On the Distortion Value of the Elections with Abstention Mohammad - - PowerPoint PPT Presentation

On the Distortion Value of the Elections with Abstention

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Mohammad Ghodsi, Mohamad Latifjan, Masoud Seddighin
 (AAAI 2019)

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Voting

• Method to make decisions
• Input: Ordinal preferences
• Output: A single winner or 


A rank list

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Voting - Formal definition

• Voters,
• Candidates,
• Voter has a preference over the candidates

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V = {v1, v2, …, vn}

C = {c1, c2, …, cm}

≺i

vi

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Voting - Good voting

• IF everyone prefers X over Y, Y should be ranked
• ver X
• X and Y ranking is independent of other

candidates

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Voting - Good voting

• IF everyone prefers X over Y, Y should be ranked
• ver X
• X and Y ranking is independent of other

candidates

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Arrow’s Impossibility Theorem:

The only way to achieve desired conditions is dictatorship.

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Utilitarian view

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Utilitarian view

• Voters and candidates are embedded in a metric

space

• Each voter votes for her nearest candidate

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Utilitarian view

• Voters and candidates are embedded in a metric

space

• Each voter votes for her nearest candidate

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Utilitarian view

• Voters and candidates are embedded in a metric

space

• Each voter votes for her nearest candidate

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x

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Utilitarian view

• Voters and candidates are embedded in a metric

space

• Each voter votes for her nearest candidate

5

x

x votes for A

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Utilitarian view

• Voters and candidates are embedded in a metric

space

• Each voter votes for her nearest candidate

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Utilitarian view

• Voters and candidates are embedded in a metric

space

• Each voter votes for her nearest candidate

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Utilitarian view

• Voters and candidates are embedded in a metric

space

• Each voter votes for her nearest candidate

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• The distortion value can be a good factor to

compare different voting systems.

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Line Metric

• At first we consider a line as our metric

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Line Metric

• At first we consider a line as our metric

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p d

Cost( ) = p . d

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A Bad Example

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A Bad Example

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A bad example (Cont’d)

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A bad example (Cont’d)

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A bad example (Cont’d)

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A bad example (Cont’d)

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This is the worst case but it’s not realistic

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Abstention

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The Costs of Voting

• Time spent on gathering information
• Processing them
• Finding the best alternative
• Standing in a queue to vote.

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Sources of Ideological Distance-based Abstention

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• Indifference-based Abstention (IA)

Sources of Ideological Distance-based Abstention

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• Indifference-based Abstention (IA)
• Alienation-based Abstention (AA)

Sources of Ideological Distance-based Abstention

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Model

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• Voters , Candidates

embedded in a line

Model

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V = {v1, v2, …, vn}

C = {𝖬, 𝖲}

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• Voters , Candidates

embedded in a line

• = distance between voter and candidate X

Model

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V = {v1, v2, …, vn}

C = {𝖬, 𝖲}

di,X

vi

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• Voters , Candidates

embedded in a line

• = distance between voter and candidate X

Abstention model: Every voter votes for his

nearest candidate, say X, with probability
 
 
 
 
 where is the other candidate.

Model

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pi = ζβ(di,X, di, ¯

X) = (

|di,X − di, ¯

X|

di,X + di, ¯

X )

¯ X

V = {v1, v2, …, vn}

C = {𝖬, 𝖲}

di,X

vi vi

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• Voters , Candidates

embedded in a line

• = distance between voter and candidate X

Abstention model: Every voter votes for his

nearest candidate, say X, with probability
 
 
 
 
 where is the other candidate.

Model

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pi = ζβ(di,X, di, ¯

X) = (

|di,X − di, ¯

X|

di,X + di, ¯

X )

¯ X

V = {v1, v2, …, vn}

C = {𝖬, 𝖲}

di,X

vi vi

ζβ for diffrent values of β

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• A voter at point votes for with probability


• If there are voters at , the expected number of votes

gets is 


An Example

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( d′− d d′+ d)

β

p ( d′− d d′+ d)

β

x

𝖬 p

x

𝖬

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How to Measure the Distortion

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• Distortion of the Expected Winner

= Distortion of the candidate with the maximum expected number of votes

How to Measure the Distortion

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· D(𝔽)

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• Distortion of the Expected Winner

= Distortion of the candidate with the maximum expected number of votes

• Expected Distortion of the Winner

How to Measure the Distortion

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·· D(𝔽) = 𝖰𝖬 . D(𝖬) + 𝖰𝖲 . D(𝖲)

· D(𝔽)

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Distortion of the Expected Winner

Theorem:

There exists an election , such that the distortion

• f is maximum, and the voters in are located in

at most three difgerent locations.

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𝔽 𝔽 𝔽

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Proof

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Proof

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Valid displacement: We can displace the position of the voters on the

• line. We call a displacement valid, if the distortion

value does not decrease and the expected winner does not change.

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Proof (Cont’d)

Lemma 1

Moving a voter to point x is a valid displacement.


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v ∈ A

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Proof (Cont’d)

Lemma 1

Moving a voter to point x is a valid displacement.


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v ∈ A

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Proof (Cont’d)

Lemma 2

Consider voters respectively at xv and xv′′. Moving v to xv+ε and v′′ to xv′-ε is a valid displacement.
 


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v, v′ ∈ D

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Proof (Cont’d)

Lemma 2

Consider voters respectively at xv and xv′′. Moving v to xv+ε and v′′ to xv′-ε is a valid displacement.
 


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v, v′ ∈ D

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Proof (Cont’d)

Lemma 3

Consider voter v ∈ B and voter v′ ∈ C respectively at xv and xv′ . Moving v to xv + ε and v′ to xv′ − ε or moving v to xv - ε and v′ to xv′ + ε is a valid displacement.
 


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Proof (Cont’d)

Lemma 3

Consider voter v ∈ B and voter v′ ∈ C respectively at xv and xv′ . Moving v to xv + ε and v′ to xv′ − ε or moving v to xv - ε and v′ to xv′ + ε is a valid displacement.
 


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Proof (Cont’d)

Lemma 4

Consider voters v, v′ ∈ B respectively at xv and xv′′. Moving v to xv+ε and v′′ to xv′-ε is a valid displacement. 
 


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Proof (Cont’d)

Lemma 4

Consider voters v, v′ ∈ B respectively at xv and xv′′. Moving v to xv+ε and v′′ to xv′-ε is a valid displacement. 
 


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Proof (Cont’d)

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Lemma 1 Lemma 2 Lemma 3 Lemma 4

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Distortion of the Expected Winner

Theorem:

There exists an election E, such that the distortion of E is maximum, and the voters in E are located in at most three difgerent locations.

• One can Solve a Convex Program to fjnd the worst case

For this value is upper-bounded by

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(1 + 2)2 1 + 2 2 ≃ 1.522

β = 1

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Other Values of β

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Expected Distortion of the Winner

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Expected Distortion of the Winner

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·· D(𝔽) = 𝖰𝖬 . D(𝖬) + 𝖰𝖲 . D(𝖲)

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Expected Distortion of the Winner

Theorem:

There exists an election , such that is maximum, and the voters are located in at most four difgerent locations.

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·· D(𝔽) = 𝖰𝖬 . D(𝖬) + 𝖰𝖲 . D(𝖲)

𝔽

·· D(𝔽)

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Expected Distortion of the Winner

Theorem:

There exists an election , such that is maximum, and the voters are located in at most four difgerent locations.

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·· D(𝔽) = 𝖰𝖬 . D(𝖬) + 𝖰𝖲 . D(𝖲)

𝔽

·· D(𝔽)

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Expected Distortion of the Winner

Theorem:

There exists an election , such that is maximum, and the voters are located in at most four difgerent locations.

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·· D(𝔽) = 𝖰𝖬 . D(𝖬) + 𝖰𝖲 . D(𝖲)

𝔽

·· D(𝔽)

ε 1 + 2ε × 1 + ε ε + 1 + ε 1 + 2ε = 2 + 2ε 1 + 2ε ≃ 2 (ε → 0)

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Expected Distortion of the Winner

Theorem:


For any and , the expected distortion value of every election whose candidates receive at least 
 expected number of votes, is at most

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α > 0

β ∈ (0,1]

9(1 + 1 + α)(1 + α) 8(2 + α − 2 1 + α)α

(1 + 2α) · D(𝔽)*

β

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Expected Distortion of the Winner

Theorem:


For any and , the expected distortion value of every election whose candidates receive at least 
 expected number of votes, is at most

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α > 0

β ∈ (0,1]

9(1 + 1 + α)(1 + α) 8(2 + α − 2 1 + α)α

(1 + 2α) · D(𝔽)*

β

α = 1 6 , β = 1 ⇒

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Expected Distortion of the Winner

Theorem:


For any and , the expected distortion value of every election whose candidates receive at least 
 expected number of votes, is at most

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α > 0

β ∈ (0,1]

9(1 + 1 + α)(1 + α) 8(2 + α − 2 1 + α)α

(1 + 2α) · D(𝔽)*

β

α = 1 6 , β = 1 ⇒ Every election whose candidates receive

at least 2552 the expected distortion value is upper-bounded by 2.

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• .
• . (triangle inequality)

General Metric Space

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di,𝖬, di,𝖲 ≥ 0

di,𝖬 + di,𝖲 ≥ d𝖬,𝖲

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• .
• . (triangle inequality)

General Metric Space

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di,𝖬, di,𝖲 ≥ 0

di,𝖬 + di,𝖲 ≥ d𝖬,𝖲

Theorem:


For any election in general metric space, we can construct an election instance in line metric with greater distortion.

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Future Works

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• Exact upper-bound on the expected distortion

value for elections in any size

• Improve the provided bound
• Consider the stochastic model where both voters

and candidates come from a distribution