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Cake Cutting is Not a Piece of Cake. Malik Magdon-Ismail Costas Busch M. S. Krishnamoorthy . Rensselaer Polytechnic Institute. users wish to share a cake. Fair portion : th of cake. The problem is interesting when people have different preferences. Example:. Meg Prefers

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cake cutting is not a piece of cake

Cake Cutting is Not a Piece of Cake

Malik Magdon-Ismail

Costas Busch

M. S. Krishnamoorthy

Rensselaer Polytechnic Institute

slide2

users wish to share a cake

Fair portion :th of cake

slide3

The problem is interesting when

people have different preferences

Example:

Meg Prefers

Yellow Fish

Tom Prefers

Cat Fish

slide4

Happy

Happy

CUT

Meg’s Piece

Tom’s Piece

Meg Prefers

Yellow Fish

Tom Prefers

Cat Fish

slide5

Unhappy

Unhappy

CUT

Tom’s Piece

Meg’s Piece

Meg Prefers

Yellow Fish

Tom Prefers

Cat Fish

slide6

The cake represents some resource:

  • Property which will be shared or divided
  • The Bandwidth of a communication line
  • Time sharing of a multiprocessor
slide7

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 doesn’t need to know
  • the user’s preferences
slide8

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

slide9

Our contribution:

We show that cuts are required

for the following classes of algorithms:

  • Phased Algorithms

(many algorithms)

  • Labeled Algorithms

(all known algorithms)

slide10

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

slide11

Talk Outline

Cake Cutting Algorithms

Lower Bound for Phased Algorithms

Lower Bound for Labeled Algorithms

Lower Bound for Envy-Free Algorithms

Conclusions

slide12

Cake

knife

slide13

Cake

cut

knife

slide14

Cake

Utility Function for user

slide15

Cake

Value of piece:

slide16

Cake

Value of piece:

slide17

Cake

Utility Density Function for user

slide18

“I cut you choose”

Step 1:

User 1 cuts at

Step 2:

User 2 chooses a piece

slide19

“I cut you choose”

Step 1:

User 1 cuts at

slide20

“I cut you choose”

User 2

Step 2:

User 2 chooses a piece

slide21

“I cut you choose”

User 1

User 2

Both users get at least of the cake

Both are happy

slide22

Algorithm

users

Each user cuts at

Phase 1:

slide23

Algorithm

users

Each user cuts at

Phase 1:

slide24

Algorithm

users

Phase 1:

Give the leftmost piece to the

respective user

slide25

Algorithm

users

Each user cuts at

Phase 2:

slide26

Algorithm

users

Each user cuts at

Phase 2:

slide27

Algorithm

users

Phase 2:

Give the leftmost piece to the

respective user

slide28

Algorithm

users

Each user cuts at

Phase 3:

And so on…

slide29

Algorithm

Total number of phases:

Total number of cuts:

slide30

Algorithm

users

Each user cuts at

Phase 1:

slide31

Algorithm

users

Each user cuts at

Phase 1:

slide32

Algorithm

users

users

Find middle cut

Phase 1:

slide33

Algorithm

users

Each user cuts at

Phase 2:

slide34

Algorithm

users

Each user cuts at

Phase 2:

slide35

Algorithm

users

Find middle cut

Phase 2:

slide36

Algorithm

users

Each user cuts at

Phase 3:

And so on…

slide37

Algorithm

user

The user is assigned the piece

Phase log N:

slide38

Algorithm

Total number of phases:

Total number of cuts:

slide39

Talk Outline

Cake Cutting Algorithms

Lower Bound for Phased Algorithms

Lower Bound for Labeled Algorithms

Lower Bound for Envy-Free Algorithms

Conclusions

slide40

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

slide41

Observation:

Algorithms and are

phased

slide42

We show:

cuts are required

to assign positive valued pieces

slide43

1

1

1

1

Phase 1:

Each user cuts according

to some ratio

slide44

1

There exist utility functions

such that the cuts overlap

slide45

2

2

1

2

2

Phase 2:

Each user cuts according

to some ratio

slide46

2

1

2

There exist utility functions

such that the cuts in each piece overlap

slide47

3

2

3

1

3

2

3

number of pieces

at most are doubled

Phase 3:

And so on…

slide48

Phase k:

Number of pieces at most

slide49

For users:

we need at least pieces

we need at least phases

slide50

Phase

Users

Pieces

Cuts

(min)

(min)

(max)

……

……

……

……

Total Cuts:

slide51

Talk Outline

Cake Cutting Algorithms

Lower Bound for Phased Algorithms

Lower Bound for Labeled Algorithms

Lower Bound for Envy-Free Algorithms

Conclusions

slide52

Labels:

00

010

011

10

11

Labeled algorithms:

each piece has a label

slide53

Labels:

00

010

011

10

11

Labeling Tree:

1

0

1

0

0

1

00

10

11

0

1

010

011

slide55

0

1

1

0

1

0

slide56

00

01

1

1

0

1

0

1

00

01

slide57

00

010

011

1

1

0

1

0

1

00

0

1

010

011

slide58

00

010

011

10

11

1

0

1

0

0

1

10

11

00

0

1

010

011

slide59

Sorting Labels

00

010

011

10

11

Users receive pieces in arbitrary order:

We would like to sort the pieces:

slide60

Sorting Labels

00

010

011

10

11

Labels will help to sort the pieces

slide61

Sorting Labels

000

010

011

100

110

Normalize the labels

slide62

Sorting Labels

000

010

011

100

110

0

1

2

3

4

5

6

7

slide63

Sorting Labels

000

010

011

100

110

011

0

1

2

3

4

5

6

7

slide64

Sorting Labels

000

010

011

100

110

010

011

0

1

2

3

4

5

6

7

slide65

Sorting Labels

000

010

011

100

110

010

011

110

0

1

2

3

4

5

6

7

slide66

Sorting Labels

000

010

011

100

110

000

010

011

110

0

1

2

3

4

5

6

7

slide67

Sorting Labels

000

010

011

100

110

Labels and pieces are ordered!

000

010

011

100

110

0

1

2

3

4

5

6

7

slide68

Sorting Labels

000

010

011

100

110

Time needed:

000

010

011

100

110

0

1

2

3

4

5

6

7

slide69

linearly-labeled &

comparison-bounded algorithms:

Require comparisons

(including handling and sorting labels)

slide70

Observation:

Algorithms and are

linearly-labeled &

comparison-bounded

Conjecture:

All known algorithms are

linearly-labeled

& comparison-bounded

slide71

We will show that cuts

are needed for

linearly-labeled & comparison-bounded

algorithms

slide72

Reduction of Sorting to Cake Cutting

Input:

distinct positive integers:

Output:

Sorted order:

Using a cake-cutting algorithm

slide73

distinct positive integers:

utility functions:

users:

slide77

Cake

cannot be satisfied!

slide79

Cake

Piece:

Rightmost positive valued pieces

slide80

Labels:

Sorted labels:

Sorted pieces:

Sorted integers:

slide82

Sorting integers:

comparisons

comparisons

Cake Cutting:

slide83

Linearly-labeled &

comparison-bounded algorithms:

Require comparisons

comparisons

require

cuts

slide84

Talk Outline

Cake Cutting Algorithms

Lower Bound for Phased Algorithms

Lower Bound for Labeled Algorithms

Lower Bound for Envy-Free Algorithms

Conclusions

slide85

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

slide86

Super Envy-free:

A user feels she gets a fair portion,

and every other user gets less than fair

slide87

Lower Bounds

cuts

Strong Envy-free:

Super Envy-free:

cuts

slide93

Strong Envy-Free, Lower Bound

must get a piece from each

of the other user’s gap

slide94

Strong Envy-Free, Lower Bound

A user needs distinct pieces

Total number of pieces:

Total number of cuts:

slide95

Talk Outline

Cake Cutting Algorithms

Lower Bound for Phased Algorithms

Lower Bound for Labeled Algorithms

Lower Bound for Envy-Free Algorithms

Conclusions

slide96

We presented new lower bounds

for several classes of

fair cake-cutting algorithms

slide97

Open problems:

  • Prove or disprove that every algorithm
  • is linearly-labeled and comp.-bounded
  • An improved lower bound for
  • envy-free algorithms