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Cake Cutting is and is not a Piece of Cake. Jeff Edmonds , York University Kirk Pruhs, University of Pittsburgh Jaisingh Solanki, York University. Toronto Star. Informal Problem Statement. Resource allocation between n self possibly deceitful players . I like . I like . 0. 1.

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cake cutting is and is not a piece of cake

Cake Cutting is and is not a Piece of Cake

Jeff Edmonds, York University

Kirk Pruhs, University of Pittsburgh

Jaisingh Solanki, York University

informal problem statement
Informal Problem Statement

Resource allocation between n self possibly deceitful players

classic problem definition
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
classic problem definition5
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]
classic problem definition6
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
classic problem definition7
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.

classic problem definition8
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.

history
History
  • Originated in 1940’s school of Polish mathematics
  • Picked up by social scientists interested in fair allocation of resources
  • The subject of a handful of books (e.g Brams and Taylor, and Robertson and Webb) and many papers
  • A quick Google search reveals cake cutting is used as a teaching example in many algorithms courses, e.g CMU CS 15-451 and CMU CS 15-750
continuous moving knife algorithm
We did not likethat piece anyway.

StopI’ll take it.

Continuous Moving Knife Algorithm
  • Protocol moves the knife continuously across the cake until the some player say stop.
  • This player gets this piece.
  • The rest of the players continue
  • Moving knife considered cheating by discrete alg.
  • We do not consider them here.
classic alg cut and choose n 2
I like

I like

I’ll take the right.

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

Careful.

My half cut is here.

I am a little jealous

but I am happywith the left.

o n log n divide and conquer algorithm even and paz
I like

I like

O(n log n) Divide and Conquer Algorithm: Even and Paz
  • Yp = cut(p, 0, 1/2) for p = 1 … n

My half cut is here.

My half cut is here.

o n log n divide and conquer algorithm even and paz13
I like O(n log n) Divide and Conquer Algorithm: Even and Paz
  • Yp = cut(p, 0, 1/2) for p = 1 … n

My half cut is here.

o n log n divide and conquer algorithm even and paz14
O(n log n) Divide and Conquer Algorithm: Even and Paz
  • Yp = cut(p, 0, 1/2) for p = 1 … n
  • m = median(y1, … , yn)
o n log n divide and conquer algorithm even and paz15
I like

I like

so am happy with the left.

so am happy with the right.

O(n log n) Divide and Conquer Algorithm: Even 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)
problem variations
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 variations17
Problem Variations

(Approximate)

failure prob O(1) vs 1/nO(1)

Open: Remove contiguous requirement for randomized alg

outline
Outline
  • Deterministic Ω(n log n) Lower Bound
    • Definition of Thin-Rich Problem
    • Ω(log n) lower bound for Thin-Rich
  • Randomized with Approximate Cuts Ω(n log n) Lower Bound
  • Randomized with Exact Cuts O(n) Upper Bound
thin rich game
At least n/2 players require thin rich pieceThin-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).
slide20
Alg

Adv

  • I dynamically choose how to answer
  • and the value function
  • so after O(logn) operationsAlg cant find a thin rich piece.
  • I give sequence of Eval[x,y] & Cut[x,v]operations.
  • Oh!
slide21
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.
slide22
Alg

Adv

...

...

...

...

...

...

  • I build a complete 3-ary treewith the n pieces as leaves
slide23
Alg

Adv

½

¼

½

¼

¼

¼

...

...

...

...

...

...

  • For each node,
  • I label edges
  • <½,¼,¼> or <¼,¼,½>
slide24
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.
slide25
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

slide26
Alg

Adv

0.4398

0

y

  • I need to find a yso that V[0,y] = 0.4398.
  • Cut[0,0.4398]?
slide27
Alg

Adv

1/2

0

1

0.4398

0.4398

  • I do binary search to find0.4398
  • Cut[0,0.4398]?

1/4

slide28
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]?
slide29
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

slide30
Alg

Adv

¼

½

½

¼

¼

¼

7/16

8/16

0.4398

  • I do binary search to find0.4398
  • I fix some edge labels
  • Cut[0,0.4398]?
slide31
Alg

Adv

¼

¼

½

½

½

¼

¼

¼

¼

0.4398

29/64

28/64

  • I do binary search to find0.4398
  • I fix some edge labels
  • Cut[0,0.4398]?
slide32
Alg

Adv

¼

¼

¼

½

½

½

½

¼

¼

¼

¼

¼

0.4398

112/256

113/256

  • I do binary search to find0.4398
  • I fix some edge labels
  • Cut[0,0.4398]?
slide33
Alg

Adv

¼

¼

¼

½

½

½

½

½

¼

¼

¼

¼

¼

¼

¼

0.4398

450/1024

451/1024

  • I do binary search to find0.4398
  • I fix some edge labels
  • Cut[0,0.4398]?
slide34
Alg

Adv

¼

¼

¼

½

½

½

½

½

¼

¼

¼

¼

¼

¼

¼

0.4398

1800/4096

1801/4096

¼

¼

½

  • I do binary search to find0.4398
  • I fix some edge labels
  • Cut[0,0.4398]?
slide35
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]?
slide36
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
slide37
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.
slide38
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

slide39
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

That is bad.To get a rich pieceI need at least 40% of the edge labels in path be ½.

  • YesAfter t operationsevery path has only t½ known labels.
deterministic log n lower bound
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)
deterministic nlog n lower bound
At least n/2 players require thin rich pieceDeterministic 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).
outline42
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

slide43
Adv
  • I define a randomized algorithm
    • I must choose the value functions
  • Show after O(nlogn) operations Alg fails whp.
  • I flip coins to give sequence of Eval[x,y] & Cut[x,v]operations.

Rand Alg

  • i.e. not dynamically
slide44
Adv

Yao

Alg

RandAdv

  • I define a randomized algorithm
    • I must choose the value functions
  • Show after O(nlogn) operations Alg fails whp.
  • I flip coins to give sequence of Eval[x,y] & Cut[x,v]operations.

Rand Alg

  • I specify a distribution on value functions.
  • I flip coins.
  • Show after O(nlogn) operations Alg fails whp.
  • I deterministically give sequence of Eval[x,y] & Cut[x,v]operations.
slide45
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>
slide46
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%!
slide47
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’

slide48
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’

slide49
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’

outline50
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

outline51
Outline
  • Randomized with O(n) Upper Bound
    • As intuition for the power of randomness, give a randomized algorithm for Thin-Rich with O(1) complexity
    • Describe “Balls and Bins” special case of cake cutting
    • Give “random graph” proof of “Balls and Bins” case
    • Give the flavor of the “random graph” proof for Cake Cutting
    • Decrease failure probability
thin rich game52
Thin-Rich Game
  • Game Definition: Single player must find a thin rich piece.
    • A piece is thin if it has width ≤ 3/n
    • A piece is rich if it has value ≥ 1/3n
  • Theorem: The deterministic complexity of Thin-Rich is Ω(log n).
  • Theorem: The randomized complexity of Thin-Rich is O(1).
  • Must make sure the player’s thin pieces don’t over lap too much.
o 1 complexity randomized protocol for thin rich
O(1) Complexity Randomized Protocol for Thin-Rich
  • Pick an i uniformly at random from 1 … 3n
  • x = Cut[0,(i-1)/3n]
  • y = Cut[ 0, i/3n]
  • Return [x,y]

value = 1/3n (ie rich)

Exp(width) = 1/3n <<3/n (ie thin)

3n Candidate Pieces

i

x=(i-1)/3n

y=i/3n

randomized protocol for cake cutting
Randomized Protocol for Cake Cutting

Protocol Description:

  • Each player randomly selects 8 candidate pieces.
  • For each player, we carefully pick one of these
randomized protocol for cake cutting55
O(1)

1/nO(1).

Randomized Protocol for Cake Cutting

Protocol Description:

  • Each player randomly selects 8 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.

Fails with probability 0(1).

balls and bins
Balls and Bins

Two Random Choices: n balls, 3n bins

each ball randomly chooses two bins.

# Hits: Whp no bin has more than O(logn) balls.

Choice: Select one of two bins for each ball.

Online: Whp no bin has more than O(loglogn) balls.

balls and bins57
Balls and Bins

Two Random Choices: n balls, 3n bins

each ball randomly chooses two bins.

# Hits: Whp no bin has more than O(logn) balls.

Choice: Select one of two bins for each ball.

Online: Whp no bin has more than O(loglogn) balls.

O(loglogn) balls per bin

balls and bins58
Balls and Bins

Two Random Choices: n balls, 3n bins

each ball randomly chooses two bins.

# Hits: Whp no bin has more than O(logn) balls.

Choice: Select one of two bins for each ball.

Online: Whp no bin has more than O(loglogn) balls.

Offline: Whp no bin has more than O(1) balls.

O(1) balls per bin

slide59
Proof on Offline: O(1)-Balls
  • 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
slide60
Proof on Offline: O(1)-Balls
  • Classic Theorem: If a graph G has n/3 independent edges, then G is acyclic with (1) prob.
  • Proof:

Prob[G contains a cycle C] ≤ ΣC Prob[C is in the graph] ~ Σi (n choose i) * (1/3n)i=(1).

cake protocol ball bin game
Cake Protocol  Ball & Bin Game

Same if

  • Each player has uniform value on cake,

so all candidate pieces are the same.

  • Each player randomly selects 2 (or 8) candidate pieces.
  • For each player, pick one of these.
  • O(1) over lap = O(1) balls per bin
cake protocol ball bin game62
I like

I like

I like

Cake Protocol  Ball & Bin Game

Different if

  • Different cake values

can make strangely overlapping candidate pieces.

cake protocol ball bin game63
Cake Protocol  Ball & Bin Game

Different if

  • Different cake values

can make strangely overlapping candidate pieces.

cake protocol ball bin game64
Cake Protocol  Ball & Bin Game

Different if

  • Different cake values

can make strangely overlapping candidate pieces.

directed graph for cake cutting choose 2 not 8 candidate peices
Vertex

Vertex

Vertex

Vertex

Directed Graph for Cake Cutting (choose 2 not 8 candidate peices)

If take

Must take

Can’t take

one difficulty
Must take

If take

Must take

Must take

One Difficulty

Theorem: Sufficient Condition

  • If there is not directed path between the two pieces of the same person
  • then maximum load is at most 1
another difficulty edges may not be independent
Another Difficulty: Edges May Not be Independent

Vertex

Vertex

Vertex

Vertex

Vertex

Vertex

Theorem: If we select 8 candidate pieces

and narrow these down to 2,

then likely, there are not many dependent edges.

example too many vees
Example: Too many Vees

Vee

  • If independent,
    • Exp(# of Vees) = n3 × 1/n × 1/n = (n)
  • Choose 2 candidate pieces
    • Exp(# of Vees) = n3 × 1/n × 1 = (n2)
example too many vees69
VeeExample: Too many Vees

n/2

n/2

  • Choose 2 candidate pieces
    • Exp(# of Vees) = n3 × 1/n × 1 = (n2)
example too many vees70
VeeExample: Too many Vees

n/2

n/2

  • Choose 2 candidate pieces
    • Exp(# of Vees) = n3 × 1/n × 1 = (n2)
  • Choose 8 candidate pieces
    • Exp(# of Vees) = … = (n)

and narrow these down to 2

too many self paths
VeeToo many Self Paths

Independent

  • If independent,
    • Exp(# of self paths) = like in balls & bins = (1)
  • Choose 2 candidate pieces
    • Exp(# of self paths) = (n)
  • Choose 8 candidate pieces and narrow these down to 2
    • Exp(# of self paths) = … = (1)

Prob(none) (1)

slide72
Conclusions

Theorem: With O(1) probability,there is not directed path between the two pieces of the same person

For each player, can pick one of these 8.

  • so that every point of cake is covered by at most O(1) pieces.

Where there is overlap, recurs.

problem variations73
A piece of cake.Problem Variations

(Approximate)

failure prob (1) vs 1/n(1)

slide74
Conclusions

Theorem: With O(1/n) probability,there is not directed path between the two pieces of player Pi. If so Pi is “bad”.

Run the algorithm twice, independently.

Each player stays in one run in which it is “good”.

Pr[Pi is bad in both runs] = O(1/n2)

Pr[Some player is bad in both runs] = n × O(1/n2)

slide75
Conclusions

Theorem: With O(1/n) probability,each run has no bad directed paths.

For each run, each of its good playerscan pick one of these 8 so that every point of cake is covered by at most O(1) pieces.

Run 1

Run 2

Where there is overlap, recurs.

slide76
Conclusions

Theorem: Alg fails with prob O(1/n).

Repeat k times.

Theorem: Alg fails k times with prob O(1/nk).

outline77
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

Skipped

Done

future directions
Future Directions
  • Determine whether randomization makes sense with cheaters
  • The complexity of several variations of fair division is still not completely nailed down
  • Complexity of envy free division is quite wide open
    • Exponential difference in upper and lower bounds
    • Current project
informal problem statement80
Informal Problem Statement

Resource allocation between n self-interested players

slide81
Adv
  • I dynamically choose error in answers
  • so after O(nlogn) operationsAlg fails whp.
  • I flip coins to give sequence of Eval[x,y] & Cut[x,v]operations.

Rand Alg

slide82
Adv

Yao

Alg

RandAdv

  • I dynamically choose error in answers
  • so after O(nlogn) operationsAlg fails whp.
  • I flip coins to give sequence of Eval[x,y] & Cut[x,v]operations.

Rand Alg

  • I flip coins to choose value function.
  • I dynamically choose error in answers
  • so after O(nlogn) operationsAlg fails whp.
  • I deterministically give sequence of Eval[x,y] & Cut[x,v]operations.
o 1 complexity randomized protocol for thin rich83
O(1) Complexity Randomized Protocol for Thin-Rich
  • For i=0…n, xi = Cut[0,i/n],
  • Cuts cake into n “candidate” pieces.
    • Each piece has value 1/n for player.
    • Different widths, but average is 1/n.
  • We will choose a random one for player.
  • It is likely has width 2/n.
    • i.e. thin and rich.

Constant time?

example too many vees84
Example: Too many Vees

n/2

n/2

Exp(# of Vees) = n3×1/n× 1/n = (n)

Exp(# of Vees) = (1) ×(n2)

example too many vees85
Example: Too many Vees

If independent,

Exp(# of Vees) = n3×1/n× 1/n = (n)

another difficulty forks
Another Difficulty: Forks

Fork of depth 3

Theorem: If we select 2d candidate pieces

and narrow these down to 2,

then likely, there are none of these.

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