[PPT] - CSC2556 Lecture 6 Kidney Exchange Cake-Cutting [Some PowerPoint Presentation

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CSC2556 Lecture 6 Kidney Exchange Cake-Cutting

[Some illustrations due to: Ariel Procaccia]

CSC2556 - Nisarg Shah 1

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Announcements

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Project proposal

➢ Due: Mar 06 by 11:59PM ➢ I’ll soon put up a few sample project ideas. ➢ If you have trouble finding a project idea, meet me.

Structure

➢ Problem space introduction ➢ High-level research question ➢ Prior work ➢ Detailed goals

Length: Ideally 1 page (2 pages max)

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Kidney Exchange

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4

Donor 2 Patient 2 Donor 1 Patient 1

Kidney Exchange

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Incentives

A decade ago kidney exchanges were carried out

by individual hospitals

Today there are nationally organized exchanges;

participating hospitals have little other interaction

It was observed that hospitals match easy-to-

match pairs internally, and enroll only hard-to- match pairs into larger exchanges

Goal: incentivize hospitals to enroll all their pairs

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The strategic model

Undirected graph, only pairwise matches

➢ Vertex = donor-patient pair ➢ Edge = compatibility

Each agent controls a subset of vertices

➢ Possible strategy: hide some vertices (match internally), and

nly reveal others

➢ Utility of agent = # its matched vertices (self-matched +

matched by mechanism)

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The strategic model

Mechanism:

➢ Input: revealed vertices by agents (edges are public) ➢ Output: matching

Target: # matched vertices
Strategyproof (SP): If no agent benefits from hiding

vertices irrespective of what other agents do.

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SLIDE 8

OPT is manipulable

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SLIDE 9

OPT is manipulable

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Approximating SW

Theorem [Ashlagi et al. 2010]: No deterministic SP

mechanism can give a 2 − 𝜗 approximation

Proof:

➢ No perfect matching exists. ➢ Any algorithm must either leave a blue node or a gray node

unmatched.

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Approximating SW

Theorem [Ashlagi et al. 2010]: No deterministic SP

mechanism can give a 2 − 𝜗 approximation

Proof:

➢ Suppose it leaves a blue node unmatched

If the blue agent hides two nodes as follows, the mechanism is forced

to return a matching of size 1 when a matching of size 2 exists.

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Approximating SW

Theorem [Ashlagi et al. 2010]: No deterministic SP

mechanism can give a 2 − 𝜗 approximation

Proof:

➢ Suppose it leaves a gray node unmatched

If the gray agent hides two nodes as follows, the mechanism is forced

to return a matching of size 1 when a matching of size 2 exists.

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Approximating SW

Theorem [Kroer and Kurokawa 2013]: No randomized

SP mechanism can give a

6 5 − 𝜗 approximation.

Proof: Homework!

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SP mechanism: Take 1

14

Assume two agents
MATCH{{1},{2}} mechanism:

➢ Consider matchings that maximize the number of

“internal edges” for each agent.

➢ Among these return, a matching with max overall

cardinality.

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Another example

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Guarantees

MATCH{{1},{2}} gives a 2-approximation

➢ Cannot add more edges to matching ➢ For each edge in optimal matching, one of the two

vertices is in mechanism’s matching

Theorem (special case): MATCH{{1},{2}} is

strategyproof for two agents.

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Proof

17

𝑁 = matching when player 1 is

honest, 𝑁′ = matching when player 1 hides vertices

𝑁Δ𝑁′ consists of paths and even-

length cycles, each consisting of alternating 𝑁, 𝑁′ edges

𝑊

1

𝑊

2

𝑁 𝑁 𝑁′ 𝑁′ 𝑁 𝑁′ 𝑁 ∩ 𝑁′ 𝑁 ∩ 𝑁′

What’s wrong with the illustration on the right?

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Proof

Consider a path in 𝑁Δ𝑁′, denote its edges in 𝑁 by

𝑄 and its edges in 𝑁′ by 𝑄′

Consider sets 𝑄

11, 𝑄22, 𝑄 12 containing edges of 𝑄

among 𝑊

1, among 𝑊 2, and between 𝑊 1- 𝑊 2

➢ Same for 𝑄′11, 𝑄′22, 𝑄′12

Note that 𝑄

11 ≥ 𝑄 11 ′

➢ Property of the algorithm

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Proof

Case 1: 𝑄

11 = 𝑄 11 ′

Agent 2’s vertices don’t change, so 𝑄

22 = 𝑄 22 ′

𝑁 is max cardinality ⇒ 𝑄

12 ≥ 𝑄 12 ′

𝑉1 𝑄 = 2 𝑄

11 + 𝑄 12

≥ 2 𝑄

11 ′

+ 𝑄

12 ′

= 𝑉1(𝑄′)

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Proof

Case 2: 𝑄

11 > 𝑄 11 ′

𝑄

12 ≥ 𝑄 12 ′

− 2

➢ Every sub-path within 𝑊

2 is of even length

➢ Pair up edges of 𝑄

12 and 𝑄 12 ′ ,

except maybe the first and the last

𝑉1 𝑄 = 2 𝑄

11 + 𝑄 12

≥ 2 𝑄

11 ′

+ 1 + 𝑄

12 ′

− 2 = 𝑉1 𝑄′ ∎

20

𝑊

1

𝑊

2

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The case of 3 players

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SP Mechanism: Take 2

22

Let Π = Π1, Π2 be a bipartition of the players
MATCH mechanism:

➢ Consider matchings that maximize the number of

“internal edges” and do not have any edges between different players on the same side of the partition

➢ Among these return a matching with max cardinality

(need tie breaking)

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Eureka?

23

Theorem [Ashlagi et al. 2010]: MATCH is

strategyproof for any number of agents and any partition Π.

Recall: For 𝑜 = 2, MATCH{{1},{2}} is a 2-approximation
Question: 𝑜 = 3, MATCH{{1},{2,3}} approximation?
1. 2
2. 3
3. 4
4. More than 4

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The Mechanism

The MIX-AND-MATCH mechanism:

➢ Mix: choose a random partition  ➢ Match: Execute MATCH

Theorem [Ashlagi et al. 2010]: MIX-AND-MATCH is

strategyproof and a 2-approximation.

We only prove the approximation ratio.

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Proof

𝑁∗ = optimal matching
Claim: I can create a matching 𝑁′ such that

➢ 𝑁′ is max cardinality on each 𝑊

𝑗, and

➢ σ𝑗 𝑁𝑗𝑗

′ + 1 2 σ𝑗≠𝑘 𝑁𝑗𝑘 ′

≥ σ𝑗 𝑁𝑗𝑗

∗ + 1 2 σ𝑗≠𝑘 |𝑁𝑗𝑘 ∗ |

➢ 𝑁∗∗ = max cardinality on each 𝑊

𝑗

➢ For each path 𝑄 in 𝑁∗Δ𝑁∗∗, add 𝑄 ∩ 𝑁∗∗ to 𝑁′ if 𝑁∗∗ has

more internal edges than 𝑁∗, otherwise add 𝑄 ∩ 𝑁∗ to 𝑁′

➢ For every internal edge 𝑁′ gains relative to 𝑁∗, it loses at

most one edge overall ∎

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Proof

Fix Π and let 𝑁Π be the output of MATCH
The mechanism returns max cardinality across Π

subject to being max cardinality internally, therefore

෍

𝑗

𝑁𝑗𝑗

Π +

෍

𝑗∈Π1,𝑘∈Π2

𝑁𝑗𝑘

Π ≥ ෍ 𝑗

𝑁𝑗𝑗

′ +

෍

𝑗∈Π1,𝑘∈Π2

𝑁𝑗𝑘

′

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Proof

𝔽 𝑁Π = 1 2𝑜 ෍

Π

෍

𝑗

𝑁𝑗𝑗

Π +

෍

𝑗∈Π1,𝑘∈Π2

𝑁𝑗𝑘

Π

≥ 1 2𝑜 ෍

Π

෍

𝑗

𝑁𝑗𝑗

′ +

෍

𝑗∈Π1,𝑘∈Π2

𝑁𝑗𝑘

′

= ෍

𝑗

𝑁𝑗𝑗

′ + 1

2𝑜 ෍

Π

෍

𝑗∈Π1,𝑘∈Π2

𝑁𝑗𝑘

′

= ෍

𝑗

𝑁𝑗𝑗

′ + 1

2 ෍

𝑗≠𝑘

𝑁𝑗𝑘

′

≥ ෍

𝑗

𝑁𝑗𝑗

∗ + 1

2 ෍

𝑗≠𝑘

𝑁𝑗𝑘

∗

≥ 1 2 ෍

𝑗

𝑁𝑗𝑗

∗ + 1

2 ෍

𝑗≠𝑘

𝑁𝑗𝑘

∗

= 1 2 𝑁∗ ∎

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SLIDE 28

Cake-Cutting

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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

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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

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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)

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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

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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?

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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.

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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

SLIDE 36

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?

SLIDE 37

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 the 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.

SLIDE 38

DUBINS-SPANIER

38

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.

SLIDE 40

DUBINS-SPANIER

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SLIDE 41

DUBINS-SPANIER

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Τ 1 3

SLIDE 42

DUBINS-SPANIER

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Τ 1 3 Τ 1 3

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DUBINS-SPANIER

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Τ 1 3 Τ 1 3 ≥ Τ 1 3

SLIDE 44

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

SLIDE 45

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.

SLIDE 46

EVEN-PAZ

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SLIDE 47

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

SLIDE 48

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

SLIDE 49

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!

SLIDE 50

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!