Cake Cutting is and is not a Piece of Cake

# Cake Cutting is and is not a Piece of Cake

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## Cake Cutting is and is not a Piece of Cake

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1. Cake Cutting is and is not a Piece of Cake Jeff Edmonds, York University Kirk Pruhs, University of Pittsburgh

2. Informal Problem Statement Resource allocation between n possibly deceitful players

3. I like I like 0 1 Classic Problem Definition = [0, 1] • n players wish to divide a cake • Each player p has an unknown value function Vp

4. v I like x y 0 1 Classic Problem Definition = [0, 1] • n players wish to divide a cake • Each player p has an unknown value function Vp • Allowed Operations : • Eval[p, x, y]: returns how much player p values piece/interval [x, y]

5. v I like x y 0 1 Classic Problem Definition = [0, 1] • n players wish to divide a cake • Each player p has an unknown value function Vp • Allowed Operations : • Eval[p, x, y]: returns how much player p values piece/interval [x, y] • Cut[p, x, v]: returns a y such Eval[p,x, y] = v

6. 1/n I like 0 1 Classic Problem Definition = [0, 1] • n players wish to divide a cake • Each player p has an unknown value function Vp • Goal: Fair cut Each honest player p is guaranteed a piece of value at least 1/n.

7. 1/n I like 0 1 Classic Problem Definition = [0, 1] • n players wish to divide a cake • Each player p has an unknown value function Vp • Goal: Fair cut Each honest player p is guaranteed a piece of value at least 1/n.

8. History • Originated in 1940’s school of Polish mathematics • Picked up by social scientists interested in fair allocation of resources • Texts by Brams and Taylor, and Robertson and Webb • A quick Google search reveals cake cutting is used as a teaching example in many algorithms courses

9. I like I like O(n log n) Divide and Conquer Algorithm: Evan and Paz • Yp = cut(p, 0, 1/2) for p = 1 … n My half cut is here. My half cut is here.

10. I like O(n log n) Divide and Conquer Algorithm: Evan and Paz • Yp = cut(p, 0, 1/2) for p = 1 … n My half cut is here.

11. O(n log n) Divide and Conquer Algorithm: Evan and Paz • Yp = cut(p, 0, 1/2) for p = 1 … n • m = median(y1, … , yn)

12. I like I like so am happy with the left. so am happy with the right. O(n log n) Divide and Conquer Algorithm: Evan and Paz • Yp = cut(p, 0, 1/2) for p = 1 … n • m = median(y1, … , yn) • Recurse on [0, m] with those n/2 players p for which yp < m • Recurse on [m, 1] with those n/2 players p for which yp > m • Time O(nlogn)

13. Problem Variations • Contiguousness: Assigned pieces must be subintervals • Approximate fairness: A protocol is c-fair if each player is a assured a piece that he gives a value of at least c/n • Approximate queries (introduced by us?): • AEval[p, ε, x, y]: returns a value v such that Vp[x, y]/(1+ε) ≤ v ≤ (1+ ε) Vp[x, y] • ACut[p, ε, x, v]: returns a y such Vp[x, y]/(1+ε) ≤ v ≤ (1+ ε) Vp[x, y]

14. Problem Variations (Approximate) * Submitted to STOC

15. Outline • Deterministic Ω(n log n) Lower Bound • Randomized with Approximate Cuts Ω(n log n) Lower Bound • Randomized with Exact Cuts O(n) Upper Bound

16. At least n/2 players require thin rich piece Thin-Rich Game • Game Definition: Single player must find a thin rich piece. • A piece is thin if it has width ≤ 2/n • A piece is rich if it has value ≥ 1/2n • Theorem: The deterministic complexity of Thin-Rich is Ω(log n). • Theorem: The deterministic complexity of cake cutting is Ω(nlog n).

17. Alg Adv • I give sequence of Eval[x,y] & Cut[x,v]operations. • I dynamically choose how to answer

18. Alg Adv • I can choose any non-continuous thin piece, • but W.L.G.I choose one of these. • I cut the cake in to n thin pieces.

19. Alg Adv ... ... ... ... ... ... • I build a complete 3-ary treewith the n pieces as leaves

20. Alg Adv ½ ¼ ½ ¼ ¼ ¼ ... ... ... ... ... ... • For each node, • I label edges • <½,¼,¼> or <¼,¼,½>

21. Alg Adv ½ ¼ ½ ¼ ¼ ¼ ¼ ¼ ½ ... ... ¼ ¼ ... ... ... ½ ... 1/1024 = ¼×¼×½×¼×¼×½ • Value of each piece isproduct of edge labelsin path.

22. Alg Adv ½ ¼ ½ ¼ ¼ ½ ¼ ¼ ½ ... ... ¼ ¼ ... ... ... ½ ... 1/256 = ½×¼×½×¼×¼×½ To get a rich pieceI need at least 40% of the edge labels in path be ½. Good luck

23. Alg Adv 0.4398 0 y • I need to find a yso that V[0,y] = 0.4398. • Cut[0,0.4398]?

24. Alg Adv 1/2 0 1 0.4398 0.4398 • I do binary search to find0.4398 • Cut[0,0.4398]? 1/4

25. Alg Adv ¼ ½ ¼ 1/4 2/4 0.4398 • I do binary search to find0.4398 • I fix some edge labels • Cut[0,0.4398]?

26. Alg Adv ¼ ½ ¼ 0.4398 0.4398 • I do binary search to find0.4398 • I fix some edge labels • Cut[0,0.4398]? 1/4 2/4 6/16 7/16 4/16 8/16

27. Alg Adv ¼ ½ ½ ¼ ¼ ¼ 7/16 8/16 0.4398 • I fix some edge labels • Cut[0,0.4398]?

28. Alg Adv ¼ ¼ ¼ ½ ½ ½ ½ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¼ 0.4398 y ¼ ¼ ½ • I find a yso that V[0,y] = 0.4398. • Cut[0,0.4398]?

29. Alg Adv ¼ ¼ ¼ ½ ½ ½ ½ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ½ • I learned a path,but all its labels are ¼ • Hee Hee

30. Alg Adv ¼ ¼ ¼ ½ ½ ½ ½ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ½ • YesAfter t operationsevery path has only t known ½ labels. • Every path has  oneknown ½ label.

31. Alg Adv ¼ ¼ ¼ ½ ½ ½ ½ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ½ y x • I fix labels in path to x & y to ¼. • and give • Eval[x,y] = 0.00928 • Eval[x,y] 0.00928

32. Alg Adv ¼ ¼ ¼ ½ ½ ½ ½ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ½ ¼ ¼ ¼ ¼ ½ • YesAfter t operationsevery path has only t½ known labels.

33. Deterministic Ω(log n) Lower Bound • Theorem: To win at Thin-Rich, the alg has to get a rich piece with at least 40% of the edge labels in path be ½. • Theorem: After t operations, every path has only  t ½ labels. • Theorem: The deterministic complexity of Thin-Rich is Ω(depth) =Ω(log n)

34. At least n/2 players require thin rich piece Deterministic Ω(nlog n) Lower Bound • Theorem: The deterministic complexity of Thin-Rich is Ω(log n). • Theorem: The deterministic complexity of cake cutting is Ω(n log n).

35. Outline Done • Deterministic Ω(n log n) Lower Bound • Randomized with Approximate Cuts Ω(n log n) Lower Bound • Randomized with Exact Cuts O(n) Upper Bound Randomized Approximate Cuts

36. Adv • I must choose the value functions • without knowing • coin flips • I define a randomized algorithm Rand Alg

37. Adv Yau Alg RandAdv • I must choose the value functions • without knowing • coin flips • I define a randomized algorithm Rand Alg • I flip coins to choose value function. • I dynamically choose error in answers • I deterministically give sequence of Eval[x,y] & Cut[x,v]operations.

38. Alg AdvRand ½ ¼ ¼ ¼ ½ ¼ ¼ ½ ¼ ... ... ... ... ... ... • For each node, • I randomly label edges • <½,¼,¼>, <¼,½,¼>, or <¼,¼,½>

39. Alg Adv ¼ ½ ¼ ¼ ¼ ½ ¼ ½ ¼ ¼ ½ ½ ¼ ¼ ¼ ½ ¼ ¼ y x • Consider path to x and y. • Flip coins for labels. • 33% of labels will be ½. • Eval[x,y] • But I need 40%!

40. Alg Adv ¼ ¼ ½ ¼ ¼ ½ ¼ ½ ¼ ¼ ½ ½ ¼ ¼ ½ ¼ ½ ¼ ¼ ¼ ¼ ½ ¼ ½ ¼ ¼ y x • I flip coins for path to x’and get 33% ½. • Cut[x’,0.4398]? x’

41. Alg Adv ¼ ¼ ½ ¼ ¼ ½ ¼ ½ ¼ ¼ ½ ¼ ½ ¼ ¼ ½ ¼ ½ ¼ ¼ ¼ ¼ ¼ ½ ¼ ½ ¼ ¼ y x • I do binary search for 0.4398, • but for some odd reasonit finds 40%½ labels. • Cut[x’,0.4398]? ½ ½ ½ x’

42. Alg Adv ¼ ¼ ½ ¼ ¼ ½ ½ ¼ ½ ¼ ½ ¼ ½ ¼ ½ ¼ ¼ ½ ½ ¼ ½ ¼ ¼ ¼ ¼ ¼ ½ ¼ ½ ¼ ¼ y x • Luckily I can give  error and this hides most of the labels. • Cut[x’,0.4398]? x’

43. Outline Done • Deterministic Ω(n log n) Lower Bound • Randomized with Approximate Cuts Ω(n log n) Lower Bound • Randomized with Exact Cuts O(n) Upper Bound Done Randomized Exact Cuts O(n) Upper

44. O(1) Complexity Randomized Protocol for Thin-Rich Protocol Description: • “Cuts” cake into n “candidate” pieces of value 1/n • Randomly chooses one. • It is likely thin and rich.

45. Randomized Protocol for Cake Cutting Protocol Description: • Each player randomly selects 2d candidate pieces. • For each player, we carefully pick one of these

46. Randomized Protocol for Cake Cutting Protocol Description: • Each player randomly selects O(1) candidate pieces. • For each player, we carefully pick one of these • so that every point of cake is covered by at most O(1) pieces. • Where there is overlap, recurs. Works with O(1) probability.

47. Balls and Bins Two Random Choices: n balls, n bins each ball randomly chooses two bins. Choice: Select one of two bins for each ball. Whp no bin has more than O(1) balls.

48. Balls and Bins Two Random Choices: n balls, n bins each ball randomly chooses two bins. Choice: Select one of two bins for each ball. Whp no bin has more than O(1) balls.

49. A piece of cake. Outline Done • Deterministic Ω(n log n) Lower Bound • Randomized with Approximate Cuts Ω(n log n) Lower Bound • Randomized with Exact Cuts O(n) Upper Bound Done More at STOC