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CUTTING A BIRTHDAY CAKE Yonatan Aumann, Bar Ilan University

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

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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

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

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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

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

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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]

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

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

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with Y. Dombb Fairness vs. Welfare

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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

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

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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

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Results Price ofProportionalityEnvy freenessEquitability Utilitarian Egalitarian 11

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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:

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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

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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

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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

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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

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

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

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

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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.

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

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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

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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

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

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

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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

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The Discrete Version Player x Player y Player z

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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

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Open Problems

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

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

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General Two dimensional cake Bounded number of pieces Chores

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Questions? Happy Birthday !

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