishisfair

share.

•Then,thepiece

goes

aroundbeinginspected,in

turn,by

persons

p2,p

3,.

..,p

n.

�Anyonewhothinksthepiece

isnottoolargejust

passes

it.

Anyonewhothinksitistoobig,may

reduce

it,putting

someback

into

themain

part.

Banach-K

nasterLast

Dim

inisherProcedure

•After

thepiece

hasbeeninspectedby

pn,thelast

personwho

reducedthepiece,takes

it.Ifthereisnosuch

person,i.e.,no

onechallenged

p1,then

thepiece

istaken

by

p1.

•Thealgorithm

continues

with

n−

1participants.

Thisprocedure

isequitablebutnotenvy-free

Dubins-SpanierMoving-K

nifeProcedure

Procedure:

Arefereeholdsaknifeattheleftedgeofthecake

andslow

lymoves

itacross

thecakeso

thatitremainsparallelto

its

startingposition.

Atanytime,anyoneofthethreeplayers(A

,B

or

C)cancall�cut�.

When

thisoccurs,theplayer

whocalled

cutreceives

thepiece

to

theleftoftheknifeandexitsthegame.

Dubins-SpanierMoving-K

nifeProcedure

Thegamenow

continues

movinguntilasecondplayer

callscut.

Thesecondplayer

receives

thesecondpiece

andthethirdplayer

getstheremainder.

Ifeither

twoorthreeplayerscallcutatthesametime,thecut

piece

isgiven

tooneofthecallersatrandom.

Thisprocedure

isequitablebutnotenvy-free

OpenQuestions

•3-person,2-cutenvy-freeprocedureshavebeenfound

(Stromquist,1980;BarbanelandBrams,2004)

OpenQuestions

•3-person,2-cutenvy-freeprocedureshavebeenfound

(Stromquist,1980;BarbanelandBrams,2004)

•4-person,3-cutenvyfree

procedure?(U

nknow

n)

�(B

arbanelandBrams,2004):

nomore

than5cuts

are

needed

to

ensure

4-personenvy-freeness.

OpenQuestions

•3-person,2-cutenvy-freeprocedureshavebeenfound

(Stromquist,1980;BarbanelandBrams,2004)

•4-person,3-cutenvyfree

procedure?(U

nknow

n)

�(B

arbanelandBrams,2004):

nomore

than5cuts

are

needed

to

ensure

4-personenvy-freeness.

•Beyond4players,

noprocedure

iskn

own

thatyieldsan

envy-freedivisionofacakeunless

anunbounded

number

ofcuts

isallow

ed(B

ramsandTaylor,1995)

Howaboutsomepie?

Acakeisalinesegmentandbecomes

apiewhen

itsendpoints

are

connectedto

form

acircle.

Thecuts

dividethepieinto

sectors

each

oneofwhichisgiven

toa

player

Gale(1993):

Isthereanallocationofthepiethatisenvy-freeand

undominated?

BarabanelandBrams:

for2playersyes,for3playersenvy-freebut

notnecessarily

undominated,for4playersno.

J.BarbanelandS.Brams.CuttingaPieIs

NotaPiece

ofCake.2005.

References

F.Su.ReviewofCake-CuttingAlgorithms:

BeFair

IfYouCan.American

Mathem

aticalMonthly

(2000).

S.Brams,M.Jones

andC.Klamler.

BetterWaysto

CutaCake.Noticesof

theAMS(2006).

S.BramsandA.Taylor.FairDivision:From

Cake-Cuttingto

Dispute

Reso-

lution.1996.

S.Brams,P.EdelmanandP.Fishburn.ParadoxesofFairDivision.

Journal

ofPhilosophy,98:6(2001).