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


Alice likes the candies
Bob likes the base
Bob
 Proportional
 Envy free
Alice cuts in the middle
Bob chooses
 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]
Example
Player 1
Player 2
Players 3,4
PlayerPlayer
1 Player
1 3 Player
Player
2 Player
2 4
Total: 1.5
Total: 2
Fairness  Maximum Utility
Social Welfare

Utilitarian: Sum of players’ utilities

Egalitarian: Minimum of players’ utilities
Fairness vs. Welfare
with Y. Dombb
The Price of Fairness

Given an instance:
PoF =
utilitarian
egalitarian
max welfare using any division
max welfare using fair division
Price of envyfreeness
Price of
equitability
Price of
proportionality
Example
Player 1
Player 2
Players 3,4
Envy-free
Total: 1.5
Utilitarian optimum
Total: 2
Utilitarian Price of Envy-Freeness:
4/3
The Price of Fairness

Given an instance:
PoF =
max welfare using any division
max welfare using fair division

Seek bounds on the Price of Fairness

First defined in [CKKK09] for non-connected divisions
Results
Price of
Proportionality
Utilitarian
Envy freeness
n
Equitability
n  O (1)
 O (1)
2
n
Egalitarian
1
2
1
Utilitarian Price of Envy Freeness
Lower Bound
Player
1
Player
2
Player
3
n
Player
3n
n
players
Best possible utilitarian:
Best proportional/envy-free utilitarian:
n
1
n

 n
Utilitarian Price of envy-freeness:

n /2

n 1  2
Utilitarian Price of Envy Freeness
Upper Bound
Envy-free piece x
newpiece:
new
piece:
piece:
new
 x  2x
 3x
Key observation:
In order to increase a player’s utility by , her new
piece must span at least (-1) cuts.
Utilitarian Price of Envy Freeness
Upper Bound
Maximize:

i
( i  1) x i

Subject to:
Always holds for
envy-free
Final utility does not
exceed 1

i
xi - utility
i – number of cuts
xi
Total number of
cuts
 n 1
xi 
1
i
n
( i  1)  x i  1
i
 i  { 0 ,1,..., n  1}
i
We bound the solution to the program by
n
2
 O (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
Throw One’s Cake and Have It Too
with O. Artzi and Y. Dombb
Example
Alice
Bob
Bob
Alice
• Utilitarian welfare: 1
• Utilitarian welfare: (1.5-)
How much can be gained by such “dumping”?
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
Pareto Improvement
Player
1
Player
2
Player
3
Player
4
Player
5
Player
6
Player
7
Player 8
Player 8
Player 1
Player 2
Player 3
Player 4
Player 5
Player 6
Player 7
Pareto Improvement
Player
1
Player 8
Player
2
Player 1
Player
3
Player 2
• Player 8: 1/n
• Players 1-7: 0.5
Player
4
Player 3
Player
5
Player 4
Player
6
Player 5
Player
7
Player 6
• Player 8: 1/n
• Player 1: 0.5
• Players 2-7: 1
Player 7
Computing Socially Optimal Divisions
with Y. Dombb and A. Hassidim
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
Happy Birthday !
Questions?