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Cake Cutting is Not a Piece of Cake

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:

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