Cake Cutting is Not a Piece of Cake


1
Cake Cutting is Not a Piece of Cake
Malik MagdonIsmail Costas Busch M. S.
Krishnamoorthy

Rensselaer Polytechnic Institute
2
users wish to share a cake
Fair portion th of cake
3
The problem is interesting when people have
different preferences
Example
Meg Prefers Yellow Fish
Tom Prefers Cat Fish
4
Happy
Happy
CUT
Megs Piece
Toms Piece
Meg Prefers Yellow Fish
Tom Prefers Cat Fish
5
Unhappy
Unhappy
CUT
Toms Piece
Megs Piece
Meg Prefers Yellow Fish
Tom Prefers Cat Fish
6
The cake represents some resource

• Property which will be shared or divided

• The Bandwidth of a communication line

• Time sharing of a multiprocessor

7
Fair CakeCutting 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

8
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
9
Our contribution
We show that cuts are
required for the following classes of algorithms

• Phased Algorithms

many algorithms

• Labeled Algorithms

all known algorithms
10
Our contribution
We show that cuts are required for
special cases of envyfree algorithms
Each user feels she gets more than the other
users
11
Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for EnvyFree
Algorithms Conclusions
12
Cake
knife
13
Cake
cut
knife
14
Cake
Utility Function for user
15
Cake
Value of piece
16
Cake
Value of piece
17
Cake
Utility Density Function for user
18
I cut you choose
Step 1
User 1 cuts at
Step 2
User 2 chooses a piece
19
I cut you choose
Step 1
User 1 cuts at
20
I cut you choose
User 2
Step 2
User 2 chooses a piece
21
I cut you choose
User 2
User 1
Both users get at least of the cake
Both are happy
22
Algorithm
users
Each user cuts at
Phase 1
23
Algorithm
users
Each user cuts at
Phase 1
24
Algorithm
users
Phase 1
Give the leftmost piece to the respective user
25
Algorithm
users
Each user cuts at
Phase 2
26
Algorithm
users
Each user cuts at
Phase 2
27
Algorithm
users
Phase 2
Give the leftmost piece to the respective user
28
Algorithm
users
Each user cuts at
Phase 3
And so on
29
Algorithm
Total number of phases
Total number of cuts
30
Algorithm
users
Each user cuts at
Phase 1
31
Algorithm
users
Each user cuts at
Phase 1
32
Algorithm
users
users
Find middle cut
Phase 1
33
Algorithm
users
Each user cuts at
Phase 2
34
Algorithm
users
Each user cuts at
Phase 2
35
Algorithm
users
Find middle cut
Phase 2
36
Algorithm
users
Each user cuts at
Phase 3
And so on
37
Algorithm
user
The user is assigned the piece
Phase log N
38
Algorithm
Total number of phases
Total number of cuts
39
Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for EnvyFree
Algorithms Conclusions
40
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
41
Observation
Algorithms and are phased
42
We show
cuts are required to assign
positive valued pieces
43
1
1
1
1
Phase 1
Each user cuts according to some ratio
44
1
There exist utility functions such that the cuts
overlap
45
2
2
2
2
1
Phase 2
Each user cuts according to some ratio
46
1
2
2
There exist utility functions such that the cuts
in each piece overlap
47
1
2
2
3
3
3
3
number of pieces at most are doubled
Phase 3
And so on
48
Phase k
Number of pieces at most
49
For users
we need at least pieces
we need at least phases
50
Phase
Users
Pieces
Cuts
min
min
max




Total Cuts
51
Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for EnvyFree
Algorithms Conclusions
52
11
10
011
010
00
Labels
each piece has a label
Labeled algorithms
53
11
10
011
010
00
Labels
Labeling Tree
1
0
1
0
0
1
00
10
11
0
1
010
011
54
No Transcript
55
0
1
1
0
1
0
56
00
1
01
1
0
1
0
1
00
01
57
00
011
010
1
1
0
1
0
1
00
0
1
010
011
58
11
10
011
010
00
1
0
1
0
0
1
10
11
00
0
1
010
011
59
Sorting Labels
11
10
011
010
00
Users receive pieces in arbitrary order
We would like to sort the pieces
60
Sorting Labels
11
10
011
010
00
Labels will help to sort the pieces
61
Sorting Labels
110
100
011
010
000
Normalize the labels
62
Sorting Labels
110
100
011
010
000
0
1
2
3
4
5
6
7
63
Sorting Labels
110
100
011
010
000
011
0
1
2
3
4
5
6
7
64
Sorting Labels
110
100
011
010
000
011
010
0
1
2
3
4
5
6
7
65
Sorting Labels
110
100
011
010
000
011
010
110
0
1
2
3
4
5
6
7
66
Sorting Labels
110
100
011
010
000
011
010
110
000
0
1
2
3
4
5
6
7
67
Sorting Labels
110
100
011
010
000
Labels and pieces are ordered
011
010
110
000
100
0
1
2
3
4
5
6
7
68
Sorting Labels
110
100
011
010
000
Time needed
011
010
110
000
100
0
1
2
3
4
5
6
7
69
linearlylabeled comparisonbounded algorithms
Require comparisons
including handling and sorting labels
70
Observation
Algorithms and are linearlylabeled
comparisonbounded
Conjecture
All known algorithms are linearlylabeled
comparisonbounded
71
We will show that cuts are
needed for linearlylabeled comparisonbounded
algorithms
72
Reduction of Sorting to Cake Cutting
Input
distinct positive integers
Output
Sorted order
Using a cakecutting algorithm
73
distinct positive integers
utility functions
users
74
Cake
75
Cake
76
Cake
77
Cake
cannot be satisfied
78
can be satisfied
79
Cake
Piece
Rightmost positive valued pieces
80
Labels
Sorted labels
Sorted pieces
Sorted integers
81
Fair cakecutting
Sorting
82
Sorting integers
comparisons
comparisons
Cake Cutting
83
Linearlylabeled comparisonbounded algorithms
Require comparisons
comparisons
require
cuts
84
Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for EnvyFree
Algorithms Conclusions
85
Variations of Fair CakeDivision
Envyfree
Each user feels she gets at least as much as the
other users
Strong Envyfree
Each user feels she gets strictly more Than the
other users
86
Super Envyfree
A user feels she gets a fair portion and every
other user gets less than fair
87
Lower Bounds
cuts
Strong Envyfree
Super Envyfree
cuts
88
Strong EnvyFree Lower Bound
89
Strong EnvyFree Lower Bound
90
Strong EnvyFree Lower Bound
91
Strong EnvyFree Lower Bound
is upset
92
Strong EnvyFree Lower Bound
is happy
93
Strong EnvyFree Lower Bound
must get a piece from each of the other
users gap
94
Strong EnvyFree Lower Bound
A user needs distinct pieces
Total number of pieces
Total number of cuts
95
Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for EnvyFree
Algorithms Conclusions
96
We presented new lower bounds for several classes
of fair cakecutting algorithms
97
Open problems

• Prove or disprove that every algorithm
• is linearlylabeled and comp.bounded

• An improved lower bound for
• envyfree algorithms