cutting a birthday cake

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CUTTING A BIRTHDAY CAKE Yonatan Aumann, Bar Ilan University

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Cutting a Birthday Cake. Yonatan Aumann, Bar Ilan University. How should the cake be divided? . “I love white decorations”. “I want lots of flowers”. “No writing on my piece at all!”. Model. The cake: 1-dimentional the interval [0,1] Valuations: Non atomic measures on [0,1] - PowerPoint PPT Presentation

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Page 1: Cutting a Birthday Cake

CUTTING A BIRTHDAY CAKEYonatan Aumann, Bar Ilan University

Page 2: Cutting a Birthday Cake

How should the cake be divided?

“I want lots of flowers”

“I love white decorations”

“No writing on my piece at all!”

Page 3: Cutting a Birthday Cake

Model The cake:

1-dimentional the interval [0,1]

Valuations: Non atomic measures on [0,1] Normalized: the entire cake is worth 1

Division: Single piece to each player, or Any number of pieces

Page 4: Cutting a Birthday Cake

How should the cake be divided?

“I want lots of flowers”

“I love white decorations”

“No writing on my piece at all!”

Page 5: Cutting a Birthday Cake

Fair Division

Proportional: Each player gets a piece worth to her

at least 1/n

Envy Free:No player prefers a piece allotted to

someone else

Equitable:All players assign the same value to

their allotted pieces

Page 6: Cutting a Birthday Cake

Cut and Choose

Alice likes the candies Bob likes the base

Alice cuts in the middleBob chooses

BobAlice

Proportional Envy free Equitable

Page 7: Cutting a Birthday Cake

Previous Work Problem first presented by H. Steinhaus (1940) Existence theorems (e.g. [DS61,Str80]) Algorithms for different variants of the

problem: Finite Algorithms (e.g. [Str49,EP84]) “Moving knife” algorithms (e.g. [Str80])

Lower bounds on the number of steps required for divisions (e.g. [SW03,EP06,Pro09])

Books: [BT96,RW98,Mou04]

Page 8: Cutting a Birthday Cake

Player 1 Player 2

Example

Players 3,4

Total: 1.5

Total: 2Player 1 Player 3 Player 2Player 4Player 1 Player 2

Fairness Maximum Utility

Page 9: Cutting a Birthday Cake

Social Welfare Utilitarian: Sum of players’

utilities

Egalitarian: Minimum of players’ utilities

Page 10: Cutting a Birthday Cake

with Y. Dombb

Fairness vs. Welfare

Page 11: Cutting a Birthday Cake

The Price of Fairness Given an instance:

max welfare using any divisionmax welfare using fair division

PoF =

Price of equitability

Price of proportionali

ty

Price of envy-

freeness

utilitarian

egalitarian

Page 12: Cutting a Birthday Cake

Player 1 Player 2

Example

Players 3,4

Total: 1.5

Total: 2

Utilitarian Price of Envy-Freeness:

4/3

Envy-free Utilitarian optimum

Page 13: Cutting a Birthday Cake

The Price of Fairness Given an instance:

max welfare using any divisionmax welfare using fair division

PoF =

Seek bounds on the Price of Fairness

First defined in [CKKK09] for non-connected divisions

Page 14: Cutting a Birthday Cake

Results

Price of Proportionality

Envy freeness

Equitability

Utilitarian

Egalitarian1 1

)1(2

On )1(On

2n

Page 15: Cutting a Birthday Cake

Utilitarian Price of Envy FreenessLower Bound

nPlayer

1Player

2Player

3Player

3

nBest possible utilitarian:

Best proportional/envy-free utilitarian:

11 nn

n

players nn

Utilitarian Price of envy-freeness: 2/n

2

Page 16: Cutting a Birthday Cake

Utilitarian Price of Envy FreenessUpper Bound

Key observation:In order to increase a player’s utility by , her new piece must span at least (-1) cuts.

Envy-free piece x

new piece: xnew piece:

2xnew piece:

3x

Page 17: Cutting a Birthday Cake

Utilitarian Price of Envy FreenessUpper Bound

inix

in

x

n

i

ii

i

i

}1,...,1,0{1)1(

11

Maximize:

Subject to:

xi - utilityi – number of cuts

Total number of cuts

Always holds for envy-freeFinal utility does not exceed 1

We bound the solution to the program by

)1(2

On

i

i ii

xx)1(

Page 18: Cutting a Birthday Cake

Trading Fairness for WelfareDefinitions: - un-proportional: exists player that gets

at most 1/n - envy: exists player that values another

player’s piece as worth at least times her own piece

- un-equale: exists player that values her allotted piece as worth more than times what another player values her allotted piece

Page 19: Cutting a Birthday Cake

Trading Fairness for Welfare Optimal utilitarian may require infinite

unfairness (under all three definitions of fairness)

Optimal egalitarian may require n-1 envy Egalitarian fairness does conflict with

proportionality or equitability

Page 20: Cutting a Birthday Cake

with O. Artzi and Y. Dombb

Throw One’s Cake and Have It Too

Page 21: Cutting a Birthday Cake

ExampleAlice

Bob

• Utilitarian welfare: 1 • Utilitarian welfare: (1.5-)

How much can be gained by such “dumping”?

Bob Alice

Page 22: Cutting a Birthday Cake

The Dumping Effect Utilitarian: dumping can increase the

utilitarian welfare by (n) Egalitarian: dumping can increase the

egalitarian welfare by n/3

Asymptotically tight

Page 23: Cutting a Birthday Cake

Pareto ImprovementPareto Improvement: No player is worse-off and

some are better-offStrict Pareto Improvement: All players are better-off

Theorem: Dumping cannot provide strict Pareto improvement

Proof: Each player that improves must get a cut. There are only n-1 cuts.

Page 24: Cutting a Birthday Cake

Pareto Improvement Dumping can provide Pareto

improvement in which: n-2 players double their utility 2 players stay the same

Page 25: Cutting a Birthday Cake

Player 2

Player 3

Player 4

Player 5

Player 6

Player 7

Pareto Improvement

Player 1

Player 8

Player 8 Player 1 Player 2 Player 3 Player 4 Player 5 Player 6 Player 7

Page 26: Cutting a Birthday Cake

Player 2

Player 3

Player 4

Player 5

Player 6

Player 7

Pareto Improvement

Player 1

Player 8 Player 1 Player 2 Player 3 Player 4 Player 5 Player 6 Player 7

• Player 8: 1/n• Players 1-7: 0.5

• Player 8: 1/n• Player 1: 0.5• Players 2-7: 1

Page 27: Cutting a Birthday Cake

with Y. Dombb and A. Hassidim

Computing Socially Optimal Divisions

Page 28: Cutting a Birthday Cake

Computing Socially Optimal Divisions Input: evaluation functions of all players

Explicit Piece-wise constant

Oracle

Find: Socially optimal division Utilitarian Egalitarian

Page 29: Cutting a Birthday Cake

Hardness It is NP-complete to decide if there is a

division which achieves a certain welfare threshold For both welfare functions Even for piece-wise constant evaluation

functions

Page 30: Cutting a Birthday Cake

The Discrete Version

Player x Player y Player z

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Approximations Hard to approximate the egalitarian

optimum to within (2-) No FPTAS for utilitarian welfare 8+o(1) approximation algorithm for

utilitarian welfare In the oracle input model

Page 32: Cutting a Birthday Cake

Open Problems

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Optimizing Social Welfare Approximating egalitarian welfare Tighter bounds for approximating

utilitarian welfare Optimizing welfare with strategic players

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Dumping Algorithmic procedures “Optimal” Pareto improvement Can dumping help in other economic

settings?

Page 35: Cutting a Birthday Cake

General Two dimensional cake Bounded number of pieces Chores

Page 36: Cutting a Birthday Cake

Questions?

Happy Birthday !