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 1dimentional ID: 272819
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Slide1
Cutting a Birthday Cake
Yonatan Aumann, Bar Ilan UniversitySlide2
How should the cake be divided?
“I want lots of flowers”
“I love white decorations”
“No writing on my piece at all!”Slide3
Model
The cake:1-dimentionalthe 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 piecesSlide4
How should the cake be divided?
“I want lots of flowers”
“I love white decorations”
“No writing on my piece at all!”Slide5
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 piecesSlide6
Cut and Choose
Alice likes the candies
Bob likes the base
Alice cuts in the middle
Bob chooses
Bob
Alice
Proportional
Envy free
EquitableSlide7
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]Slide8
Player 1
Player 2
Example
Players 3,4
Total: 1.5
Total: 2
Player 1
Player 3
Player 2
Player 4
Player 1
Player 2
Fairness
Maximum UtilitySlide9
Social Welfare
Utilitarian: Sum of players’ utilities
Egalitarian:
Minimum of players’ utilitiesSlide10
with Y. Dombb
Fairness vs. WelfareSlide11
The Price of Fairness
Given an instance:max welfare using any division
max
welfare
using
fair
division
PoF
=
Price of equitability
Price of proportionality
Price of envy-freeness
utilitarian
egalitarianSlide12
Player 1
Player 2
Example
Players 3,4
Total: 1.5
Total: 2
Utilitarian Price of Envy-Freeness:
4/3
Envy-free
Utilitarian optimumSlide13
The Price of Fairness
Given an instance:max welfare using any division
max
welfare
using
fair
division
PoF
=
Seek bounds on the
Price of Fairness
First defined in [CKKK09] for non-connected divisionsSlide14
Results
Price of
Proportionality
Envy freeness
Equitability
Utilitarian
Egalitarian
1
1Slide15
Utilitarian Price of Envy FreenessLower Bound
Player 1
Player 2
Player 3
Player
3
Best possible utilitarian:
Best proportional/envy-free utilitarian:
players
Utilitarian Price of envy-freeness: Slide16
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:
3xSlide17
Utilitarian Price of Envy FreenessUpper Bound
Maximize
:
Subject to:
x
i
- utility
i – 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 Slide18
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 pieceSlide19
Trading Fairness for Welfare
Optimal utilitarian may require infinite unfairness (under all three definitions of fairness)Optimal egalitarian may require n-1 envyEgalitarian fairness does conflict with proportionality or equitabilitySlide20
with O. Artzi and Y. Dombb
Throw One’s Cake and Have It TooSlide21
Example
Alice
Bob
Utilitarian welfare: 1
Utilitarian welfare: (1.5-
)
How much can be gained by such “dumping”?
Bob
AliceSlide22
The Dumping Effect
Utilitarian: dumping can increase the utilitarian welfare by (n)Egalitarian: dumping can increase the egalitarian welfare by n/3Asymptotically tightSlide23
Pareto Improvement
Pareto Improvement: No player is worse-off and some are better-offStrict Pareto Improvement: All players are better-offTheorem: Dumping cannot provide strict Pareto improvement
Proof:
Each player that improves must get a cut.
There are only n-1 cuts.Slide24
Pareto Improvement
Dumping can provide Pareto improvement in which: n-2 players double their utility2 players stay the sameSlide25
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 7Slide26
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: 1Slide27
with Y. Dombb and A. Hassidim
Computing Socially Optimal DivisionsSlide28
Computing Socially Optimal Divisions
Input: evaluation functions of all playersExplicitPiece-wise constantOracleFind: Socially optimal division
Utilitarian
EgalitarianSlide29
Hardness
It is NP-complete to decide if there is a division which achieves a certain welfare thresholdFor both welfare functionsEven for piece-wise constant evaluation functionsSlide30
The Discrete Version
Player x
Player y
Player zSlide31
Approximations
Hard to approximate the egalitarian optimum to within (2-)No FPTAS for utilitarian welfare8+o(1) approximation algorithm for utilitarian welfare
In the oracle input modelSlide32
Open ProblemsSlide33
Optimizing Social Welfare
Approximating egalitarian welfareTighter bounds for approximating utilitarian welfareOptimizing welfare with strategic playersSlide34
Dumping
Algorithmic procedures“Optimal” Pareto improvementCan dumping help in other economic settings?Slide35
General
Two dimensional cakeBounded number of piecesChoresSlide36
Questions?
Happy Birthday !