Lecture 6 Fair Division 1: Cake-Cutting [Some illustrations due to: - - PowerPoint PPT Presentation

CSC2556 Lecture 6 Fair Division 1: Cake-Cutting

[Some illustrations due to: Ariel Procaccia]

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Announcements

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• Reminder

➢ Project proposal due by March 1st by 12:59PM ➢ If you want to run your idea by me, this is a good time to

approach me.

• Remember to use office hours (drop me an email)

if you’re having any difficulty with homework questions.

Fair Division

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Cake-Cutting

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• A heterogeneous, divisible good

➢ Heterogeneous: it may be valued

differently by different individuals

➢ Divisible: we can share/divide

it between individuals

• Represented as [0,1]

➢ Almost without loss of generality

• Set of players 𝑂 = {1, … , 𝑜}
• Piece of cake 𝑌 ⊆ [0,1]

➢ A finite union of disjoint intervals

Agent Valuations

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• Each player 𝑗 has a valuation 𝑊

𝑗 that

is very much like a probability distribution over [0,1]

• Additive: For 𝑌 ∩ 𝑍 = ∅,

𝑊

𝑗 𝑌 + 𝑊 𝑗 𝑍 = 𝑊 𝑗 𝑌 ∪ 𝑍

• Normalized: 𝑊

𝑗

0,1 = 1

• Divisible: ∀𝜇 ∈ [0,1] and 𝑌,

∃𝑍 ⊆ 𝑌 s.t. 𝑊

𝑗 𝑍 = 𝜇𝑊 𝑗(𝑌)

𝛽 𝜇𝛽 𝛽 β β

𝛽 + 𝛾

Fairness Goals

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• An allocation is a disjoint partition 𝐵 = (𝐵1, … , 𝐵𝑜)
• f the cake
• We desire the following fairness properties from
• ur allocation 𝐵:
• Proportionality (Prop):

∀𝑗 ∈ 𝑂: 𝑊

𝑗 𝐵𝑗 ≥ 1

𝑜

• Envy-Freeness (EF):

∀𝑗, 𝑘 ∈ 𝑂: 𝑊

𝑗 𝐵𝑗 ≥ 𝑊 𝑗(𝐵𝑘)

Fairness Goals

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• Prop: ∀𝑗 ∈ 𝑂: 𝑊

𝑗 𝐵𝑗 ≥

Τ 1 𝑜

• EF: ∀𝑗, 𝑘 ∈ 𝑂: 𝑊

𝑗 𝐵𝑗 ≥ 𝑊 𝑗 𝐵𝑘

• Question: What is the relation between

proportionality and EF?

1.

Prop ⇒ EF

2.

EF ⇒ Prop

3.

Equivalent

4.

Incomparable

CUT-AND-CHOOSE

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• Algorithm for 𝑜 = 2 players
• Player 1 divides the cake into two pieces 𝑌, 𝑍 s.t.

𝑊

1 𝑌 = 𝑊 1 𝑍 =

Τ 1 2

• Player 2 chooses the piece she prefers.
• This is EF and therefore proportional.

➢ Why?

Input Model

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• How do we measure the “time complexity” of a

cake-cutting algorithm for 𝑜 players?

• Typically, time complexity is a function of the

length of input encoded as binary.

• Our input consists of functions 𝑊

𝑗, which requires

infinite bits to encode.

• We want running time just as a function of 𝑜.

Robertson-Webb Model

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• We restrict access to valuations 𝑊

𝑗’s through two

types of queries:

➢ Eval𝑗(𝑦, 𝑧) returns 𝑊

𝑗

𝑦, 𝑧

➢ Cut𝑗(𝑦, 𝛽) returns 𝑧 such that 𝑊

𝑗

𝑦, 𝑧 = 𝛽

𝑦 𝑧

𝛽

eval output cut output

Robertson-Webb Model

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• Two types of queries:

➢ Eval𝑗 𝑦, 𝑧 = 𝑊

𝑗

𝑦, 𝑧

➢ Cut𝑗 𝑦, 𝛽 = 𝑧 s.t. 𝑊

𝑗

𝑦, 𝑧 = 𝛽

• Question: How many queries are needed to find an

EF allocation when 𝑜 = 2?

• Answer: 2

➢ Why?

DUBINS-SPANIER

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• Protocol for finding a proportional allocation for 𝑜

players

• Referee starts at 0, and continuously moves knife

to the right.

• Repeat: when piece to the left of knife is worth 1

/𝑜 to a player, the player shouts “stop”, gets the piece, and exits.

• The last player gets the remaining piece.

DUBINS-SPANIER

13

1/3 1/3 ≥ 1/3

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DUBINS-SPANIER

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• Moving knife is not really needed.
• At each stage, we can ask each remaining player a

cut query to mark his 1/𝑜 point in the remaining cake.

• Move the knife to the leftmost mark.

DUBINS-SPANIER

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DUBINS-SPANIER

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Τ 1 3

DUBINS-SPANIER

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Τ 1 3 Τ 1 3

DUBINS-SPANIER

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Τ 1 3 Τ 1 3 ≥ Τ 1 3

DUBINS-SPANIER

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• Question: What is the complexity of the Dubins-

Spanier protocol in the Robertson-Webb model?

1.

Θ 𝑜

2.

Θ 𝑜 log 𝑜

3.

Θ 𝑜2

4.

Θ 𝑜2 log 𝑜

EVEN-PAZ

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• Input: Interval [𝑦, 𝑧], number of players 𝑜

➢ Assume 𝑜 = 2𝑙 for some 𝑙

• If 𝑜 = 1, give [𝑦, 𝑧] to the single player.
• Otherwise, let each player 𝑗 mark 𝑨𝑗 s.t.

𝑊

𝑗

𝑦, 𝑨𝑗 = 1 2 𝑊

𝑗

𝑦, 𝑧

• Let 𝑨∗ be the 𝑜/2 mark from the left.
• Recurse on [𝑦, 𝑨∗] with the left 𝑜/2 players, and on

[𝑨∗, 𝑧] with the right 𝑜/2 players.

EVEN-PAZ

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EVEN-PAZ

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• Theorem: EVEN-PAZ returns a Prop allocation.
• Proof:

➢ Inductive proof. We want to prove that if player 𝑗 is

allocated piece 𝐵𝑗 when [𝑦, 𝑧] is divided between 𝑜 players, 𝑊

𝑗 𝐵𝑗 ≥

Τ 1 𝑜 𝑊

𝑗

𝑦, 𝑧

• Then Prop follows because initially 𝑊

𝑗

𝑦, 𝑧 = 𝑊

𝑗

0,1 = 1

➢ Base case: 𝑜 = 1 is trivial. ➢ Suppose it holds for 𝑜 = 2𝑙−1. We prove for 𝑜 = 2𝑙. ➢ Take the 2𝑙−1 left players.

• Every left player 𝑗 has 𝑊

𝑗

𝑦, 𝑨∗ ≥ Τ 1 2 𝑊

𝑗

𝑦, 𝑧

• If it gets 𝐵𝑗, by induction, 𝑊

𝑗 𝐵𝑗 ≥ 1 2𝑙−1 𝑊 𝑗

𝑦, 𝑨∗ ≥

1 2𝑙 𝑊 𝑗

𝑦, 𝑧

EVEN-PAZ

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• Question: What is the complexity of the Even-Paz

protocol in the Robertson-Webb model?

1.

Θ 𝑜

2.

Θ 𝑜 log 𝑜

3.

Θ 𝑜2

4.

Θ 𝑜2 log 𝑜

Complexity of Proportionality

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• Theorem [Edmonds and Pruhs, 2006]: Any

proportional protocol needs Ω(𝑜 log 𝑜) operations in the Robertson-Webb model.

• Thus, the EVEN-PAZ protocol is (asymptotically)

provably optimal!

Envy-Freeness?

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• “I suppose you are also going to give such cute

algorithms for finding envy-free allocations?”

• Bad luck. For 𝑜-player EF cake-cutting:

➢ [Brams and Taylor, 1995] give an unbounded EF protocol. ➢ [Procaccia 2009] shows Ω 𝑜2 lower bound for EF. ➢ Last year, the long-standing major open question of

“bounded EF protocol” was resolved!

➢ [Aziz and Mackenzie, 2016]: 𝑃(𝑜𝑜𝑜𝑜𝑜𝑜

) protocol!

• Not a typo!

Other Desiderata

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• There are two more properties that we often

desire from an allocation.

• Pareto optimality (PO)

➢ Notion of efficiency ➢ Informally, it says that there should be no “obviously

better” allocation

• Strategyproofness (SP)

➢ No player should be able to gain by misreporting her

valuation

Strategyproofness (SP)

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• For deterministic mechanisms

➢ “Strategyproof”: No player should be able to increase her

utility by misreporting her valuation, irrespective of what

• ther players report.
• For randomized mechanisms

➢ “Strategyproof-in-expectation”: No player should be able

to increase her expected utility by misreporting.

➢ For simplicity, we’ll call this strategyproofness, and

assume we mean “in expectation” if the mechanism is randomized.

Strategyproofness (SP)

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• Deterministic

➢ Bad news! ➢ Theorem [Menon & Larson ‘17] : No deterministic SP

mechanism is (even approximately) proportional.

• Randomized

➢ Good news! ➢ Theorem [Chen et al. ‘13, Mossel & Tamuz ‘10]: There is a

randomized SP mechanism that always returns an envy- free allocation.

Perfect Partition

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• Theorem [Lyapunov ’40]:

➢ There always exists a “perfect partition” (𝐶1, … , 𝐶𝑜) of

the cake such that 𝑊

𝑗 𝐶 𝑘 = Τ 1 𝑜 for every 𝑗, 𝑘 ∈ [𝑜].

➢ Every agent values every bundle equally.

• Theorem [Alon ‘87]:

➢ There exists a perfect partition that only cuts the cake at

𝑞𝑝𝑚𝑧(𝑜) points.

➢ In contrast, Lyapunov’s proof is non-constructive, and

might need an unbounded number of cuts.

Perfect Partition

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• Q: Can you use an algorithm for computing a

perfect partition as a black-box to design a randomized SP+EF mechanism?

➢ Yes! Compute a perfect partition, and assign the 𝑜

bundles to the 𝑜 players uniformly at random.

➢ Why is this EF?

• Every agent values every bundle at Τ

1 𝑜.

➢ Why is this SP-in-expectation?

• Because an agent is assigned a random bundle, her expected

utility is Τ

1 𝑜, irrespective of what she reports.