Cake Cutting is and is not a Piece of Cake

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

2. Informal Problem Statement • n self interested players wish to divide items of value such that each player believes that they received at least 1/n of the value • Players may not agree on the values of items • Players may be deceitful, cunning, dishonest, etc.

3. An Instance of Cake Cutting From History

4. A Politically Incorrect Reference to Cake Cutting

5. Classic Problem Definition • n players wish to divide a cake = [0, 1] • Each player p has an unknown value function Vp • Vp[x, y] = how much player p values piece/interval [x, y] • The protocol’s goal is Fairness: Each honest player p is guaranteed a piece of cake of value at least Vp[0,1]/n = 1/n

6. 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

7. Classic Two Person Discrete Algorithm (n=2): Cut and Choose • Person A cuts the cake into two pieces • Person B selects one of the two pieces, and person A gets the other piece

8. Two Person Continuous Algorithm (n=2): Moving Knife • Protocol moves the knife continuously across the cake until the first player say stop • This player gets this piece, and the rest of the players continue • Moving knife algorithms are considered cheating by discrete algorithmic researchers and we do not consider them here

9. Formalizing Allowed Operations • Queries the protocol can make to each player: • Eval[p, x, y]: returns Vp[x, y] • Cut[p, x, v]: returns a y such Vp[x, y] = v • All know algorithms can be implemented with these operations

10. Two Person Algorithm (n=2):Cut and Choose • y = cut(A, 0, ½) • If eval(B, 0, y) ≤ ½ then • Player A gets [0, y] and player B gets [y, 1] • Else • Player B gets [0, y] and player A gets [y, 1]

11. Three Person Algorithm (n=3):Steinhaus • YA = cut(A, 0, 1/3) • YB = cut(B, 0, 1/3) • YC = cut(C, 0, 1/3) • Assume wlog YA ≤ YB ≤ YC • Player A gets [0, yA], and players B and C “cut and choose” on [yA, 1]

12. O(n log n) Divide and Conquer Algorithm: Evan and Paz • Yi = cut(i, 0, 1/2) for i = 1 … n • m = median(y1, … , yn) • Recurse on [0, m] with those n/2 players i for which yi < m • Recurse on [m, 1] with those n/2 players i for which yi > m

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 * The proof is currently only roughly written up at this point

15. Outline • Deterministic Ω(n log n) Lower Bound • Definition of Thin-Rich game • Sufficiency to lower bound Thin-Rich • Definition of value tree cakes • Lower bound for Thin-Rich • Hint at Randomized Ω(n log n) Lower Bound with Approximate Cuts • Randomized O(n) Upper Bound

16. Thin-Rich Game • Game Definition: Find a thin rich piece for a particular player • A piece is thin if it has width ≤ 2/n • A piece is rich if it has value ≥ 1/2n • Theorem: The complexity of cake cutting is at least n/2 times the complexity of thin-rich • Proof: In cake cutting, at least n/2 players have to end up with a thin rich piece

17. Value Tree 1/4 1/2 1/4 1/4 1/4 1/4 1/4 1/2 1/4 1/4 1/2 1/2 0 1/9 2/9 3/9 4/9 5/9 6/9 7/9 8/9 1 Value = Product of edge labels

18. Deterministic Ω(log n) Lower Bound for Thin-Rich • Theorem: To win at Thin-Rich, when the input is derived from a value tree, the protocol has to find a leaf where at least 40% of the edge labels on root to leaf path are ½ • Theorem: From each query, the protocol learns the edge labels on at most two root to leaf paths • Theorem: The deterministic complexity of Thin-Rich is Ω(log n) • Proof: Reveal edges with label ¼ on the two paths learned by the protocol

19. Randomized Lower Bound • Theorem: From each approximate query, the protocol learns the edge labels on at most two root to leaf paths, and at most one constant depth triangle • Theorem: The randomized complexity of thin-rich with approximate queries is Ω(log n) • Proof:Use Yao’s technique. For each vertex in the value tree, uniformly at random pick the edge to label ½.The expected number of labels of ½ on all known labeled paths after k queries is O( (log3 n)/3 + k)

20. Outline • Deterministic Ω(n log n) Lower Bound • Hint at Randomized Ω(n log n) Lower Bound with Approximate Cuts • Randomized O(n) Upper Bound • O(1) complexity randomized protocol for Thin-Rich • Cake cutting algorithm • Generalized offline power of two choices lemma • Non-independent random graph model

21. O(1) Complexity Randomized Protocol for for Thin-Rich • Pick an i uniformly at random from 0 … n-1 • x = Cut[0, i/n] • y = Cut[ 0, (i+1)/n] • If (y-x) ≤ 2/n then return piece [x, y] • Else goto step 1

22. Randomized Protocol for Cake Cutting • Protocol Description: • Each player repeatedly applies randomized thin-rich protocol to get 2d pieces • For each player, pick one of the two tentative pieces in such a way that every point of cake is covered by at most O(1) pieces. If this is not possible, then start over again. • Theorem: This protocol is approximately fair • We need to show that the second step of the protocol is successful with probability Ω(1)

23. Digression(1) • Power of Two Choices Setting: n balls, each of which can be put into two of n bins that are selected independently uniformly at random • Online Theorem: The online greedy assignment guarantees maximum load of O(log log n) whp • Offline Theorem: There is an assignment with maximum load O(1) whp

24. Digression(2): Proof of Offline Power of Two Choices Theorem • Consider a graph G • Vertices = bins • One edge for each ball connecting the corresponding vertices • Important: Edges are independent • Lemma: If G is acyclic then the maximum load is 1 • Classic Theorem: If a graph G has n/3 independent edges, then G is acyclic whp • Proof: Union Bound. • Prob[G contains a cycle C] ≤ ΣC Prob[C is in the graph] ~ Σi (n choose i) * (1/3n)i

25. Key Theorem for O(n) Bound: Generalized Offline Balls and Bins • Each of n players arbitrarily partition [0, 1] into n pieces • Each player picks uniformly at random 2*d pieces • Then with probability Ω(1), we can assign to each player one of its 2*d pieces so that every point is covered by at most O(1) pieces • This is equivalent to offline balls and bins if the partition is into equal sized pieces, except that: • We may need d > 1, and • We don’t get high probability bound

26. Why a High Probability result is Not Achievable . . . Probability of overlap of k ~ (n choose k) / nk

27. Problem Case: Forks • Theorem: With probability Ω(1) there is no fork of depth ω(1) • Therefore we throw out forked paths, and proceed Fork of depth 3

28. Directed Graph for Cake Cutting (d=1) Vertex Vertex Vertex Vertex

29. Sufficiency Condition • Theorem: The maximum load is at most 1 if there is not directed path between the two pieces of the same person

30. One Difficulty: Edges May Not be Independent

31. Dealing with Dependent Edges • Lemma: There are not many dependent edges • Lemma: Each possible path, between two pieces of the same player, can have at most two dependent edges • Lemma: With probability Ω(1) there is no path between two pieces of the same player

32. Conclusions • Generalized offline balls and bins theorem may be useful elsewhere • The model of random graphs, where there are some dependencies on the edges, and our analysis may be useful elsewhere • Is dependent random graph model novel ?