Cake Cutting is Not a Piece of Cake


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Cake Cutting is Not a Piece of Cake
Malik Magdon-Ismail Costas Busch M. S.
Krishnamoorthy

Rensselaer Polytechnic Institute
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users wish to share a cake
Fair portion th of cake
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The problem is interesting when people have
different preferences
Example
Meg Prefers Yellow Fish
Tom Prefers Cat Fish
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Happy
Happy
CUT
Megs Piece
Toms Piece
Meg Prefers Yellow Fish
Tom Prefers Cat Fish
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Unhappy
Unhappy
CUT
Toms Piece
Megs Piece
Meg Prefers Yellow Fish
Tom Prefers Cat Fish
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The cake represents some resource

• Property which will be shared or divided

• The Bandwidth of a communication line

• Time sharing of a multiprocessor

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Fair Cake-Cutting Algorithms

• Each user gets what she considers
• to be th of the cake

• Specify how each user cuts the cake

• The algorithm doesnt need to know
• the users preferences

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For users it is known how to divide the
cake fairly with cuts
Steinhaus 1948 The problem of fair division
It is not known if we can do better than
cuts
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Our contribution
We show that cuts are
required for the following classes of algorithms

• Phased Algorithms

(many algorithms)

• Labeled Algorithms

(all known algorithms)
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Our contribution
We show that cuts are required for
special cases of envy-free algorithms
Each user feels she gets more than the other
users
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Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for Envy-Free
Algorithms Conclusions
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Cake
knife
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Cake
cut
knife
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Cake
Utility Function for user
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Cake
Value of piece
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Cake
Value of piece
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Cake
Utility Density Function for user
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I cut you choose
Step 1
User 1 cuts at
Step 2
User 2 chooses a piece
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I cut you choose
Step 1
User 1 cuts at
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I cut you choose
User 2
Step 2
User 2 chooses a piece
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I cut you choose
User 2
User 1
Both users get at least of the cake
Both are happy
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Algorithm
users
Each user cuts at
Phase 1
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Algorithm
users
Each user cuts at
Phase 1
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Algorithm
users
Phase 1
Give the leftmost piece to the respective user
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Algorithm
users
Each user cuts at
Phase 2
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Algorithm
users
Each user cuts at
Phase 2
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Algorithm
users
Phase 2
Give the leftmost piece to the respective user
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Algorithm
users
Each user cuts at
Phase 3
And so on
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Algorithm
Total number of phases
Total number of cuts
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Algorithm
users
Each user cuts at
Phase 1
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Algorithm
users
Each user cuts at
Phase 1
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Algorithm
users
users
Find middle cut
Phase 1
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Algorithm
users
Each user cuts at
Phase 2
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Algorithm
users
Each user cuts at
Phase 2
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Algorithm
users
Find middle cut
Phase 2
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Algorithm
users
Each user cuts at
Phase 3
And so on
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Algorithm
user
The user is assigned the piece
Phase log N
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Algorithm
Total number of phases
Total number of cuts
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Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for Envy-Free
Algorithms Conclusions
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Phased algorithm
consists of a sequence of phases
At each phase
Each user cuts a piece which is defined in
previous phases
A user may be assigned a piece in any phase
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Observation
Algorithms and are phased
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We show
cuts are required to assign
positive valued pieces
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1
1
1
1
Phase 1
Each user cuts according to some ratio
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1
There exist utility functions such that the cuts
overlap
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2
2
2
2
1
Phase 2
Each user cuts according to some ratio
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1
2
2
There exist utility functions such that the cuts
in each piece overlap
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1
2
2
3
3
3
3
number of pieces at most are doubled
Phase 3
And so on
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Phase k
Number of pieces at most
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For users
we need at least pieces
we need at least phases
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Phase
Users
Pieces
Cuts
(min)
(min)
(max)




Total Cuts
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Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for Envy-Free
Algorithms Conclusions
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10
011
010
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Labels
each piece has a label
Labeled algorithms
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11
10
011
010
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Labels
Labeling Tree
1
0
1
0
0
1
00
10
11
0
1
010
011
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0
1
1
0
1
0
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00
1
01
1
0
1
0
1
00
01
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00
011
010
1
1
0
1
0
1
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0
1
010
011
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11
10
011
010
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1
0
1
0
0
1
10
11
00
0
1
010
011
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Sorting Labels
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10
011
010
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Users receive pieces in arbitrary order
We would like to sort the pieces
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Sorting Labels
11
10
011
010
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Labels will help to sort the pieces
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Sorting Labels
110
100
011
010
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Normalize the labels
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Sorting Labels
110
100
011
010
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Sorting Labels
110
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011
010
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011
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Sorting Labels
110
100
011
010
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011
010
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Sorting Labels
110
100
011
010
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011
010
110
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Sorting Labels
110
100
011
010
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011
010
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Sorting Labels
110
100
011
010
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Labels and pieces are ordered!
011
010
110
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100
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Sorting Labels
110
100
011
010
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Time needed
011
010
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100
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linearly-labeled comparison-bounded algorithms
Require comparisons
(including handling and sorting labels)
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Observation
Algorithms and are linearly-labeled
comparison-bounded
Conjecture
All known algorithms are linearly-labeled
comparison-bounded
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We will show that cuts are
needed for linearly-labeled comparison-bounded
algorithms
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Reduction of Sorting to Cake Cutting
Input
distinct positive integers
Output
Sorted order
Using a cake-cutting algorithm
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distinct positive integers
utility functions
users
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Cake
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Cake
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Cake
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Cake
cannot be satisfied!
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can be satisfied!
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Cake
Piece
Rightmost positive valued pieces
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Labels
Sorted labels
Sorted pieces
Sorted integers
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Fair cake-cutting
Sorting
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Sorting integers
comparisons
comparisons
Cake Cutting
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Linearly-labeled comparison-bounded algorithms
Require comparisons
comparisons
require
cuts
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Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for Envy-Free
Algorithms Conclusions
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Variations of Fair Cake-Division
Envy-free
Each user feels she gets at least as much as the
other users
Strong Envy-free
Each user feels she gets strictly more Than the
other users
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Super Envy-free
A user feels she gets a fair portion and every
other user gets less than fair
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Lower Bounds
cuts
Strong Envy-free
Super Envy-free
cuts
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Strong Envy-Free Lower Bound
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Strong Envy-Free Lower Bound
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Strong Envy-Free Lower Bound
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Strong Envy-Free Lower Bound
is upset!
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Strong Envy-Free Lower Bound
is happy!
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Strong Envy-Free Lower Bound
must get a piece from each of the other
users gap
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Strong Envy-Free Lower Bound
A user needs distinct pieces
Total number of pieces
Total number of cuts
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Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for Envy-Free
Algorithms Conclusions
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We presented new lower bounds for several classes
of fair cake-cutting algorithms
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Open problems

• Prove or disprove that every algorithm
• is linearly-labeled and comp.-bounded

• An improved lower bound for
• envy-free algorithms