Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and...
Transcript of Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and...
![Page 1: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/1.jpg)
CakeCuttingAlgorithms
EricPacuit
January
7,2007
ILLC,University
ofAmsterdam
staff.science.uva.nl/∼epacuit
![Page 2: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/2.jpg)
PlanforToday
Discuss
somefairdivisionalgorithms
•Whatdoes
itmeanto
�fairly�dividegoods?
•IndivisibleGoods
•DivisibleGoods(C
uttingaCake)
�DivideandChoose
�SurplusProcedure
�Banach-K
naster
Last
Dim
inisher
�Dubins-SpanierMovingKnifeProcedure
![Page 3: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/3.jpg)
Main
Question
How
dowecu
tacake
fairly?
![Page 4: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/4.jpg)
Main
Question
How
dowecu
tacake
fairly?
•anydesirablesetofgood
s(orchoresormixtures)
•each
may
bedivisibleorindivisible
•theremay
berestrictions(such
asthenumber
ofgoodsaplayer
may
receive)
![Page 5: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/5.jpg)
Main
Question
How
dowecu
tacake
fairly?
•discreteprocedure
•continuousmovingknifeprocedures
•compensationprocedures(usingmoney
asadivisiblemedium
forindivisibleobjects)
![Page 6: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/6.jpg)
Main
Question
How
dowecu
tacake
fairly?
•Interested
notonly
intheexistence
ofa(fair)divisionbutalso
aconstructiveprocedure
(analgorithm)for�ndingit
![Page 7: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/7.jpg)
Main
Question
How
dowecu
tacake
fairly?
•Di�erentresultsknow
nfor2,3,4,.
..cutters!
![Page 8: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/8.jpg)
Main
Question
How
dowecu
tacake
fairly?
•Manywaysto
makethisprecise!
![Page 9: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/9.jpg)
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
![Page 10: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/10.jpg)
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
![Page 11: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/11.jpg)
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
![Page 12: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/12.jpg)
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
![Page 13: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/13.jpg)
Truthfulness
Someproceduresask
playersto
representtheirpreferences.
This
representationneednotbe
�truthful�
Typically,itisassumed
thatagents
willfollow
amaxim
instrategy
(maxim
izethesetofitem
sthatare
guaranteed)
![Page 14: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/14.jpg)
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.
![Page 15: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/15.jpg)
IndivisibleGoods
S.Brams,P.EdelmanandP.Fishburn.ParadoxesofFairDivision.
Journal
ofPhilosophy,98:6(2001).
![Page 16: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/16.jpg)
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.
![Page 17: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/17.jpg)
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}
![Page 18: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/18.jpg)
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.
![Page 19: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/19.jpg)
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
![Page 20: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/20.jpg)
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
![Page 21: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/21.jpg)
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
![Page 22: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/22.jpg)
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
![Page 23: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/23.jpg)
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
![Page 24: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/24.jpg)
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.
![Page 25: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/25.jpg)
IndivisibleGoods:
Envy-Freeness
andE�ciency
Theremaybe
noen
vy-freedivision,even
when
allplayers
have
di�eren
tpreference
rankings
![Page 26: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/26.jpg)
IndivisibleGoods:
Envy-Freeness
andE�ciency
Theremaybe
noen
vy-freedivision,even
when
allplayers
have
di�eren
tpreference
rankings
Trivialifallplayershavethesamepreference.
![Page 27: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/27.jpg)
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.
![Page 28: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/28.jpg)
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.
![Page 29: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/29.jpg)
2Players,1Cake
Twoplayers
Aand
B
Thecakeistheunitinterval[0
,1]
Only
parallel,verticalcuts,perpendicularto
thehorizontalx-axis
are
made
![Page 30: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/30.jpg)
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)
![Page 31: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/31.jpg)
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
).
![Page 32: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/32.jpg)
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.
![Page 33: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/33.jpg)
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.
![Page 34: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/34.jpg)
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.
![Page 35: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/35.jpg)
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)
![Page 36: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/36.jpg)
SurplusProcedure
![Page 37: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/37.jpg)
SurplusProcedure
1.Independently,
Aand
Breport
theirvaluefunctions
f Aand
f B
over
[0,1
]to
areferee.
Theseneednotbethesameas
v Aand
v B.
![Page 38: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/38.jpg)
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.
![Page 39: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/39.jpg)
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.
![Page 40: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/40.jpg)
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.
![Page 41: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/41.jpg)
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.
![Page 42: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/42.jpg)
3Players,2Cuts
FactIt
isnotalwayspossibleto
divideacakeamongthreeplayers
intoenvy-freeandequitable
portionsusing2cuts.
![Page 43: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/43.jpg)
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).
![Page 44: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/44.jpg)
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.
![Page 45: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/45.jpg)
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
![Page 46: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/46.jpg)
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.
![Page 47: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/47.jpg)
Dubins-SpanierMoving-K
nifeProcedure
Thegamenow
continues
movinguntilasecondplayer
callscut.
Thesecondplayer
receives
thesecondpiece
andthethirdplayer
getstheremainder.
Ifeither
twoorthreeplayerscallcutatthesametime,thecut
piece
isgiven
tooneofthecallersatrandom.
Thisprocedure
isequitablebutnotenvy-free
![Page 48: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/48.jpg)
OpenQuestions
•3-person,2-cutenvy-freeprocedureshavebeenfound
(Stromquist,1980;BarbanelandBrams,2004)
![Page 49: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/49.jpg)
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.
![Page 50: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/50.jpg)
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)
![Page 51: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/51.jpg)
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.
![Page 52: Algorithms - Artificial Intelligenceai.stanford.edu/~epacuit/classes/cakecutlec.pdfcedure 1., A and B functions f A and f B er [0, 1] s v A and v B. 2. ts a and b of A and B to f A](https://reader034.fdocuments.in/reader034/viewer/2022050503/5f94e98bfa1cbb509b15f540/html5/thumbnails/52.jpg)
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).