Cake Cutting is and is not a Piece of Cake

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# Cake Cutting is and is not a Piece of Cake - PowerPoint PPT Presentation

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

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

Jeff Edmonds, York University

Kirk Pruhs, University of Pittsburgh

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.
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
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
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
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
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
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]
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]
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
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]
Problem Variations

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

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
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
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

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
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)
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
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
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)
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
• 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
• 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
Why a High Probability result is Not Achievable

.

.

.

Probability of overlap of k ~ (n choose k) / nk

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

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