Cutting a Birthday Cake

1. Cutting a Birthday Cake Yonatan Aumann, Bar Ilan University

2. How should the cake be divided? “I love white decorations” “I want lots of flowers” “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 love white decorations” “I want lots of flowers” “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 Bob Alice • 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. Example Player 1 Player 2 Players 3,4 Player 1 Player 1 Player 3 Player 2 Player 2 Player 4 Total: 1.5 Total: 2 Fairness  Maximum Utility

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

10. Fairness vs. Welfare with Y. Dombb

11. The Price of Fairness • Given an instance: max welfare using any division PoF = max welfare using fairdivision Price of equitability utilitarian Price of envy-freeness Price of proportionality egalitarian

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

13. The Price of Fairness • Given an instance: max welfare using any division PoF = max welfare using fairdivision • Seek bounds on the Price of Fairness • First defined in [CKKK09] for non-connected divisions

14. Results

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

16. 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: 3x

17. Utilitarian Price of Envy FreenessUpper Bound xi - utility i – number of cuts Maximize: Total number of cuts Subject to: 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. Throw One’s Cake and Have It Too with O. Artzi and Y. Dombb

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

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. 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

26. Pareto Improvement Player 1 Player 2 Player 3 Player 4 Player 5 Player 6 Player 7 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: 1

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

28. Computing Socially Optimal Divisions • 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 y Player z Player x

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. Happy Birthday ! Questions?