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Cake Cutting is Not a Piece of Cake. Malik Magdon-Ismail Costas Busch M. S. Krishnamoorthy . Rensselaer Polytechnic Institute. users wish to share a cake. Fair portion : th of cake. The problem is interesting when people have different preferences. Example:. Meg Prefers

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Cake Cutting is Not a Piece of Cake


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    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 Example: Meg Prefers Yellow Fish Tom Prefers Cat Fish

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

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

    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: • Each user gets what she considers • to be th of the cake • Specify how each user cuts 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 Steinhaus 1948:“The problem of fair division” It is not known if we can do better than cuts

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

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

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

    12. Cake knife

    13. Cake cut knife

    14. Cake Utility Function for user

    15. Cake 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” User 2 Step 2: User 2 chooses a piece

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

    22. Algorithm users Each user cuts at Phase 1:

    23. Algorithm users Each user cuts at Phase 1:

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

    25. Algorithm users Each user cuts at Phase 2:

    26. Algorithm users Each user cuts at Phase 2:

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

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

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

    30. Algorithm users Each user cuts at Phase 1:

    31. Algorithm users Each user cuts at Phase 1:

    32. Algorithm users users Find middle cut Phase 1:

    33. Algorithm users Each user cuts at Phase 2:

    34. Algorithm users Each user cuts at Phase 2:

    35. Algorithm users Find middle cut Phase 2:

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

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

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

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

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

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

    45. 2 2 1 2 2 Phase 2: Each user cuts according to some ratio

    46. 2 1 2 There exist utility functions such that the cuts in each piece overlap

    47. 3 2 3 1 3 2 3 number of pieces at most are doubled Phase 3: 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. Phase Users Pieces Cuts (min) (min) (max) …… …… …… …… Total Cuts: