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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 l.jpg

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

Meg’s Piece

Tom’s Piece

Meg Prefers

Yellow Fish

Tom Prefers

Cat Fish


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Unhappy

Unhappy

CUT

Tom’s Piece

Meg’s 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 doesn’t need to know

  • the user’s 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 1

User 2

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

1

2

2

Phase 2:

Each user cuts according

to some ratio


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2

1

2

There exist utility functions

such that the cuts in each piece overlap


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3

2

3

1

3

2

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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Labels:

00

010

011

10

11

Labeled algorithms:

each piece has a label


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Labels:

00

010

011

10

11

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

01

1

1

0

1

0

1

00

01


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00

010

011

1

1

0

1

0

1

00

0

1

010

011


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00

010

011

10

11

1

0

1

0

0

1

10

11

00

0

1

010

011


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Sorting Labels

00

010

011

10

11

Users receive pieces in arbitrary order:

We would like to sort the pieces:


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Sorting Labels

00

010

011

10

11

Labels will help to sort the pieces


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Sorting Labels

000

010

011

100

110

Normalize the labels


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Sorting Labels

000

010

011

100

110

0

1

2

3

4

5

6

7


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Sorting Labels

000

010

011

100

110

011

0

1

2

3

4

5

6

7


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Sorting Labels

000

010

011

100

110

010

011

0

1

2

3

4

5

6

7


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Sorting Labels

000

010

011

100

110

010

011

110

0

1

2

3

4

5

6

7


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Sorting Labels

000

010

011

100

110

000

010

011

110

0

1

2

3

4

5

6

7


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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


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Sorting Labels

000

010

011

100

110

Time needed:

000

010

011

100

110

0

1

2

3

4

5

6

7


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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

cannot 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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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


Slide85 l.jpg

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

must get a piece from each

of the other user’s gap


Slide94 l.jpg

Strong Envy-Free, Lower Bound

A user needs distinct pieces

Total number of pieces:

Total number of cuts:


Slide95 l.jpg

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


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