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 !