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 middle  Bob 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 1Player 2 Example Players 3,4 Total: 1.5Total: 2 Player 1 Player 3Player 2Player 4Player 1Player 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 division max welfare using fair division PoF = Price of equitability Price of proportionality Price of envy- freeness utilitarian egalitarian

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

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

Results Price ofProportionalityEnvy freenessEquitability Utilitarian Egalitarian 11

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:

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

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

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 1Player 2Player 3Player 4Player 5Player 6Player 7

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

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

Questions? Happy Birthday !