Title: Cake Cutting is Not a Piece of Cake
1Cake 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
3The problem is interesting when people have
different preferences
Example
Meg Prefers Yellow Fish
Tom Prefers Cat Fish
4Happy
Happy
CUT
Megs Piece
Toms Piece
Meg Prefers Yellow Fish
Tom Prefers Cat Fish
5Unhappy
Unhappy
CUT
Toms Piece
Megs Piece
Meg Prefers Yellow Fish
Tom Prefers Cat Fish
6The cake represents some resource
- Property which will be shared or divided
- The Bandwidth of a communication line
- Time sharing of a multiprocessor
7Fair 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
8For 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
9Our contribution
We show that cuts are
required for the following classes of algorithms
many algorithms
all known algorithms
10Our contribution
We show that cuts are required for
special cases of envyfree algorithms
Each user feels she gets more than the other
users
11Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for EnvyFree
Algorithms Conclusions
12Cake
knife
13Cake
cut
knife
14Cake
Utility Function for user
15Cake
Value of piece
16Cake
Value of piece
17Cake
Utility Density Function for user
18I cut you choose
Step 1
User 1 cuts at
Step 2
User 2 chooses a piece
19I cut you choose
Step 1
User 1 cuts at
20I cut you choose
User 2
Step 2
User 2 chooses a piece
21I cut you choose
User 2
User 1
Both users get at least of the cake
Both are happy
22Algorithm
users
Each user cuts at
Phase 1
23Algorithm
users
Each user cuts at
Phase 1
24Algorithm
users
Phase 1
Give the leftmost piece to the respective user
25Algorithm
users
Each user cuts at
Phase 2
26Algorithm
users
Each user cuts at
Phase 2
27Algorithm
users
Phase 2
Give the leftmost piece to the respective user
28Algorithm
users
Each user cuts at
Phase 3
And so on
29Algorithm
Total number of phases
Total number of cuts
30Algorithm
users
Each user cuts at
Phase 1
31Algorithm
users
Each user cuts at
Phase 1
32Algorithm
users
users
Find middle cut
Phase 1
33Algorithm
users
Each user cuts at
Phase 2
34Algorithm
users
Each user cuts at
Phase 2
35Algorithm
users
Find middle cut
Phase 2
36Algorithm
users
Each user cuts at
Phase 3
And so on
37Algorithm
user
The user is assigned the piece
Phase log N
38Algorithm
Total number of phases
Total number of cuts
39Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for EnvyFree
Algorithms Conclusions
40Phased 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
41Observation
Algorithms and are phased
42We show
cuts are required to assign
positive valued pieces
431
1
1
1
Phase 1
Each user cuts according to some ratio
441
There exist utility functions such that the cuts
overlap
452
2
2
2
1
Phase 2
Each user cuts according to some ratio
461
2
2
There exist utility functions such that the cuts
in each piece overlap
471
2
2
3
3
3
3
number of pieces at most are doubled
Phase 3
And so on
48Phase k
Number of pieces at most
49For users
we need at least pieces
we need at least phases
50Phase
Users
Pieces
Cuts
min
min
max
Total Cuts
51Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for EnvyFree
Algorithms Conclusions
5211
10
011
010
00
Labels
each piece has a label
Labeled algorithms
5311
10
011
010
00
Labels
Labeling Tree
1
0
1
0
0
1
00
10
11
0
1
010
011
54No Transcript
550
1
1
0
1
0
5600
1
01
1
0
1
0
1
00
01
5700
011
010
1
1
0
1
0
1
00
0
1
010
011
5811
10
011
010
00
1
0
1
0
0
1
10
11
00
0
1
010
011
59Sorting Labels
11
10
011
010
00
Users receive pieces in arbitrary order
We would like to sort the pieces
60Sorting Labels
11
10
011
010
00
Labels will help to sort the pieces
61Sorting Labels
110
100
011
010
000
Normalize the labels
62Sorting Labels
110
100
011
010
000
0
1
2
3
4
5
6
7
63Sorting Labels
110
100
011
010
000
011
0
1
2
3
4
5
6
7
64Sorting Labels
110
100
011
010
000
011
010
0
1
2
3
4
5
6
7
65Sorting Labels
110
100
011
010
000
011
010
110
0
1
2
3
4
5
6
7
66Sorting Labels
110
100
011
010
000
011
010
110
000
0
1
2
3
4
5
6
7
67Sorting Labels
110
100
011
010
000
Labels and pieces are ordered
011
010
110
000
100
0
1
2
3
4
5
6
7
68Sorting Labels
110
100
011
010
000
Time needed
011
010
110
000
100
0
1
2
3
4
5
6
7
69linearlylabeled comparisonbounded algorithms
Require comparisons
including handling and sorting labels
70Observation
Algorithms and are linearlylabeled
comparisonbounded
Conjecture
All known algorithms are linearlylabeled
comparisonbounded
71We will show that cuts are
needed for linearlylabeled comparisonbounded
algorithms
72Reduction of Sorting to Cake Cutting
Input
distinct positive integers
Output
Sorted order
Using a cakecutting algorithm
73distinct positive integers
utility functions
users
74Cake
75Cake
76Cake
77Cake
cannot be satisfied
78can be satisfied
79Cake
Piece
Rightmost positive valued pieces
80Labels
Sorted labels
Sorted pieces
Sorted integers
81Fair cakecutting
Sorting
82Sorting integers
comparisons
comparisons
Cake Cutting
83Linearlylabeled comparisonbounded algorithms
Require comparisons
comparisons
require
cuts
84Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for EnvyFree
Algorithms Conclusions
85Variations 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
86Super Envyfree
A user feels she gets a fair portion and every
other user gets less than fair
87Lower Bounds
cuts
Strong Envyfree
Super Envyfree
cuts
88Strong EnvyFree Lower Bound
89Strong EnvyFree Lower Bound
90Strong EnvyFree Lower Bound
91Strong EnvyFree Lower Bound
is upset
92Strong EnvyFree Lower Bound
is happy
93Strong EnvyFree Lower Bound
must get a piece from each of the other
users gap
94Strong EnvyFree Lower Bound
A user needs distinct pieces
Total number of pieces
Total number of cuts
95Talk Outline
Cake Cutting Algorithms Lower Bound for Phased
Algorithms Lower Bound for Labeled
Algorithms Lower Bound for EnvyFree
Algorithms Conclusions
96We presented new lower bounds for several classes
of fair cakecutting algorithms
97Open problems
- Prove or disprove that every algorithm
- is linearlylabeled and comp.bounded
- An improved lower bound for
- envyfree algorithms