Resolute vs. irresolute rules The usual way of defining voting rules: ◮ we first define an irresolute rule F ◮ a resolute rule is implicitly defined from F by using a tie-breaking priority ◮ usual assumption: break neutrality ◮ F + tie-breaking priority > over X �→ F > voting rule ◮ F > ( P ) = max( >, F ( P )) Example: ◮ P = � a ≻ b , b ≻ a � ◮ Maj irresolute rule: Maj ( P ) = { a , b } ◮ Maj a > b and Maj b > a resolute rules ◮ Maj a > b ( P ) = a In the rest of the talk, we usually define irresolute rules, and resolute rules are defined implicitly by a tie-breaking priority.
Majority When there are only two candidates a and b , the only “reasonable” irresolute rule is majority : { a } if a strict majority of voters prefer a to b Maj ( V 1 , . . . , V n ) = { b } if a strict majority of voters prefer b to a { a , b } otherwise (tie) Exact characterization of majority in (May, 1952). ◮ Anonymity : voters should be treated symmetrically ◮ Neutrality : candidates should be treated symmetrically ◮ Positive Responsiveness : if a (sole or tied) winner receives increased support, then she should become the sole winner. May’s Theorem (1952) An irresolute rule for two candidates satisfies anonymity, neutrality and positive responsiveness if and only if it is Maj .
FAQ Q: Why don’t we allow voters to express indifferences? A: most voting rules can be easily and naturally generalised to profiles consisting of weak orders (instead of linear orders). We just don’t do that today because we don’t have enough time. Q: Why don’t we allow voters to express incomparabilities? A: this is less easy. Later in the talk we’ll talk about voting with profiles consisting of partial orders, even if our interpretation then won’t be that voters are indifferent but that we have an incomplete knowledge of their preferences. Q: Wouldn’t be simpler to ask voters to give numbers? A: sometimes, yes; in most cases, no. Numbers raise the issue of interpersonal comparison (is my 7 really better than your 6?), and it is sometimes difficult for voters to report numbers. Back on this later.
Voting ◮ P = ( V 1 , . . . , V n ) voting profile ◮ V i = linear order over X = vote expressed by voter i . Here is a 100-voter profile over X = { a , b , c , d , e } 33 votes: a ≻ b ≻ c ≻ d ≻ e 16 votes: b ≻ d ≻ c ≻ e ≻ a 3 votes: c ≻ d ≻ b ≻ a ≻ e 8 votes: c ≻ e ≻ b ≻ d ≻ a 18 votes: d ≻ e ≻ c ≻ b ≻ a e ≻ c ≻ b ≻ d ≻ a 22 votes: Who should be elected?
Voting with more than three candidates Generalizing simple majority: pairwise majority given any two alternatives x , y ∈ X , use simple majority to determine whether the group prefers x to y or vice versa. Does this work? Sometimes yes: associated majority graph 33 votes: a ≻ b ≻ c ≻ d ≻ e a d 16 votes: b ≻ d ≻ c ≻ e ≻ a 3 votes: c ≻ d ≻ b ≻ a ≻ e c 8 votes: c ≻ e ≻ b ≻ d ≻ a d ≻ e ≻ c ≻ b ≻ a 18 votes: 22 votes: e ≻ c ≻ b ≻ d ≻ a e b
Voting with more than three candidates Generalizing simple majority: pairwise majority given any two alternatives x , y ∈ X , use simple majority to determine whether the group prefers x to y or vice versa. Does this work? Sometimes yes: associated majority graph 33 votes: a ≻ b ≻ c ≻ d ≻ e a d b ≻ d ≻ c ≻ e ≻ a 16 votes: 3 votes: c ≻ d ≻ b ≻ a ≻ e c c ≻ e ≻ b ≻ d ≻ a 8 votes: 18 votes: d ≻ e ≻ c ≻ b ≻ a e 22 votes: e ≻ c ≻ b ≻ d ≻ a b Collective preference relation: c ≻ b ≻ d ≻ e ≻ a Winner: c
Voting with more than three candidates Generalizing simple majority: pairwise majority given any two alternatives x , y ∈ X , use simple majority to determine whether the group prefers x to y or vice versa. Does this work? Sometimes no: associated majority graph 33 votes: a ≻ b ≻ d ≻ c ≻ e a d b ≻ d ≻ c ≻ e ≻ a 16 votes: 3 votes: c ≻ d ≻ b ≻ a ≻ e c c ≻ e ≻ b ≻ d ≻ a 8 votes: 18 votes: d ≻ e ≻ c ≻ b ≻ a e 22 votes: e ≻ c ≻ b ≻ d ≻ a b
Voting with more than three candidates Generalizing simple majority: pairwise majority given any two alternatives x , y ∈ X , use simple majority to determine whether the group prefers x to y or vice versa. Does this work? Sometimes no: associated majority graph a ≻ b ≻ d ≻ c ≻ e 33 votes: a d 16 votes: b ≻ d ≻ c ≻ e ≻ a c ≻ d ≻ b ≻ a ≻ e 3 votes: c 8 votes: c ≻ e ≻ b ≻ d ≻ a 18 votes: d ≻ e ≻ c ≻ b ≻ a 22 votes: e ≻ c ≻ b ≻ d ≻ a e b Collective preference relation: { b ≻ d ≻ c ≻ b ≻ ... } ≻ e ≻ a ; Winner: ?
Condorcet winner ◮ N ( x , y ) = # { i , x ≻ i y } number of voters who prefer x to y . ◮ x Condorcet winner if for all y � = x , N ( x , y ) > n 2 a a d d c c e e b b c Condorcet winner no Condorcet winner ◮ sometimes there is no Condorcet winner ◮ when there is a Condorcet winner, it is unique ◮ a rule is Condorcet-consistent if it outputs the Condorcet winner whenever there is one.
Single-peakedness ◮ O : x 1 > x 2 > . . . > x n voter-independent axis on which alternatives are located. ◮ left-right axis (election of the next French president) M´ elenchon > Jadot > Hollande > Bayrou > Fillon > Le Pen ◮ numerical axis (number of breaks during the tutorial): 0 > 1 > 2 > 3 ◮ peak ( ≻ ) preferred alternative according to ≻ . ◮ ≻ is single-peaked with respect to O if for all x , y : ◮ if x < y < peak ( ≻ ) then y ≻ x ◮ if peak ( ≻ ) < x < y then x ≻ y ◮ �≻ 1 , . . . , ≻ n � single-peaked with respect to O if every ≻ i is. ◮ P = � 1 ≻ 2 ≻ 0 ≻ 3 , 2 ≻ 3 ≻ 0 ≻ 1 , 0 ≻ 1 ≻ 2 ≻ 3 � ◮ Q = � 1 ≻ 2 ≻ 0 ≻ 3 , 2 ≻ 3 ≻ 1 ≻ 0 , 0 ≻ 1 ≻ 2 ≻ 3 � ◮ Are P and Q single-peaked?
Single-peakedness ◮ Black’s theorem: if P is single-peaked then ◮ the pairwise majority relation associated with P is transitive ◮ if n is odd, then P has is a Condorcet winner, which is the median of { peak ( ≻ 1 ) , . . . , peak ( ≻ n ) } ◮ Example: ◮ Q = � 1 ≻ 2 ≻ 0 ≻ 3 , 2 ≻ 3 ≻ 1 ≻ 0 , 0 ≻ 1 ≻ 2 ≻ 3 � ◮ collective preference: 1 ≻ 2 ≻ 0 ≻ 3 ◮ Condorcet winner: 1 ◮ So far: everything is ok when ◮ we have only two alternatives ◮ (more generally) P is single-peaked ◮ What can we do when |X| ≥ 3 and P is not single-peaked?
Arrow’s theorem What can we do when |X| ≥ 3 and P is not single-peaked? We would like to have a way of aggregating preferences (and/or to select a set of cowinners) satisfying some desirable conditions. ◮ Condition 1: unrestricted domain (UR) F maps every collection of linear orders �≻ 1 , . . . , ≻ n � into a collective linear order ≻ c ◮ no domain restriction such as single-peakedness ◮ no ties, no incomparabilities, no randomization ◮ Condition 2: Pareto efficiency for any x , y ∈ X , if for every i we have x ≻ i y then x ≻ c y ◮ also called unanimity : if everyone prefers x to y , so does the group
Arrow’s theorem ◮ Condition 3: independence of irrelevant alternatives (IIA) ◮ Let P = �≻ 1 , . . . , ≻ n � , Q = �≻ ′ 1 , . . . , ≻ ′ n � , F ( P ) = ≻ c , F ( Q ) = ≻ ′ c . If for every i we have x ≻ i y if and only if x ≻ ′ i y then x ≻ c y if and only if x ≻ ′ c y (The collective preference between two alternatives x and y depends only on the individual preferences between x and y .) ◮ Condition 4: nondictatorship ◮ It is not the case that there exists a voter i such that for every profile P = �≻ 1 , . . . , ≻ n � we have F ( P ) = ≻ i . Arrow’s theorem (1951) If |X| ≥ 3 , there exists no aggregation function satisfying conditions 1, 2, 3 and 4.
Arrow’s theorem: reformulation for irresolute voting rules An irresolute rule F is ◮ Pareto-efficient if for all P = �≻ 1 , . . . , ≻ n � and x , y ∈ X : if for every i we have x ≻ i y then y / ∈ F ( P ) (no Pareto-dominated alternative should be a cowinner). ◮ independence of losing alternatives (ILA) if for all P = �≻ 1 , . . . , ≻ n � , P ′ = �≻ ′ 1 , . . . , ≻ ′ n � , and x , y ∈ X : (a) for all i , x ≻ i y ⇔ x ≻ ′ i y , (b) x ∈ F ( P ) and ∈ F ( P ) (c) y / imply (d) y / ∈ F ( P ′ ). (If x wins in P , y loses in P , and the relative order of x and y is the same in every vote of P and P ′ , then y must lose in P ′ .) ◮ nondictatorial if it is not the case that there exists a voter i such that for every profile P , F ( P ) = { top ( ≻ i ) } . Arrow’s theorem, voting version (Taylor, 2005) If |X| ≥ 3 , no irresolute voting rule satisfies Pareto-efficiency, ILA and nondictatorship.
Escaping Arrow’s theorem ◮ Relaxing nondictatorship is not considered an option ◮ Relaxing the unrestricted domain property ◮ (1) domain restriction such as single-peakedness ◮ (2) output a collective preference relation with cycles or incomparabilities ◮ (3) different input ( numerical or dichotomous preferences ) ◮ Relaxing Pareto-efficiency ◮ Exercise: define a voting rule satisfying all properties except Pareto. ◮ not really interesting. ◮ (4) Relaxing IIA ◮ lots of interesting voting rules satisfying all properties except IIA. ◮ Exercise: define a few such voting rules.
Escaping Arrow’s theorem ◮ Relaxing nondictatorship is not considered an option ◮ Relaxing the unrestricted domain property ◮ (1) domain restriction such as single-peakedness ◮ (2) output a collective preference relation with cycles or incomparabilities ◮ (3) different input ( such as numerical preferences ) ◮ Relaxing Pareto-efficiency ◮ Exercise: define a voting rule satisfying all properties except Pareto. ◮ not really interesting. ◮ (4) Relaxing IIA ◮ lots of interesting voting rules satisfying all properties except IIA. ◮ Exercise: define a few such voting rules.
Escaping Arrow’s theorem: dichotomous preferences Approval voting ◮ approval vote = a subset of (approved) candidates A ⊆ X ◮ approval profile = a collection of approval votes P = � A 1 , . . . , A n � ◮ Winner(s): candidate(s) approved most often. ◮ X = { a , b , c , d , e } ◮ n = 5 ◮ P = �{ a , c } , { b , c , d } , ∅ , { a , b , c , d , e } , { d }� ◮ c and d approved by 3 voters: a and b by 2 voters; e by 1. ◮ cowinners: { c , d } ◮ Arrow’s theorem does not apply.
Escaping Arrow’s theorem: numerical preferences ◮ profile : P = � u 1 , . . . , u n � , u i : X → L utility function , L linearly ordered scale. ◮ tripadvisor scale: L = { 1 , . . . , 5 } ◮ EasyChair scale: L = { strong reject, weak reject, borderline, weak accept, strong accept } ◮ ⋆ aggregation function on L ◮ winner(s): maximize ⋆ ( u 1 ( x ) , . . . , u n ( x )) ◮ Arrow’s theorem does not apply.
Escaping Arrow’s theorem: numerical preferences Range voting ◮ voters evaluate alternatives using numbers (the higher, the more preferred) ◮ the global score of an alternative is the sum of all scores ◮ the winner is the alternative with the highest global score ◮ candidates: X = { a , b , c } ◮ n = 3 voters ◮ voter 1: a �→ 90; b �→ 60; c �→ 30 ◮ voter 2: a �→ 60; b �→ 70; c �→ 40 ◮ voter 3: a �→ 0; b �→ 30; c �→ 100 ◮ global scores: a �→ 150; b �→ 160; c �→ 170 ◮ winner: d .
Escaping Arrow’s theorem: numerical preferences Range voting ◮ Similar to range voting, except that the sum of the scores given by a voter must have a fixed value Majority judgment (Balinski and Laraki, 2010) ◮ Similar to range voting, except that the global score is not the sum of all scores but the median
Subsets, numbers, or rankings? ◮ dichotomous preferences: + simple – weak expressivity (cannot express intensities of preference) ◮ numerical preferences: + very expressive – interpersonal comparison of preference (does a 7 given by me mean the same thing as a 7 given by you?) – difficulty of elicitation ◮ ordinal preferences: + good expressivity trade-off – Arrow’s theorem. + can be escaped by relaxing ILA: define specific voting/aggregation rules and see how good they are.
Plan 1. Social choice and computational social choice 2. Preference aggregation, Arrow’s theorem, how to escape it 3. Voting rules: informational basis 4. Voting rules: properties 5. Voting rules: computation 6. Combinatorial domains 7. Strategic behaviour 8. Voting with incomplete preferences 9. Other issues
Informational basis of voting rules and SWFs ◮ the informational basis of F is the minimal necessary information I ( P ) from the profile P to determine F ( P ) ? : I ( P ) = ∅ for all P ◮
Informational basis of voting rules and SWFs ◮ the informational basis of F is the minimal necessary information I ( P ) from the profile P to determine F ( P ) ◮ constant rules: I ( P ) = ∅ for all P ? : I ( P ) = V j for some j . ◮
Informational basis of voting rules and SWFs ◮ the informational basis of F is the minimal necessary information I ( P ) from the profile P to determine F ( P ) ◮ constant rules: I ( P ) = ∅ for all P ◮ dictatorial rules, among others: I ( P ) = V j for some j .
Informational basis of voting rules and SWFs ◮ the informational basis of F is the minimal necessary information I ( P ) from the profile P to determine F ( P ) ◮ constant rules: I ( P ) = ∅ for all P ◮ dictatorial rules, among others: I ( P ) = V j for some j . ◮ rank-based rules: ◮ I ( P ): for each i and x , N ( x , P , i ) number of votes V i that rank x in position i ◮ positional scoring rules (such as Borda) and a few others ◮ rules based on the majority graph: ◮ I ( P ) = M P = directed graph containing x → y if and only if a majority of votes in P prefers x to y ◮ (Condorcet winner), Copeland, Slater, top cycle, etc. ◮ rules based on the weighted majority graph: ◮ I ( P ) = W P x , y �→ W P ( x , y ) = numbers of votes in P that prefer x to y ◮ maximin, Kemeny, Borda, etc. ◮ other rules.
Informational basis, 1: rank-based rules Positional scoring rules ◮ m candidates ◮ fixed list of m integers s 1 ≥ . . . ≥ s m , with s 1 > s m ◮ if voter i ranks candidate x in position j then score i ( x ) = s j ◮ winner(s): candidate(s) maximizing n � s ( x ) = score i ( x ) i =1 Three important examples: plurality s 1 = 1, s 2 = . . . = s m = 0. (Informational basis: N ( x , P , 1) for all x .) antiplurality (or veto) s 1 = s 2 = . . . = s m − 1 = 1, s m = 0 (more generally) k -approval s 1 = . . . = s k = 1 , s k +1 = . . . = s m = 0. 1-approval = plurality; ( m − 1)-approval = veto Borda s 1 = m − 1, s 2 = m − 2, . . . s m = 0
Informational basis, 1: rank-based rules Positional scoring rules 33 a ≻ b ≻ c ≻ d ≻ e b ≻ d ≻ c ≻ e ≻ a 16 3 c ≻ d ≻ b ≻ a ≻ e c ≻ e ≻ b ≻ d ≻ a 8 18 d ≻ e ≻ c ≻ b ≻ a 22 e ≻ c ≻ b ≻ d ≻ a ◮ plurality: a �→ 33, b �→ 16, c �→ 11, d �→ 18, e �→ 22 winner: a ◮ Borda: a �→ (33 × 4) + (3 × 1) = 135 b �→ 247; c �→ 244; d �→ 192; e �→ 182 winner: b ◮ antiplurality: a �→ 36, b �→ 100, c �→ 100, d �→ 100, e �→ 64 cowinners: b , c , d ◮ 3-approval: a �→ 33, b �→ 82, c �→ 100, d �→ 37, e �→ 48 winner: c
Informational basis, 1: rank-based rules Bucklin ◮ S k ( P , x ) = number of voters who rank x in the first k positions ◮ k ∗ = min { k , there exists a x such that S k ( P , x ) > n 2 } ◮ Bucklin winner(s) = k ∗ -approval winner(s) 33 a ≻ b ≻ c ≻ d ≻ e b ≻ d ≻ c ≻ e ≻ a 16 3 c ≻ d ≻ b ≻ a ≻ e c ≻ e ≻ b ≻ d ≻ a 8 18 d ≻ e ≻ c ≻ b ≻ a e ≻ c ≻ b ≻ d ≻ a 22 ◮ k ∗ = 3 ◮ Bucklin winner: c
Informational basis, 2: majority graph pairwise majority given any two alternatives x , y ∈ X , x is majority-preferred to y if a majority of votes in P prefers x to y : x ≻ maj y P M P = { x → y | x ≻ maj y } P ◮ x Condorcet winner if M P contains x → y for all y � = x . a a d d c c e e b b c Condorcet winner no Condorcet winner ◮ sometimes there is no Condorcet winner ◮ when there is a Condorcet winner, it is unique ◮ a rule is Condorcet-consistent if it outputs the Condorcet
Informational basis, 2: majority graph ◮ M P majority graph associated with P ◮ A voting rule F is based on the majority graph if F ( P ) = f ( M P ) for some function f . ◮ For the sake of simplicity, we assume an odd number of voters; in this case the majority graph is a complete asymmetric graph: a tournament. Copeland ◮ C ( x ) = number of candidates y such that M P contains x − → y . ◮ Copeland winner(s): maximize(s) C . C ( a ) = 0 a d C ( b ) = 3 C ( c ) = 3 c C ( d ) = 3 C ( e ) = 1 e b b ∼ c ∼ d ≻ e ≻ a
Informational basis, 2: majority graph ◮ M P majority graph associated with P ◮ A voting rule F is based on the majority graph if F ( P ) = f ( M P ) for some function f . ◮ For the sake of simplicity, we assume an odd number of voters; in this case the majority graph is a complete asymmetric graph: a tournament. Slater ◮ distance between two rankings = number of disagreeing pairs ◮ Slater ranking for P = ranking minimising distance to M P ◮ Slater winner: best candidate in some Slater ranking a d c c winner (plus two others) e b
Informational basis, 3: weighted majority graph ◮ P profile ◮ W P ( x , y ) = # { i , x ≻ i y } − # { i , y ≻ i x } number of voters who prefer x to y minus number of voters who prefer x to y (pairwise majority matrix / weighted majority graph) ◮ A voting rule F is based on the weighted majority graph if F ( P ) = g ( W P ) for some function g .
Informational basis, 3: weighted majority graph ◮ P profile ◮ W P ( x , y ) = # { i , x ≻ i y } − # { i , y ≻ i x } number of voters who prefer x to y minus number of voters who prefer x to y (pairwise majority matrix / weighted majority graph) ◮ A voting rule F is based on the weighted majority graph if F ( P ) = g ( W P ) for some function g . Maximin ◮ maximize S m ( x ) = min y � = x W P ( x , y ) W P a b c d e S m ( . ) a 0 − 34 − 34 − 34 − 30 − 34 − 2 − 2 b +34 0 +58 +4 c +34 +2 0 − 34 +20 − 34 − 58 − 58 d +34 +34 0 +40 e +30 − 4 − 20 − 40 0 − 40 Winner: b
Informational basis, 3: weighted majority graph Ranked Pairs 1. G := graph with X as vertices and no edge. 2. order the pairs ( x , y ) by non-increasing order of W P ( x , y ), using some tie-breaking priority when necessary 3. take the first pair ( x , y ) in the list 4. if adding x − → y to G does not produce any cycle then add it to G 5. remove ( x , y ) from the list 6. iterate until the graph is complete W P a b c d e a d a 0 − 34 − 34 − 34 − 30 − 2 b +34 0 +58 +4 c c +34 +2 0 − 34 +20 d +34 − 58 +34 0 +40 e b e +30 − 4 − 20 − 40 0
Informational basis, 3: weighted majority graph Ranked Pairs 1. G := graph with X as vertices and no edge. 2. order the pairs ( x , y ) by non-increasing order of N P ( x , y ), using some tie-breaking priority when necessary 3. take the first pair ( x , y ) in the list 4. if adding x − → y to G does not produce any cycle then add it to G 5. remove ( x , y ) from the list 6. iterate until the graph is complete W P a b c d e a d a 0 − 34 − 34 − 34 − 30 − 2 b +34 0 +58 +4 c c +34 +2 0 − 34 +20 d +34 − 58 +34 0 +40 e b e +30 − 4 − 20 − 40 0
Informational basis, 3: weighted majority graph Ranked Pairs 1. G := graph with X as vertices and no edge. 2. order the pairs ( x , y ) by non-increasing order of W P ( x , y ), using some tie-breaking priority when necessary 3. take the first pair ( x , y ) in the list 4. if adding x − → y to G does not produce any cycle then add it to G 5. remove ( x , y ) from the list 6. iterate until the graph is complete W P a b c d e a d a 0 − 34 − 34 − 34 − 30 − 2 b +34 0 +58 +4 c c +34 +2 0 − 34 +20 d +34 − 58 +34 0 +40 e b e +30 − 4 − 20 − 40 0
Informational basis, 3: weighted majority graph Ranked Pairs 1. G := graph with X as vertices and no edge. 2. order the pairs ( x , y ) by non-increasing order of W P ( x , y ), using some tie-breaking priority when necessary 3. take the first pair ( x , y ) in the list 4. if adding x − → y to G does not produce any cycle then add it to G 5. remove ( x , y ) from the list 6. iterate until the graph is complete W P a b c d e a d a 0 − 34 − 34 − 34 − 30 − 2 b +34 0 +58 +4 c c +34 +2 0 − 34 +20 d +34 − 58 +34 0 +40 e b e +30 − 4 − 20 − 40 0
Informational basis, 3: weighted majority graph Ranked Pairs 1. G := graph with X as vertices and no edge. 2. order the pairs ( x , y ) by non-increasing order of W P ( x , y ), using some tie-breaking priority when necessary 3. take the first pair ( x , y ) in the list 4. if adding x − → y to G does not produce any cycle then add it to G 5. remove ( x , y ) from the list 6. iterate until the graph is complete W P a b c d e a d a 0 − 34 − 34 − 34 − 30 − 2 b +34 0 +58 +4 c c +34 +2 0 − 34 +20 d +34 − 58 +34 0 +40 e b e +30 − 4 − 20 − 40 0
Informational basis, 3: weighted majority graph Ranked Pairs 1. G := graph with X as vertices and no edge. 2. order the pairs ( x , y ) by non-increasing order of W P ( x , y ), using some tie-breaking priority when necessary 3. take the first pair ( x , y ) in the list 4. if adding x − → y to G does not produce any cycle then add it to G 5. remove ( x , y ) from the list 6. iterate until the graph is complete W P a b c d e a d a 0 − 34 − 34 − 34 − 30 − 2 b +34 0 +58 +4 c c +34 +2 0 − 34 +20 d +34 − 58 +34 0 +40 e b e +30 − 4 − 20 − 40 0
Informational basis, 3: weighted majority graph Ranked Pairs 1. G := graph with X as vertices and no edge. 2. order the pairs ( x , y ) by non-increasing order of W P ( x , y ), using some tie-breaking priority when necessary 3. take the first pair ( x , y ) in the list 4. if adding x − → y to G does not produce any cycle then add it to G 5. remove ( x , y ) from the list 6. iterate until the graph is complete W P a b c d e a d a 0 − 34 − 34 − 34 − 30 − 2 b +34 0 +58 +4 c c +34 +2 0 − 34 +20 d +34 − 58 +34 0 +40 e b e +30 − 4 − 20 − 40 0
Informational basis, 3: weighted majority graph Ranked Pairs 1. G := graph with X as vertices and no edge. 2. order the pairs ( x , y ) by non-increasing order of W P ( x , y ), using some tie-breaking priority when necessary 3. take the first pair ( x , y ) in the list 4. if adding x − → y to G does not produce any cycle then add it to G 5. remove ( x , y ) from the list 6. iterate until the graph is complete W P a b c d e a d a 0 − 34 − 34 − 34 − 30 − 2 b +34 0 +58 +4 c c +34 +2 0 − 34 +20 d +34 − 58 +34 0 +40 e b e +30 − 4 − 20 − 40 0
Informational basis, 3: weighted majority graph Ranked Pairs 1. G := graph with X as vertices and no edge. 2. order the pairs ( x , y ) by non-increasing order of W P ( x , y ), using some tie-breaking priority when necessary 3. take the first pair ( x , y ) in the list 4. if adding x − → y to G does not produce any cycle then add it to G 5. remove ( x , y ) from the list 6. iterate until the graph is complete W P a b c d e a d a 0 − 34 − 34 − 34 − 30 − 2 b +34 0 +58 +4 c c +34 +2 0 − 34 +20 d +34 − 58 +34 0 +40 e b e +30 − 4 − 20 − 40 0
Informational basis, 3: weighted majority graph Ranked Pairs 1. G := graph with X as vertices and no edge. 2. order the pairs ( x , y ) by non-increasing order of W P ( x , y ), using some tie-breaking priority when necessary 3. take the first pair ( x , y ) in the list 4. if adding x − → y to G does not produce any cycle then add it to G 5. remove ( x , y ) from the list 6. iterate until the graph is complete W P a b c d e a d a 0 − 34 − 34 − 34 − 30 − 2 b +34 0 +58 +4 c c +34 +2 0 − 34 +20 d +34 − 58 +34 0 +40 e b e +30 − 4 − 20 − 40 0
Informational basis, 3: weighted majority graph Ranked Pairs 1. G := graph with X as vertices and no edge. 2. order the pairs ( x , y ) by non-increasing order of W P ( x , y ), using some tie-breaking priority when necessary 3. take the first pair ( x , y ) in the list 4. if adding x − → y to G does not produce any cycle then add it to G 5. remove ( x , y ) from the list 6. iterate until the graph is complete W P a b c d e a d a 0 − 34 − 34 − 34 − 30 − 2 b +34 0 +58 +4 c c +34 +2 0 − 34 +20 d +34 − 58 +34 0 +40 e b e +30 − 4 − 20 − 40 0
Informational basis, 3: weighted majority graph Ranked Pairs 1. G := graph with X as vertices and no edge. 2. order the pairs ( x , y ) by non-increasing order of W P ( x , y ), using some tie-breaking priority when necessary 3. take the first pair ( x , y ) in the list 4. if adding x − → y to G does not produce any cycle then add it to G 5. remove ( x , y ) from the list 6. iterate until the graph is complete W P a b c d e a d a 0 − 34 − 34 − 34 − 30 − 2 b +34 0 +58 +4 c c +34 +2 0 − 34 +20 d +34 − 58 +34 0 +40 e b e +30 − 4 − 20 − 40 0
Informational basis, 3: weighted majority graph Ranked Pairs 1. G := graph with X as vertices and no edge. 2. order the pairs ( x , y ) by non-increasing order of W P ( x , y ), using some tie-breaking priority when necessary 3. take the first pair ( x , y ) in the list 4. if adding x − → y to G does not produce any cycle then add it to G 5. remove ( x , y ) from the list 6. iterate until the graph is complete a W P a b c d e d − 34 − 34 − 34 − 30 a 0 b +34 0 − 2 +58 +4 c − 34 c +34 +2 0 +20 d +34 − 58 +34 0 +40 e b e +30 − 4 − 20 − 40 0 winner: b
Informational basis, 3: weighted majority graph Kemeny ◮ for two rankings R , R ′ ; d K ( R , R ′ ) = number of ( x , y ) on which R and R ′ disagree ◮ d K ( R , � V 1 , . . . , V n � ) = � i =1 ,..., n d K ( R , V i ) ◮ Kemeny consensus = ranking R ∗ minimizing d K ( R ∗ , � V 1 , . . . , V n � ) ◮ Kemeny winner = candidate ranked first in a Kemeny consensus
Informational basis, 3: weighted majority graph Kemeny Equivalent definition: ◮ for any ranking R define K ( R ) = � ( x , y ) ∈ R W P ( x , y ) ◮ x ≻ y in R corresponds to W P ( x , y ) “agreements minus disagreements” ◮ R ∗ is a Kemeny consensus iff K ( R ∗ ) is maximum. W P a b c d e a 0 − 34 − 34 − 34 − 30 b +34 0 − 2 +58 +4 c +34 +2 0 − 34 +20 − 58 d +34 +34 0 +40 e +30 − 4 − 20 − 40 0 K ( b ≻ d ≻ c ≻ e ≻ a ) = 286 (maximum value) Kemeny winner: b
Informational basis, 3: weighted majority graph Borda, again ◮ maximize S b ( x ) = � y � = x W P ( x , y ) W P a b c d e S b ( . ) − 34 − 34 − 34 − 30 − 132 a 0 b +34 0 − 2 +58 +4 +94 c +34 +2 0 − 34 +20 +22 d +34 − 58 +34 0 +40 +50 e +30 − 4 − 20 − 40 0 − 34
Informational basis, 4: Other Plurality with runoff ◮ let x , y the two candidates with the highest plurality score (use tie-breaking rule if necessary) ◮ winner: majority winner between x and y ◮ informational basis: M V + { N ( x , V , 1) | x ∈ A }
Informational basis, 4: Other Single transferable vote (STV) Repeat x := candidate ranked first by the fewest voters; eliminate x from all ballots { votes for x transferred to the next best remaining candidate } Until there remains a single candidate; ◮ when there are only 3 candidates, STV coincides with plurality with runoff.
Informational basis, 4: Other Single transferable vote (STV) 33 a ≻ b ≻ c ≻ d ≻ e 33 a ≻ b ≻ d ≻ e b ≻ d ≻ c ≻ e ≻ a b ≻ d ≻ e ≻ a 16 16 3 c ≻ d ≻ b ≻ a ≻ e 3 d ≻ b ≻ a ≻ e c ≻ e ≻ b ≻ d ≻ a e ≻ b ≻ d ≻ a 8 8 18 d ≻ e ≻ c ≻ b ≻ a 18 d ≻ e ≻ b ≻ a 22 e ≻ c ≻ b ≻ d ≻ a 22 e ≻ b ≻ d ≻ a 33 a ≻ d ≻ e 33 a ≻ d 16 d ≻ e ≻ a 16 d ≻ a 3 d ≻ a ≻ e 3 d ≻ a 8 e ≻ d ≻ a 8 d ≻ a 18 d ≻ e ≻ a 18 d ≻ a 22 e ≻ d ≻ a 22 d ≻ a d ≻ a ≻ e ≻ b ≻ c
Informational basis, 4: Other Dodgson ◮ elementary change : swap of two adjacent candidates in a voter’s ranking ◮ Dodgson score of a candidate x : minimal number of elementary changes needed to make x the Condorcet winner ◮ Dodgson winner(s): the candidate(s) with the smallest Dodgson score ◮ variant for social welfare functions Dodgson score of a ranking R : minimal number of elementary changes needed to obtain R as majority graph ◮ Informational basis; quite complicated.
Plan 1. Social choice and computational social choice 2. Preference aggregation, Arrow’s theorem, how to escape it 3. Voting rules: informational basis 4. Voting rules: properties 5. Voting rules: computation 6. Combinatorial domains 7. Strategic behaviour 8. Voting with incomplete preferences 9. Other issues
Voting rules: some important properties ◮ Anonymity: all voters treated equally Formally: the winner does not change if we apply a permutation of the voters.
Voting rules: some important properties ◮ Anonymity ◮ Neutrality: all candidates treated equally . Formally: if the winner is x and we apply a permutation σ of the candidates’ names then the resulting winner will be σ ( x ).
Voting rules: some important properties ◮ Anonymity ◮ Neutrality ◮ Condorcet-consistency: the Condorcet winner is elected whenever there is one.
Properties of voting rules: Condorcet-consistency ◮ Copeland, Slater, maximin, ranked pairs, Kemeny, Dodgson: all Condorcet-consistent ◮ No positional scoring rule is Condorcet-consistent (Fishburn, 73) 6 a ≻ b ≻ c c ≻ a ≻ b 3 4 b ≻ a ≻ c b ≻ c ≻ a 4 Without loss of generality, let s 3 = 0. ◮ S ( a ) = 6 s 1 + 7 s 2 ◮ S ( b ) = 8 s 1 + 6 s 2 ◮ S ( b ) − S ( a ) = 2 s 1 − s 2 = s 1 + ( s 1 − s 2 ) > 0 ◮ S ( b ) > S ( a ) whatever the value of s 1 and s 2 ◮ but a is a Condorcet winner! ◮ Plurality with runoff and STV are not Condorcet-consistent (prove it!)
Voting rules: some important properties ◮ Anonymity ◮ Neutrality ◮ Condorcet-consistency ◮ Pareto-efficiency: if every voter prefers x to y then y cannot be a winner.
Properties of voting rules: Pareto-efficiency ◮ a positional scoring rule is Pareto-efficient if its scoring vector does not have two identical non-null values ◮ plurality, Borda: yes ◮ k -approval for k > 2, and in particular veto: no ◮ Copeland, Slater, maximin, ranked pairs, Kemeny, Dodgson: yes
Voting rules: some important properties ◮ Anonymity ◮ Neutrality ◮ Condorcet-consistency ◮ Pareto-efficiency ◮ Monotonicity: if the winner for profile P is x and P ′ is obtained from P by raising x in a vote without changing anything else, then the winner for P ′ is still x .
Properties of voting rules: monotonicity ◮ positional scoring rules: yes ◮ Copeland, Slater, maximin, Kemeny, ranked pairs, Dodgson: yes ◮ plurality with runoff and STV: no! a ≻ b ≻ c 8 7 b ≻ c ≻ a c ≻ a ≻ b 6 ◮ finalists a , b ; ◮ winner a ◮ two voters change their vote from b ≻ c ≻ a to a ≻ b ≻ c a ≻ b ≻ c 10 5 b ≻ c ≻ a 6 c ≻ a ≻ b ◮ finalists a , c ◮ winner c
Voting rules: some important properties ◮ Anonymity ◮ Neutrality ◮ Condorcet-consistency ◮ Pareto-efficiency ◮ Monotonicity ◮ Participation if the winner for profile P is x and P ′ = P ∪ {≻ n +1 } , then the winner for P ′ is either x , or a candidate y such that y ≻ n +1 x .
Properties of voting rules: participation ◮ positional scoring rules: yes ◮ for m ≥ 4, no Condorcet-consistent rule satisfies participation (Moulin, 86) Proof for maximin: 3 a ≻ d ≻ c ≻ b 3 a ≻ d ≻ c ≻ b a ≻ d ≻ b ≻ c 3 a ≻ d ≻ b ≻ c 3 5 d ≻ c ≻ b ≻ a 5 d ≻ c ≻ b ≻ a 4 b ≻ c ≻ a ≻ d 4 b ≻ c ≻ a ≻ d 4 c ≻ a ≻ b ≻ d S m ( a ) = W P ( a , c ) = − 3; S m ( a ) = − 7; S m ( b ) = − 5; b , d : − 5; c : − 7; etc. maximin winner: a maximin winner: b The four new voters had rather stayed home! Also called no-show paradox ◮ STV, plurality with runoff: no
Voting rules: some important properties ◮ Anonymity ◮ Neutrality ◮ Condorcet-consistency ◮ Pareto-efficiency ◮ Monotonicity ◮ Participation ◮ Reinforcement (or consistency): if P and Q are two profiles (on disjoint electorates) and x is the winner for P and the winner for Q , then it is also the winner for P ∪ Q .
Properties of voting rules: reinforcement ◮ positional scoring rules: yes ◮ if m ≥ 3 then no Condorcet-consistent rule satisfies reinforcement (Young, 75) ◮ STV, plurality with runoff: no
Properties of voting rules: reinforcement Reinforcement is in fact a key property of positional scoring rules. Continuity if electorate N 1 elects x and electorate N 2 does not, adding sufficiently many replicates of N 1 to N 2 leads to elect x Axiomatic characterisation of positional scoring rules (Young, 75): an irresolute voting rule is a positional scoring rule if and only if it satisfies anonymity, neutrality, reinforcement, and continuity.
Voting rules: some important properties ◮ Anonymity ◮ Neutrality ◮ Condorcet-consistency ◮ Pareto-efficiency ◮ Monotonicity ◮ Participation ◮ Reinforcement (or consistency) ◮ Clone-proofness: assume a candidate x is “cloned” into several candidates (ranked contiguously in every vote). If the winner was not x then the new winner remains the same; if it was x then it is now one of the clones of x .
Properties of voting rules: clone-proofness ◮ most rules (including all positional scoring rules) are not clone-proof ◮ but ranked pairs is clone-proof
Plan 1. Social choice and computational social choice 2. Preference aggregation, Arrow’s theorem, how to escape it 3. Voting rules: informational basis 4. Voting rules: properties 5. Voting rules: computation 6. Combinatorial domains 7. Strategic behaviour 8. Voting with incomplete preferences 9. Fair division 10. Other issues
Computing voting rules: easy rules What is the complexity of winner determination for the voting rules we have seen so far? ◮ scoring rules, plurality with runoff, approval: O ( nm ) ◮ Copeland, maximin, STV( ∗ ), ranked pairs ∗ , O ( nm 2 ). But some voting rules are NP -hard...
Computing: parallel universes (*) How do we handle ties in STV and ranked pairs? STV T ties are broken immediately using a tie-breaking priority T : polynomial STV PU exploring all possibilities and possible use tie-breaking at the very last moment: NP-complete ◮ break ties immediately: c 4 a ≻ d ≻ b ≻ c eliminated, then b , winner: d 3 b ≻ c ≻ d ≻ a ◮ parallel universes: 2 c ≻ d ≻ a ≻ b ◮ branch 1 (above): winner: d d ≻ b ≻ c ≻ a 2 ◮ branch 2: d eliminated, then c , winner: a Tie-breaking : ◮ cowinners { a , d } , winner: a . a > b > d > c ◮ Conitzer, Rognlie and Xia (09): winner determination for STV PU is NP-complete. ◮ Brill and Fischer (12): winner determination for parallel universe ranked pairs is NP-complete.
Computing voting rules: Kemeny Recall: Kemeny is based on the weighted majority graph. 4 a ≻ b ≻ c b ≻ c ≻ a 3 2 c ≻ a ≻ b Computing d ( a ≻ b ≻ c , �≻ 1 , . . . , ≻ 9 � ): ◮ 3 voters disagree with a ≻ b N a b c ◮ 5 voters disagree with a ≻ c a − 6 4 ◮ 2 voters disagree with b ≻ c b 3 − 7 ◮ hence c 5 2 − d ( a ≻ b ≻ c , �≻ 1 , . . . , ≻ 9 � ) = 10. Kemeny scores: abc acb bac bca cab cba 10 15 13 12 14 17 Kemeny consensus: abc ; Kemeny winner: a
Computing voting rules: Kemeny ◮ NP-hard (Bartholdi et al. , 89; Hudry, 89) ◮ exact complexity: deciding whether a candidate is a Kemeny winner is Θ P 2 -complete (Hemaspaandra et al. , 04) ◮ a 4 / 3-approximation algorithm based on linear programming (Ailon et al. , 08) ◮ good heuristics
Computing voting rules Dodgson: ◮ deciding whether x is a Dodgson winner is Θ P 2 -complete (Hemaspaandra, Hemaspaandra & Rothe, 97) ◮ Caragiannis, Kaklamanis, Karanikolas & Procaccia (10): socially desirable approximations of Dodgson . ◮ Example: monotonic approximations = voting rules: ◮ satisfying monotonicity ◮ close enough to Dodgson ◮ computable in polynomial time ◮ the approximation of a voting rule is a new voting rule that may be interesting per se ! Slater: ◮ straightforward reduction from feedback arc set ◮ Slater’s rule is NP-hard (but maybe not in NP), even under the restriction that pairwise ties cannot occur
Computing voting rules: Banks ◮ M P majority graph induced by P : ◮ maximal subtournament of M P : maximal subset of X such that the restriction of M P to X is transitive. ◮ x is a Banks winner if x is undominated in some maximal subtournament of M P . ◮ deciding whether x is a Banks winner is NP-complete (Woeginger, 2003) ◮ however, it is possible to find an arbitrary Banks winner in polynomial time (Hudry, 2004) A := { x } where x is an arbitrary candidate; repeat find y such that the restriction of M P to A ∪ { y } is cycle-free; add y to A until it is no longer possible to do so; return the maximal element in A
Computing voting rules Discussion ◮ winner determination is in P: easy to compute positional scoring rules, Bucklin, Copeland, maximin, plurality with runoff, STV T , ranked pairs T , and others ◮ winner determination is NP-complete: not easy to compute but easy to verify a solution using a succinct certificate not so many: Banks, STV PU , ranked pairs PU ◮ winner determination is beyond NP: not even easy to verify. Kemeny, Young, Dodgson (and probably Slater), and others
Is there a life after NP-hardness? ◮ efficient computation : design algorithms that do as well as possible, possibly using heuristics, or translations into well-known frameworks (such as integer linear programming). ◮ fixed-parameter complexity : isolate the components of the problem and find the main cause(s) of hardness ◮ approximation : design algorithms that produce a (generally suboptimal) result, with some performance guarantee. ◮ The approximation of a voting rule is a new voting rule that may be interesting per se .
Plan 1. Social choice and computational social choice 2. Preference aggregation, Arrow’s theorem, how to escape it 3. Voting rules: informational basis 4. Voting rules: properties 5. Voting rules: computation 6. Combinatorial domains 7. Strategic behaviour 8. Voting with incomplete preferences 9. Fair division 10. Other issues
Voting in combinatorial domains Key question: structure of the set X of candidates. Example 1 choosing a common menu: X = { asparagus risotto, foie gras } × { roasted chicken, vegetable curry } × { white wine, red wine } Example 2 multiple referendum : a local community has to decide on several interrelated issues (should we build a swimming pool or not? should we build a tennis court or not?) Example 3 recruiting committee (3 positions, 6 candidates): X = { A | A ⊆ { a , b , c , d , e , f } , | A | ≤ 3 } . Combinatorial domains: ◮ P = { X 1 , . . . , X p } set of variables , or issues ; ◮ X = D 1 × ... × D p ◮ for each i , D i is a finite value domain for variable X i
Voting in combinatorial domains ◮ Two binary variables: ◮ S (build a new swimming pool or npt) ◮ T (build a new tennis court or not) ◮ 5 voters with their preferences: voters 1 and 2 ST ≻ ST ≻ ST ≻ ST voters 3 and 4 ST ≻ ST ≻ ST ≻ ST voter 5 ST ≻ ST ≻ ST ≻ ST
Voting in combinatorial domains ◮ Two binary variables: ◮ S (build a new swimming pool or npt) ◮ T (build a new tennis court or not) ◮ 5 voters with their preferences: voters 1 and 2 ST ≻ ST ≻ ST ≻ ST voters 3 and 4 ST ≻ ST ≻ ST ≻ ST voter 5 ST ≻ ST ≻ ST ≻ ST ◮ Problem 1 : voters 1-4 feel ill at ease reporting a preference on { S , S } and { T , T }
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