1. Cake Cutting is Not a Piece of Cake Malik Magdon-Ismail Costas Busch M. S. Krishnamoorthy Rensselaer Polytechnic Institute

2. users wish to share a cake Fair portion : th of cake

3. The problem is interesting when people have different preferences Meg Prefers Yellow Fish Tom Prefers Cat Fish Example:

4. Meg Prefers Yellow Fish Tom Prefers Cat Fish CUT Meg’s PieceTom’s Piece Happy

5. Meg Prefers Yellow Fish Tom Prefers Cat Fish CUT Tom’s PieceMeg’s Piece Unhappy

6. The cake represents some resource: Property which will be shared or divided The Bandwidth of a communication line Time sharing of a multiprocessor

7. Fair Cake-Cutting Algorithms: Specify how each user cuts the cake Each user gets what she considers to be th of the cake The algorithm doesn’t need to know the user’s preferences

8. For users it is known how to divide the cake fairly with cuts It is not known if we can do better than cuts Steinhaus 1948: “The problem of fair division”

9. We show that cuts are required for the following classes of algorithms: Phased Algorithms Labeled Algorithms (many algorithms) (all known algorithms) Our contribution:

10. We show that cuts are required for special cases of envy-free algorithms: Each user feels she gets more than the other users Our contribution:

11. Cake Cutting Algorithms Lower Bound for Phased Algorithms Lower Bound for Labeled Algorithms Lower Bound for Envy-Free Algorithms Conclusions Talk Outline

12. Cake knife

13. Cake knife cut

14. Utility Function for user Cake

15. Value of piece:

16. Cake Value of piece:

17. Cake Utility Density Function for user

18. “I cut you choose” Step 1:User 1 cuts at Step 2:User 2 chooses a piece

19. “I cut you choose” Step 1:User 1 cuts at

20. “I cut you choose” Step 2:User 2 chooses a piece User 2

21. “I cut you choose” User 2User 1 Both users get at least of the cake Both are happy

22. Algorithm users Phase 1: Each user cuts at

23. Algorithm users Phase 1: Each user cuts at

24. Algorithm users Phase 1: Give the leftmost piece to the respective user

25. Algorithm users Phase 2: Each user cuts at

26. Algorithm users Phase 2: Each user cuts at

27. Algorithm users Phase 2: Give the leftmost piece to the respective user

28. Algorithm users Phase 3: Each user cuts at And so on…

29. Algorithm Total number of phases: Total number of cuts:

30. Algorithm users Phase 1: Each user cuts at

31. Algorithm users Phase 1: Each user cuts at

32. Algorithm users Phase 1: Find middle cut users

33. Algorithm users Phase 2: Each user cuts at

34. Algorithm users Phase 2: Each user cuts at

35. Algorithm Phase 2: Find middle cut users

36. Algorithm users Phase 3: Each user cuts at And so on…

37. Algorithm Phase log N: user The user is assigned the piece

38. Algorithm Total number of phases: Total number of cuts:

39. Cake Cutting Algorithms Lower Bound for Phased Algorithms Lower Bound for Labeled Algorithms Lower Bound for Envy-Free Algorithms Conclusions Talk Outline

40. Phased algorithm:consists of a sequence of phases At each phase: Each user cuts a piece which is defined in previous phases A user may be assigned a piece in any phase

41. Observation: Algorithms and are phased

42. We show: cuts are required to assign positive valued pieces

43. Phase 1:Each user cuts according to some ratio 1111

44. There exist utility functions such that the cuts overlap 1

45. Phase 2:Each user cuts according to some ratio 22221

46. There exist utility functions such that the cuts in each piece overlap 122

47. 122 Phase 3: 3333 number of pieces at most are doubled And so on…

48. Phase k:Number of pieces at most

49. For users: we need at least pieces we need at least phases

50. PhaseUsersPiecesCuts (min) (max) (min) …… Total Cuts:

51. Cake Cutting Algorithms Lower Bound for Phased Algorithms Lower Bound for Labeled Algorithms Lower Bound for Envy-Free Algorithms Conclusions Talk Outline

52. 111001101000Labels: each piece has a labelLabeled algorithms:

53. 111001101000Labels: 0 1 01 0 1 01 00 010011 1011 Labeling Tree:

55. 0 1 0 1 01

56. 0 1 00 1 101 01

57. 0 1 00 1 1 01 011010 01 011

58. 111001101000 0 1 01 0 1 01 010011 1011

59. 1001101000 Sorting Labels Users receive pieces in arbitrary order: We would like to sort the pieces:

60. 111001101000 Sorting Labels Labels will help to sort the pieces

61. 110100011010000 Sorting Labels Normalize the labels

62. 110100011010000 Sorting Labels 01234567

63. 110100011010000 Sorting Labels 01234567 011

64. 110100011010000 Sorting Labels 01234567 011010

65. 110100011010000 Sorting Labels 01234567 011010110

66. 100011010000 Sorting Labels 01234567 011010110000

67. 110100011010000 Sorting Labels 01234567 011010110000100 Labels and pieces are ordered!

68. 110100011010000 Sorting Labels 01234567 011010110000100 Time needed:

69. linearly-labeled & comparison-bounded algorithms: Require comparisons (including handling and sorting labels)

70. Conjecture:All known algorithms are linearly-labeled & comparison-bounded Observation:Algorithms and are linearly-labeled & comparison-bounded

71. We will show that cuts are needed for linearly-labeled & comparison-bounded algorithms

72. distinct positive integers: Sorted order: Reduction of Sorting to Cake Cutting Input: Output: Using a cake-cutting algorithm

73. distinct positive integers: utility functions: users:

74. Cake

77. cannot be satisfied!

78. can be satisfied!

79. Cake Rightmost positive valued pieces Piece:

80. Labels: Sorted labels: Sorted pieces: Sorted integers:

81. Fair cake-cutting Sorting

82. Sorting integers:comparisons Cake Cutting: comparisons

83. Linearly-labeled & comparison-bounded algorithms: Require comparisons comparisons cuts require

84. Cake Cutting Algorithms Lower Bound for Phased Algorithms Lower Bound for Labeled Algorithms Lower Bound for Envy-Free Algorithms Conclusions Talk Outline

85. Envy-free:Each user feels she gets at least as much as the other users Variations of Fair Cake-Division Strong Envy-free: Each user feels she gets strictly more Than the other users

86. Super Envy-free: A user feels she gets a fair portion, and every other user gets less than fair

87. Lower Bounds Strong Envy-free: Super Envy-free: cuts

88. Strong Envy-Free, Lower Bound

91. is upset!

92. Strong Envy-Free, Lower Bound is happy!

93. Strong Envy-Free, Lower Bound must get a piece from each of the other user’s gap

94. Strong Envy-Free, Lower Bound A user needs distinct pieces Total number of cuts: Total number of pieces:

95. Cake Cutting Algorithms Lower Bound for Phased Algorithms Lower Bound for Labeled Algorithms Lower Bound for Envy-Free Algorithms Conclusions Talk Outline

96. We presented new lower bounds for several classes of fair cake-cutting algorithms

97. Open problems: Prove or disprove that every algorithm is linearly-labeled and comp.-bounded An improved lower bound for envy-free algorithms