Download presentation

Presentation is loading. Please wait.

Published byShyanne Rotherham Modified over 2 years ago

1
CUTTING A BIRTHDAY CAKE Yonatan Aumann, Bar Ilan University

2
How should the cake be divided? “I want lots of flowers” “I love white decorations” “No writing on my piece at all!”

3
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

4
How should the cake be divided? “I want lots of flowers” “I love white decorations” “No writing on my piece at all!”

5
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

6
Cut and Choose Alice likes the candies Bob likes the base Alice cuts in the middle Bob chooses BobAlice Proportional Envy free Equitable

7
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]

8
Player 1Player 2 Example Players 3,4 Total: 1.5Total: 2 Player 1 Player 3Player 2Player 4Player 1Player 2 Fairness Maximum Utility

9
Social Welfare Utilitarian: Sum of players’ utilities Egalitarian: Minimum of players’ utilities

10
with Y. Dombb Fairness vs. Welfare

11
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 egalitarian

12
Player 1Player 2 Example Players 3,4 Total: 1.5Total: 2 Utilitarian Price of Envy-Freeness: 4/3 Envy-freeUtilitarian optimum

13
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 divisions

14
Results Price ofProportionalityEnvy freenessEquitability Utilitarian Egalitarian 11

15
Utilitarian Price of Envy Freeness Lower Bound Player 1 Player 2 Player 3 Best possible utilitarian: Best proportional/envy-free utilitarian: players Utilitarian Price of envy-freeness:

16
Utilitarian Price of Envy Freeness Upper 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

17
Utilitarian Price of Envy Freeness Upper 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

18
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

19
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

20
with O. Artzi and Y. Dombb Throw One’s Cake and Have It Too

21
Example Alice Bob Utilitarian welfare: 1 Utilitarian welfare: (1.5- ) How much can be gained by such “dumping”? Bob Alice

22
The Dumping Effect Utilitarian: dumping can increase the utilitarian welfare by ( n) Egalitarian: dumping can increase the egalitarian welfare by n/3 Asymptotically tight

23
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.

24
Pareto Improvement Dumping can provide Pareto improvement in which: n-2 players double their utility 2 players stay the same

25
Player 2 Player 3 Player 4 Player 5 Player 6 Player 7 Pareto Improvement Player 1 Player 8 Player 1Player 2Player 3Player 4Player 5Player 6Player 7

26
Player 2 Player 3 Player 4 Player 5 Player 6 Player 7 Pareto Improvement Player 1 Player 8Player 1Player 2Player 3Player 4Player 5Player 6Player 7 Player 8: 1/n Players 1-7: 0.5 Player 8: 1/n Player 1: 0.5 Players 2-7: 1

27
with Y. Dombb and A. Hassidim Computing Socially Optimal Divisions

28
Input: evaluation functions of all players Explicit Piece-wise constant Oracle Find: Socially optimal division Utilitarian Egalitarian

29
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

30
The Discrete Version Player x Player y Player z

31
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

32
Open Problems

33
Optimizing Social Welfare Approximating egalitarian welfare Tighter bounds for approximating utilitarian welfare Optimizing welfare with strategic players

34
Dumping Algorithmic procedures “Optimal” Pareto improvement Can dumping help in other economic settings?

35
General Two dimensional cake Bounded number of pieces Chores

36
Questions? Happy Birthday !

Similar presentations

OK

Not Everyone Likes Mushrooms: Fair Division and Degrees of Guaranteed Envy-Freeness* Second GASICS Meeting Computational Foundations of Social Choice Aachen,

Not Everyone Likes Mushrooms: Fair Division and Degrees of Guaranteed Envy-Freeness* Second GASICS Meeting Computational Foundations of Social Choice Aachen,

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on trade fair circular Ppt on business model canvas Ppt on internet banking system project Ppt on ethical hacking and cybercrime Ppt on power line communication wiki Ppt on introduction to object-oriented programming definition Ppt on digital media Ppt on power generation using footsteps sound Ppt on 10 famous personalities of india Ppt on op amp circuits analysis