The Communication Complexity of Fair Division Simina Brnzei and - - PowerPoint PPT Presentation

Noam Nisan

The Communication Complexity

• f

Fair Division

Simina Brânzei and Noam Nisan

Hebrew University

Cake Cutting

n Metaphor for fair division:

q Cake = [0,1] q Each player i has a non-atomic probability

measure vi on [0,1]

q Challenge: partition the cake to the n players so

that each feels that he has been treated “fairly”.

q Many applications, from land to divorces to cloud

computing…

q Long research history (Steinhaus 1948 … Aziz

Mackenzie 17)

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Some Fairness Criteria

n Proportional: vi(Si) ≥ 1/n. n Envy Free: vi(Si) ≥ vi(Sj). n Equitable: vi(Si) = vj(Sj). n Perfect: vi(Sj) = 1/n.

Theorem: For any tuple of atomless player valuations, each of these exists. Proof: Usually some fixed-point theorem

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Example: Perfect, n=2 players Austin

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v1([x,f(x)])=1/2 x f(x)

Example: Perfect, n=2 players

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v1([x,f(x)])=1/2 x f(x) v2([x,f(x)]) >? 1/2

Example: Perfect, n=2 players

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v1([x*,f(x*)])=1/2 x* f(x*) v2([x*,f(x*)]) = 1/2

Models for Cake Cutting Algorithms

n Atom-less Input valuations n Infinite-precision output n Infinite-precision Operations/Queries:

q Robertson-Webb model:

n Value Query: vi([a,b])? n Cut Query: Find x such that vi([0,x])=c.

q Moving Knife Procedures:

n Various examples where players continuously move

knives as a function of their valuation

n No single accepted formal model

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Discrete Models For Cake Cutting

n Discrete Queries n Output to satisfy fairness to within e:

q Proportional: vi(Si) ≥ 1/n - e. q Envy Free: vi(Si) ≥ vi(Sj) - e. q Equitable: |vi(Si) - vj(Sj)| ≤ e. q Perfect: |vi(Sj) - 1/n| ≤ e.

n Input valuations have bounded density

q Otherwise nothing is possible

n Complexity is a function of e (and n)

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Communication Complexity

Which discrete query model should we choose?

q vi([a,b]) > c? q Any query that depends on a single valuation

n

But “charge” for the length of the answer in bits

Communication Complexity model (Yao 79)

q Each player “knows” his valuation q They exchange messages according to a fixed protocol q Equivalent to “any query on a single valuation” model

n A message from i corresponds to a query on vi n Number of bits corresponds to answer length

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Simulating infinite-precision queries

Proposition: For any notion of fairness that can be found using O(1) Robertson-Webb queries, an e-fair cutting can be found with O(1) rounds

• f communication of O(log e-1) bits.

Proof: “Round” the valuations (carefully). Comment: Simulation only works for “notions of fairness”

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Simulating Moving Knife Procedures

Definition: Moving Knife Procedure….

(proofs that use intermediate value theorem…)

Theorem: A moving knife procedure can be simulated with O(log e-1) rounds of communication each using O(log e-1) bits.

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Proof of Theorem (by example)

Binary search over knife positions. Requires O(loge-1) “queries” each requiring O(loge-1) bits of communication.

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v1([x,f(x)])=1/2 x f(x) v2([x,f(x)]) >? 1/2

Deduced CC of popular fairness notions

Easy (RW protocols): Q(log e-1)

q Contiguous Proportional (any fixed n) Even-Paz q Envy Free (any fixed n) Aziz-Mackenzie

Medium (Moving Knife Protocol): O(log2 e-1)

q Contiguous Envy Free (n=3)

Stromquist

q Contiguous Equitable (any fixed n) q Perfect, 2 cuts (n=2)

Austin

Hard: O(e-1 log e-1)

q Everything else

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New Upper bounds

Easy (RW protocols): Q(log e-1)

q Contiguous Proportional (any fixed n) q Envy Free (any fixed n)

Medium (Moving Knife Protocol):

q Contiguous Envy Free (n=3) q Contiguous Equitable (n=2) q Perfect, 2 cuts (n=2)

Hard: O(e-1 log e-1)

q Everything else

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Q(log e-1)

O(log e-1 loglog e-1) randomized

New Lower Bounds

Easy (RW protocols): Q(log e-1) (with O(1) rounds)

q Contiguous Proportional (any fixed n) q Envy Free (any fixed n)

Medium (Moving Knife Protocol):

q Contiguous Envy Free (n=3) q Contiguous Equitable (n=2) q Perfect, 2 cuts (n=2)

Hard: O(e-1 log e-1)

q Everything else

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Main Open Problem Require many rounds

The Crossing Problem

n Alice gets x0, x1, …, xm Î {0, 1, … m} n Bob gets y0, y1, …, ym Î {0, 1, … m} n x0 ≤ y0 ; xm ≥ ym n Find i so that xi ≤ yi ; xi+1 ≥ yi+1

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The Crossing Problem

n Alice gets x0, x1, …, xm Î {0, 1, … m} n Bob gets y0, y1, …, ym Î {0, 1, … m} n x0 ≤ y0 ; xm ≥ ym n Find i so that xi ≤ yi ; xi+1 ≥ yi+1

Monotone Variant:

q 0 = x0 ≤ x1 ≤ … ≤ xm = m q m = y0 ≥ y1 ≥ … ≥ ym = 0

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Monotone Crossing ≈ Equitable(n=2)

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v1([0,x*]) = v2([x*,1]) xi=v1([0,i/m]) yi=v2([i/m,1])

Crossing ≈ Perfect(n=2, 2 cuts)

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i/m xi

xi chosen so that v1([i/m,xi]) ≈ 1/2 yi chosen so that v2([i/m,yi]) ≈ 1/2

1/2

Monotone Crossing problem

Lemma: The Monotone Crossing problem has an O(log m) bit communication protocol. Corollary: Contiguous equitable cutting for n=2 can be done with O(loge-1) bits. Contiguous Envy Free cutting for n=3 can be done with O(loge-1) bits. Proof: Consider crossing with m numbers in range 0…k.

n Query xm/2>k/2? And ym/2>k/2? n Answers will allow to either cut m by factor of 2 or cut k by

factor of 2, giving a total of O(log m + log k) queries.

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General Crossing Problem

Lemma: The Crossing problem has a randomized communication protocol that uses O(log m loglog m) bits. Corollary: Perfect cutting with 2-cuts for n=2 can be done with O(loge-1 logloge-1) bits (randomized). Proof: Binary search using “xi<yi?” queries. Randomized CC of each “x<y?” is only loglogm (Nisan-Safra 1993) .

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x0 x1 . . xi . . . xm y0 y1 . . yi . . . ym

≤ ≥

?

Communication Rounds

Recall: CC rounds correspond to queries and number of bits corresponds to length of answers.

n General simulation of Robertson-Webb protocols gives

r=O(1) rounds and t=O(loge-1) bits for the “easy problems”.

n General simulation of moving-knife protocols, as well as

• ur new upper bounds require r=O(loge-1) ) rounds for

the “medium problems”. Theorem: The monotone crossing problem requires r=W(log m/log t) rounds of most t bits each. Corollary: For n=2, Perfect and Equitable division require r=W(loge-1/logloge-1) rounds of t=polyloge-1 bits

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Round-elimination proofs

Main Lemma: r-round protocol for input size m è (r-1)-round protocol for input size W(m/t). Proof Structure:

Step 1: “Encode” k instances of problem of size m as one instance of problem of size km. Step 2: If t<<k then first round of t bits of communication cannot “help” all k instances.

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Round elimination lemma (Miltersen et al 1998, Sen-Venkatesh 2008)

n Let f(x,y) be an arbitrary communication

problem.

n fk is defined as follows:

q Alice gets x1…xk q Bob gets zÎ{1…k} and y q Need to solve f(xz,y)

Lemma: A randomized communication protocol for fk where Alice sends first round of t<<k bits of communication implies a similar protocol for f with the first round removed.

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Encoding multiple instances

Notation: MC(m) – the monotone crossing problem on inputs of size m.

q 0 = x0 ≤ x1 ≤ … ≤ xm = m q m = y0 ≥ y1 ≥ … ≥ ym = 0 q Goal: Find i so that xi ≤ yi ; xi+1 ≥ yi+1

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Encoding multiple instances

Notation: MC(m) – the monotone crossing problem on inputs of size m.

q 0 = x0 ≤ x1 ≤ … ≤ xm = m q m = y0 ≥ y1 ≥ … ≥ ym = 0 q Goal: Find i so that xi ≤ yi ; xi+1 ≥ yi+1

Lemma: Protocol for MC(mk) è similar protocol for (MC(m))k

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Encoding multiple instances

Notation: MC(m) – the monotone crossing problem on inputs of size m.

q 0 = x0 ≤ x1 ≤ … ≤ xm = m q m = y0 ≥ y1 ≥ … ≥ ym = 0 q Goal: Find i so that xi ≤ yi ; xi+1 ≥ yi+1

Lemma: Protocol for MC(mk) è similar protocol for (MC(m))k Proof:

Alice: Bob:

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… x1 xz xk y z …

Summary

n Study communication requirements of fair division

notions

n Main open problem: Prove a super-logarithmic CC

lower bound for some fairness notion

n Other open problems:

q Determine the exact randomized/deterministic CC of the general

crossing problem.

q Lower bound for contiguous envy-free cutting for n=3?

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Thanks

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