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

On the Distortion Value of the Elections with Abstention Mohammad Ghodsi, Mohamad Lati fj an, Masoud Seddighin 
 (AAAI 2019) � 1

Voting • Method to make decisions • Input: Ordinal preferences • Output: A single winner or 
 A rank list � 2 / 28

Voting - Formal definition • Voters, V = { v 1 , v 2 , …, v n } • Candidates, C = { c 1 , c 2 , …, c m } • Voter has a preference over the candidates v i ≺ i � 3 / 28

Voting - Good voting • IF everyone prefers X over Y, Y should be ranked over X • X and Y ranking is independent of other candidates � 4 / 28

Voting - Good voting • IF everyone prefers X over Y, Y should be ranked over X • X and Y ranking is independent of other candidates Arrow’s Impossibility Theorem: The only way to achieve desired conditions is dictatorship. � 4 / 28

Utilitarian view � 5 / 28

Utilitarian view • Voters and candidates are embedded in a metric space • Each voter votes for her nearest candidate � 5 / 28

Utilitarian view • Voters and candidates are embedded in a metric space • Each voter votes for her nearest candidate � 5 / 28

Utilitarian view • Voters and candidates are embedded in a metric space • Each voter votes for her nearest candidate x � 5 / 28

Utilitarian view • Voters and candidates are embedded in a metric space • Each voter votes for her nearest candidate x x votes for A � 5 / 28

Utilitarian view • Voters and candidates are embedded in a metric space • Each voter votes for her nearest candidate � 5 / 28

Utilitarian view • Voters and candidates are embedded in a metric space • Each voter votes for her nearest candidate � 5 / 28

Utilitarian view • Voters and candidates are embedded in a metric space • Each voter votes for her nearest candidate • The distortion value can be a good factor to compare different voting systems. � 5 / 28

Line Metric • At first we consider a line as our metric � 6 / 28

Line Metric • At first we consider a line as our metric d p Cost ( ) = p . d � 6 / 28

A Bad Example � 7 / 28

A Bad Example � 7 / 28

A bad example (Cont ’ d) � 8 / 28

A bad example (Cont ’ d) � 8 / 28

A bad example (Cont ’ d) � 9 / 28

A bad example (Cont ’ d) This is the worst case but it’s not realistic � 9 / 28

Abstention � 10 / 28

The Costs of Voting • Time spent on gathering information • Processing them • Finding the best alternative • Standing in a queue to vote. � 11 / 28

Sources of Ideological Distance-based Abstention � 12 / 28

Sources of Ideological Distance-based Abstention • Indifference-based Abstention (IA) � 12 / 28

Sources of Ideological Distance-based Abstention • Indifference-based Abstention (IA) • Alienation-based Abstention (AA) � 12 / 28

Model � 13 / 28

Model • Voters , Candidates V = { v 1 , v 2 , …, v n } C = { 𝖬 , 𝖲 } embedded in a line � 13 / 28

Model • Voters , Candidates V = { v 1 , v 2 , …, v n } C = { 𝖬 , 𝖲 } embedded in a line • = distance between voter and candidate X d i , X v i � 13 / 28


 
 
 
 Model • Voters , Candidates V = { v 1 , v 2 , …, v n } C = { 𝖬 , 𝖲 } embedded in a line • = distance between voter and candidate X d i , X v i Abstention model: Every voter votes for his v i nearest candidate, say X, with probability 
 X ) = ( X ) | d i , X − d i , ¯ X | p i = ζ β ( d i , X , d i , ¯ d i , X + d i , ¯ ¯ where is the other candidate. X � 13 / 28


 
 
 
 Model • Voters , Candidates V = { v 1 , v 2 , …, v n } C = { 𝖬 , 𝖲 } embedded in a line • = distance between voter and candidate X d i , X v i Abstention model: Every voter votes for his v i nearest candidate, say X, with probability 
 X ) = ( X ) | d i , X − d i , ¯ X | ζ β for di ff rent values of β p i = ζ β ( d i , X , d i , ¯ d i , X + d i , ¯ ¯ where is the other candidate. X � 13 / 28


 
 An Example • A voter at point votes for with probability 
 x 𝖬 β ( d ′ � + d ) d ′ � − d • If there are voters at , the expected number of votes x p 𝖬 gets is 
 p ( β d ′ � + d ) d ′ � − d � 14 / 28

How to Measure the Distortion � 15 / 28

How to Measure the Distortion • Distortion of the Expected Winner · = Distortion of the candidate with the D ( 𝔽 ) maximum expected number of votes � 15 / 28

How to Measure the Distortion • Distortion of the Expected Winner · = Distortion of the candidate with the D ( 𝔽 ) maximum expected number of votes • Expected Distortion of the Winner ·· D ( 𝔽 ) = 𝖰 𝖬 . D ( 𝖬 ) + 𝖰 𝖲 . D ( 𝖲 ) � 15 / 28

Distortion of the Expected Winner Theorem: There exists an election , such that the distortion 𝔽 of is maximum, and the voters in are located in 𝔽 𝔽 at most three di fg erent locations . � 16 / 28

Proof � 17 / 28

Proof 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. � 17 / 28

Proof (Cont ’ d) Lemma 1 v ∈ A Moving a voter to point x is a valid displacement. 
 � 18 / 28

Proof (Cont ’ d) Lemma 1 v ∈ A Moving a voter to point x is a valid displacement. 
 � 18 / 28


 Proof (Cont ’ d) Lemma 2 Consider voters respectively at x v and x v ′ ′ . v , v ′ � ∈ D Moving v to x v + ε and v ′ ′ to x v ′ - ε is a valid displacement. 
 � 19 / 28


 Proof (Cont ’ d) Lemma 2 Consider voters respectively at x v and x v ′ ′ . v , v ′ � ∈ D Moving v to x v + ε and v ′ ′ to x v ′ - ε is a valid displacement. 
 � 19 / 28


 Proof (Cont ’ d) Lemma 3 Consider voter v ∈ B and voter v ′ ∈ C respectively at x v and x v ′ . Moving v to x v + ε and v ′ to x v ′ − ε or moving v to x v - ε and v ′ to x v ′ + ε is a valid displacement. 
 � 20 / 28


 Proof (Cont ’ d) Lemma 3 Consider voter v ∈ B and voter v ′ ∈ C respectively at x v and x v ′ . Moving v to x v + ε and v ′ to x v ′ − ε or moving v to x v - ε and v ′ to x v ′ + ε is a valid displacement. 
 � 20 / 28


 Proof (Cont ’ d) Lemma 4 Consider voters v, v ′ ∈ B respectively at x v and x v ′ ′ . Moving v to x v + ε and v ′ ′ to x v ′ - ε is a valid displacement. 
 � 21 / 28


 Proof (Cont ’ d) Lemma 4 Consider voters v, v ′ ∈ B respectively at x v and x v ′ ′ . Moving v to x v + ε and v ′ ′ to x v ′ - ε is a valid displacement. 
 � 21 / 28

Proof (Cont ’ d) Lemma 1 Lemma 2 Lemma 3 Lemma 4 � 22 / 28

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 di fg erent locations . • One can Solve a Convex Program to fj nd the worst case 2) 2 (1 + For this value is upper-bounded by β = 1 ≃ 1.522 1 + 2 2 � 23 / 28

Other Values of β � 24 / 28

Expected Distortion of the Winner � 25 / 28

Expected Distortion of the Winner ·· D ( 𝔽 ) = 𝖰 𝖬 . D ( 𝖬 ) + 𝖰 𝖲 . D ( 𝖲 ) � 25 / 28

Expected Distortion of the Winner ·· D ( 𝔽 ) = 𝖰 𝖬 . D ( 𝖬 ) + 𝖰 𝖲 . D ( 𝖲 ) Theorem: ·· 𝔽 D ( 𝔽 ) There exists an election , such that is maximum, and the voters are located in at most four di fg erent locations . � 25 / 28

Expected Distortion of the Winner ·· D ( 𝔽 ) = 𝖰 𝖬 . D ( 𝖬 ) + 𝖰 𝖲 . D ( 𝖲 ) Theorem: ·· 𝔽 D ( 𝔽 ) There exists an election , such that is maximum, and the voters are located in at most four di fg erent locations . � 25 / 28

Expected Distortion of the Winner ·· D ( 𝔽 ) = 𝖰 𝖬 . D ( 𝖬 ) + 𝖰 𝖲 . D ( 𝖲 ) Theorem: ·· 𝔽 D ( 𝔽 ) There exists an election , such that is maximum, and the voters are located in at most four di fg erent locations . 1 + 2 ε × 1 + ε ε + 1 + ε 1 + 2 ε = 2 + 2 ε 1 + 2 ε ≃ 2 ( ε → 0) ε � 25 / 28


 
 Expected Distortion of the Winner Theorem: 
 For any and , the expected distortion β ∈ (0,1] α > 0 value of every election whose candidates receive at least 9(1 + 1 + α )(1 + α ) 8(2 + α − 2 1 + α ) α (1 + 2 α ) · D ( 𝔽 )* expected number of votes, is at most β � 26 / 28


 
 Expected Distortion of the Winner Theorem: 
 For any and , the expected distortion β ∈ (0,1] α > 0 value of every election whose candidates receive at least 9(1 + 1 + α )(1 + α ) 8(2 + α − 2 1 + α ) α (1 + 2 α ) · D ( 𝔽 )* expected number of votes, is at most β α = 1 6 , β = 1 ⇒ � 26 / 28