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 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 !
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