CUTTING A BIRTHDAY CAKE

Yonatan Aumann, Bar Ilan University

How should the cake be divided?

“I want lots of flowers”

“I love white decorations”

“No writing on my piece at all!”

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

How should the cake be divided?

“I want lots of flowers”

“I love white decorations”

“No writing on my piece at all!”

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

Cut and Choose

Alice likes the candies Bob likes the base

Alice cuts in the middleBob chooses

BobAlice

Proportional

Envy free

Equitable

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]

Player 1 Player 2

Example

Players 3,4

Total: 1.5

Total: 2

Player 1 Player 3 Player 2Player 4Player 1 Player 2

Fairness Maximum Utility

Social Welfare

Utilitarian: Sum of players’ utilities

Egalitarian: Minimum of players’ utilities

with Y. Dombb

Fairness vs. Welfare

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

Player 1 Player 2

Example

Players 3,4

Total: 1.5

Total: 2

Utilitarian Price of Envy-Freeness:

4/3

Envy-free Utilitarian optimum

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

Results

Price of Proportionality

Envy freeness

Equitability

Utilitarian

Egalitarian1 1

)1(2

On

)1(On

2

n

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

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: x

new piece: 2x

new piece: 3x

Utilitarian Price of Envy FreenessUpper Bound

in

ix

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-free

Final utility does not exceed 1

We bound the solution to the program by

)1(2

On

i

i ii

x

x)1(

Trading Fairness for Welfare

Definitions: - 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

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

with O. Artzi and Y. Dombb

Throw One’s Cake and Have It Too

Example

Alice

Bob

• Utilitarian welfare: 1

• Utilitarian welfare: (1.5-)

How much can be gained by such “dumping”?

Bob Alice

The Dumping Effect

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

Egalitarian: dumping can increase the egalitarian welfare by n/3

Asymptotically tight

Pareto Improvement

Pareto Improvement: No player is worse-off and some are better-off

Strict 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.

Pareto Improvement

Dumping can provide Pareto improvement in which:

n-2 players double their utility

2 players stay the same

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

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

with Y. Dombb and A. Hassidim

Computing Socially Optimal Divisions

Computing Socially Optimal Divisions Input: evaluation functions of all players

Explicit Piece-wise constant

Oracle

Find: Socially optimal division Utilitarian Egalitarian

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

The Discrete Version

Player x Player y Player z

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

Open Problems

Optimizing Social Welfare

Approximating egalitarian welfare Tighter bounds for approximating

utilitarian welfare Optimizing welfare with strategic players

Dumping

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

settings?

General

Two dimensional cake Bounded number of pieces Chores

Questions?

Happy Birthday !