CMU 15-896 Matching 1: Kidney exchange Teacher: Ariel Procaccia - - PowerPoint PPT Presentation

CMU 15-896

Matching 1: Kidney exchange

Teacher: Ariel Procaccia

15896 Spring 2016: Lecture 13

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Donor 2 Patient 2 Donor 1 Patient 1

Kidney Exchange

15896 Spring 2016: Lecture 13

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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15896 Spring 2016: Lecture 13

The strategic model

• Undirected graph (only pairwise matches!)
• Vertices = donor-patient pairs
• Edges = compatibility
• Each player controls subset of vertices
• Mechanism receives a graph and returns a matching
• Utility of player = # its matched vertices
• Target: # matched vertices
• Strategy: subset of revealed vertices
• But edges are public knowledge
• Mechanism is strategyproof (SP) if it is a dominant

strategy to reveal all vertices

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15896 Spring 2016: Lecture 13

OPT is manipulable

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15896 Spring 2016: Lecture 13

OPT is manipulable

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15896 Spring 2016: Lecture 13

Approximating SW

• Theorem [Ashlagi et al. 2010]: No

deterministic SP mechanism can give a approximation

• Proof: We just proved it!
• Theorem [Kroer and Kurokawa 2013]: No

randomized SP mechanism can give a

• approximation
• Proof: Homework 2

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15896 Spring 2016: Lecture 13

SP mechanism: Take 1

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• Assume two players
• The MATCH{{1},{2}} mechanism:
• Consider matchings that maximize the

number of “internal edges”

• Among these return a matching with max

cardinality

15896 Spring 2016: Lecture 13

Another example

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15896 Spring 2016: Lecture 13

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 players

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15896 Spring 2016: Lecture 13

Proof of theorem

• matching when player 1 is

honest, = matching when player 1 hides vertices

• consists of paths and even-

length cycles, each consisting of alternating edges

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• ′

′

• ′

∩ ′

• ∩

′

What’s wrong with the illustration on the right?

15896 Spring 2016: Lecture 13

Proof of theorem

• Consider a path in

, denote its edges in by and its edges in by

• For
• , suppose
• It holds that
• is max cardinality
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15896 Spring 2016: Lecture 13

Proof of theorem

• Suppose
• Every subpath within is of

even length

• We can pair the edges of

and

, except maybe the first

and the last

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15896 Spring 2016: Lecture 13

The case of 3 players

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15896 Spring 2016: Lecture 13

SP mechanism: Take 2

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• Let
• be a bipartition of the

players

• The 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)

15896 Spring 2016: Lecture 13

Eureka?

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

strategyproof for any number of players and any partition

• Recall: for

, MATCH{{1},{2}} guarantees a 2-approx

15896 Spring 2016: Lecture 13

Eureka?

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Poll 1: approximation guarantees given by MATCH for and ?

1. 2. 3. 4.

More than

15896 Spring 2016: Lecture 13

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 guarantees a 2-approximation

• We only prove the approximation ratio

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15896 Spring 2016: Lecture 13

Proof of theorem

• ∗
• ptimal matching
• Create a matching

such that

is max cardinality on each , and

• ∗
• ∗
• ∗∗ max cardinality on each
• For each path in ∗Δ∗∗, add ∩ ∗∗ to ′ if

∗∗ has more internal edges than ∗, otherwise add ∩ ∗ to ′

• For every internal edge ′ gains relative to ∗, it

loses at most one edge overall ∎

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15896 Spring 2016: Lecture 13

Proof of theorem

• Fix

and let

be the output of

MATCH

• The mechanism returns max cardinality

across subject to being max cardinality internally, therefore

• ∈,∈
• ∈,∈
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15896 Spring 2016: Lecture 13

Proof of theorem

1 2

• ∈,∈
• 1

2

• ∈,∈
• 1

2

• ∈,∈
• 1

2

• ∗ 1

2

∗

• 1

2

∗ 1

2

∗

• 1

2 ∗ ∎

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