Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Majority Judgment vs Majority Rule Rida Laraki CNRS, (Lamsade University of Dauphine) and Economics Department (Ecole Polytechnique)
Based on joint work with Michel Balinski CNRS-Polytechnique Séminaire Roy, PSE, November 28, 2016
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
1
Majority judgment method Inspired by practice Majority judgment for small jury Majority Judgment for a large electorate
2
May’s axioms for n = 2 candidates
3
Extending May’s Axioms to n ≥ 3 [based on comparions] Condorcet and Arrow Paradoxes Arrow’s Theorem
4
Extending May’s axioms to n ≥ 1 candidates [based on measures] Dahl’s intensity problem Ranking methods based on measures Strategy proofness and second characterization of MJ
5
Scale and language dependency
6
Conclusion and references
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Actual rules in diving The rules of the Fédération Internationale de Natation (FINA) are as follows :
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Actual rules in diving The rules of the Fédération Internationale de Natation (FINA) are as follows : Each dive has a degree of difficulty.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Actual rules in diving The rules of the Fédération Internationale de Natation (FINA) are as follows : Each dive has a degree of difficulty. Judges grade each dive on a scale of :
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Actual rules in diving The rules of the Fédération Internationale de Natation (FINA) are as follows : Each dive has a degree of difficulty. Judges grade each dive on a scale of : 0 “completely failed” 1 to 2 ; “unsatisfactory” 2 2 12 to 4 12 “deficient” 5 to 6 “satisfactory” 6 12 to 8 “good” 8 12 to 10 “very good”
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Actual rules in diving The rules of the Fédération Internationale de Natation (FINA) are as follows : Each dive has a degree of difficulty. Judges grade each dive on a scale of : 0 “completely failed” 1 to 2 ; “unsatisfactory” 2 2 12 to 4 12 “deficient” 5 to 6 “satisfactory” 6 12 to 8 “good” 8 12 to 10 “very good”
There are either 5 or 7 judges. To minimize manipulability : If 5, the highest and lowest scores of a dive are eliminated leaving 3 scores. If 7, the 2 highest and 2 lowest scores are eliminated, leaving 3 scores.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Actual rules in diving The rules of the Fédération Internationale de Natation (FINA) are as follows : Each dive has a degree of difficulty. Judges grade each dive on a scale of : 0 “completely failed” 1 to 2 ; “unsatisfactory” 2 2 12 to 4 12 “deficient” 5 to 6 “satisfactory” 6 12 to 8 “good” 8 12 to 10 “very good”
There are either 5 or 7 judges. To minimize manipulability : If 5, the highest and lowest scores of a dive are eliminated leaving 3 scores. If 7, the 2 highest and 2 lowest scores are eliminated, leaving 3 scores.
The sum of the 3 remaining scores is multiplied by the degree of difficulty to obtain the score of the dive.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Actual rules in diving The rules of the Fédération Internationale de Natation (FINA) are as follows : Each dive has a degree of difficulty. Judges grade each dive on a scale of : 0 “completely failed” 1 to 2 ; “unsatisfactory” 2 2 12 to 4 12 “deficient” 5 to 6 “satisfactory” 6 12 to 8 “good” 8 12 to 10 “very good”
There are either 5 or 7 judges. To minimize manipulability : If 5, the highest and lowest scores of a dive are eliminated leaving 3 scores. If 7, the 2 highest and 2 lowest scores are eliminated, leaving 3 scores.
The sum of the 3 remaining scores is multiplied by the degree of difficulty to obtain the score of the dive. There are many other instances that use measures—well defined scales of grades—to grades, to rank and or to designate winners : guide Michelin, figure skating, gymnastics, concours Chopin, wine competitions, etc.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
A real use of Majority Judgment : small jury
Opinion profile : LAMSADE Jury ranking PhD candidates for a grant, 2015 A B C D E F
: : : : : :
J1 Excellent Excellent Passable V. Good Good V. Good
J2 Excellent V. Good Excellent Good Passable Passable
J3 V. Good V. Good Good Passable V. Good Insufficient
Rida Laraki
J4 Excellent V. Good V. Good Good Good Passable
J5 Excellent Good V. Good Good Good Passable
Majority Judgment vs Majority Rule
J6 Excellent V. Good Excellent Good Good Good
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
A real use of Majority Judgment : small jury
Opinion profile : LAMSADE Jury ranking PhD candidates for a grant, 2015 A B C D E F
: : : : : :
J1 Excellent Excellent Passable V. Good Good V. Good
J2 Excellent V. Good Excellent Good Passable Passable
J3 V. Good V. Good Good Passable V. Good Insufficient
Excellent V. Good Excellent Good Good Good
Excellent V. Good V. Good Good Good Passable
J4 Excellent V. Good V. Good Good Good Passable
J5 Excellent Good V. Good Good Good Passable
J6 Excellent V. Good Excellent Good Good Good
Merit profile : A B C D E F
: : : : : :
Excellent Excellent Excellent V. Good V. Good V. Good
Rida Laraki
Excellent V. Good V. Good Good Good Passable
Excellent V. Good Good Good Good Passable
V. Good Good Passable Passable Passable Insufficent
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
A succinct, simple description of MJ A B C D E F
: : : : : :
Excellent 5 1 2
Very Good 1 4 2 1 1 1
Good
Passable
Insufficient
1 1 4 4 1
1 1 1 3
1
Merit profile (counts), LAMSADE Jury.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
A succinct, simple description of MJ A B C D E F
: : : : : :
Excellent 5 1 2
Very Good 1 4 2 1 1 1
Good
Passable
Insufficient
1 1 4 4 1
1 1 1 3
1
Merit profile (counts), LAMSADE Jury. For each pair of competitors ignore as many equal numbers of highest and lowest grades of their merit profiles as possible until
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
A succinct, simple description of MJ A B C D E F
: : : : : :
Excellent 5 1 2
Very Good 1 4 2 1 1 1
Good
Passable
Insufficient
1 1 4 4 1
1 1 1 3
1
Merit profile (counts), LAMSADE Jury. For each pair of competitors ignore as many equal numbers of highest and lowest grades of their merit profiles as possible until first order domination or consensus=second order dominance ranks them.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
A succinct, simple description of MJ A B C D E F
: : : : : :
Excellent 5 1 2
Very Good 1 4 2 1 1 1
Good
Passable
Insufficient
1 1 4 4 1
1 1 1 3
1
Merit profile (counts), LAMSADE Jury. For each pair of competitors ignore as many equal numbers of highest and lowest grades of their merit profiles as possible until first order domination or consensus=second order dominance ranks them. Ranking PhD candidates B and C by LAMSADE Jury : B : C :
Excellent Excellent
V. Good Excellent
V. Good V. Good
Rida Laraki
V. Good V. Good
V. Good Good
Good Passable
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
A succinct, simple description of MJ A B C D E F
: : : : : :
Excellent 5 1 2
Very Good 1 4 2 1 1 1
Good
Passable
Insufficient
1 1 4 4 1
1 1 1 3
1
Merit profile (counts), LAMSADE Jury. For each pair of competitors ignore as many equal numbers of highest and lowest grades of their merit profiles as possible until first order domination or consensus=second order dominance ranks them. Ranking PhD candidates B and C by LAMSADE Jury : B : C :
Excellent Excellent B : C :
V. Good Excellent
V. Good V. Good
V. Good V. Good
V. Good Good
V. Good Excellent
V. Good V. Good
V. Good V. Good
V. Good Good
Rida Laraki
Good Passable
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
A succinct, simple description of MJ A B C D E F
: : : : : :
Excellent 5 1 2
Very Good 1 4 2 1 1 1
Good
Passable
Insufficient
1 1 4 4 1
1 1 1 3
1
Merit profile (counts), LAMSADE Jury. For each pair of competitors ignore as many equal numbers of highest and lowest grades of their merit profiles as possible until first order domination or consensus=second order dominance ranks them. Ranking PhD candidates B and C by LAMSADE Jury : B : C :
Excellent Excellent B : C :
V. Good Excellent
V. Good V. Good
V. Good V. Good
V. Good Good
V. Good Excellent
V. Good V. Good
V. Good V. Good
V. Good Good
Good Passable
For all pairs (except between B and C ), first order domination decides ! Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Comparison with Borda and Condorcet rankings Assuming the scale of grades sufficiently rich to faithfully represent the preferences
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Comparison with Borda and Condorcet rankings Assuming the scale of grades sufficiently rich to faithfully represent the preferences (meaning : same grade <=> indifference),
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Comparison with Borda and Condorcet rankings Assuming the scale of grades sufficiently rich to faithfully represent the preferences (meaning : same grade <=> indifference), we can deduce the majority rule pairwise votes :
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Comparison with Borda and Condorcet rankings Assuming the scale of grades sufficiently rich to faithfully represent the preferences (meaning : same grade <=> indifference), we can deduce the majority rule pairwise votes :
A C B D E F
A – 1 1 0 0.5 0
C 5 – 2.5 1 2 1
B 5 3.5 – 0.5 1 0
D 6 5 5.5 – 2.5 1
Rida Laraki
E 5.5 4 5 3.5 – 2
F 6 5 6 5 4 –
Borda score 5.5 3.7 4.0 2.0 2.0 0.8
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Comparison with Borda and Condorcet rankings Assuming the scale of grades sufficiently rich to faithfully represent the preferences (meaning : same grade <=> indifference), we can deduce the majority rule pairwise votes :
A C B D E F
A – 1 1 0 0.5 0
C 5 – 2.5 1 2 1
B 5 3.5 – 0.5 1 0
D 6 5 5.5 – 2.5 1
E 5.5 4 5 3.5 – 2
F 6 5 6 5 4 –
Borda score 5.5 3.7 4.0 2.0 2.0 0.8
Condorcet ranking is A Condo C Condo B Condo D Condo E Condo F .
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Comparison with Borda and Condorcet rankings Assuming the scale of grades sufficiently rich to faithfully represent the preferences (meaning : same grade <=> indifference), we can deduce the majority rule pairwise votes :
A C B D E F
A – 1 1 0 0.5 0
C 5 – 2.5 1 2 1
B 5 3.5 – 0.5 1 0
D 6 5 5.5 – 2.5 1
E 5.5 4 5 3.5 – 2
F 6 5 6 5 4 –
Borda score 5.5 3.7 4.0 2.0 2.0 0.8
Condorcet ranking is A Condo C Condo B Condo D Condo E Condo F . Borda ranking = MJ ranking = A Borda B Borda C Borda D ≈Borda E Borda F
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Comparison with Borda and Condorcet rankings Assuming the scale of grades sufficiently rich to faithfully represent the preferences (meaning : same grade <=> indifference), we can deduce the majority rule pairwise votes :
A C B D E F
A – 1 1 0 0.5 0
C 5 – 2.5 1 2 1
B 5 3.5 – 0.5 1 0
D 6 5 5.5 – 2.5 1
E 5.5 4 5 3.5 – 2
F 6 5 6 5 4 –
Borda score 5.5 3.7 4.0 2.0 2.0 0.8
Condorcet ranking is A Condo C Condo B Condo D Condo E Condo F . Borda ranking = MJ ranking = A Borda B Borda C Borda D ≈Borda E Borda F Majority judgment (and Borda) disagree with majority rule.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Comparison with Borda and Condorcet rankings Assuming the scale of grades sufficiently rich to faithfully represent the preferences (meaning : same grade <=> indifference), we can deduce the majority rule pairwise votes :
A C B D E F
A – 1 1 0 0.5 0
C 5 – 2.5 1 2 1
B 5 3.5 – 0.5 1 0
D 6 5 5.5 – 2.5 1
E 5.5 4 5 3.5 – 2
F 6 5 6 5 4 –
Borda score 5.5 3.7 4.0 2.0 2.0 0.8
Condorcet ranking is A Condo C Condo B Condo D Condo E Condo F . Borda ranking = MJ ranking = A Borda B Borda C Borda D ≈Borda E Borda F Majority judgment (and Borda) disagree with majority rule. This is a major criticism against MJ (and Borda since the 18th century).
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Comparison with Borda and Condorcet rankings Assuming the scale of grades sufficiently rich to faithfully represent the preferences (meaning : same grade <=> indifference), we can deduce the majority rule pairwise votes :
A C B D E F
A – 1 1 0 0.5 0
C 5 – 2.5 1 2 1
B 5 3.5 – 0.5 1 0
D 6 5 5.5 – 2.5 1
E 5.5 4 5 3.5 – 2
F 6 5 6 5 4 –
Borda score 5.5 3.7 4.0 2.0 2.0 0.8
Condorcet ranking is A Condo C Condo B Condo D Condo E Condo F . Borda ranking = MJ ranking = A Borda B Borda C Borda D ≈Borda E Borda F Majority judgment (and Borda) disagree with majority rule. This is a major criticism against MJ (and Borda since the 18th century). One of the main objectives of this paper is to address this criticism. Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
The majority judgement ballot (large electorate)
Ballot : Election of the President of France 2012 To be president of France, having taken into account all considerations, I judge, in conscience, that this candidate would be : Outstanding
Excellent
Very Good
Good
Accepable
Insufficient
François Hollande François Bayrou Nicolas Sarkozy Jean-Luc Mélenchon Nicolas Dupont-Aignan Eva Joly Philippe Poutou Marine Le Pen Nathalie Arthaud Jacques Cheminade
Rida Laraki
Majority Judgment vs Majority Rule
Reject
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Pool OpinionWay-Terra Nova, April 12-16 2012
Hollande Bayrou Sarkozy Mélenchon Dupont-Aignan Joly Poutou Le Pen Arthaud Cheminade
Outstanding 12.48% 2.58% 9.63% 5.43% 0.54% 0.81% 0.14% 5.97% 0.00% 0.41%
Excellent 16.15% 9.77% 12.35% 9.50% 2.58% 2.99% 1.36% 7.33% 1.36% 0.81%
Very Good 16.42% 21.71% 16.28% 12.89% 5.97% 6.51% 4.48% 9.50% 3.80% 2.44%
Rida Laraki
Good 11.67% 25.24% 10.99% 14.65% 11.26% 11.80% 7.73% 9.36% 6.51% 5.83%
Accepable 14.79% 20.08% 11.13% 17.10% 20.22% 14.65% 12.48% 13.98% 13.16% 11.67%
Insufficient 14.25% 11.94% 7.87% 15.06% 25.51% 24.69% 28.09% 6.24% 25.24% 26.87%
Majority Judgment vs Majority Rule
Reject 14.24% 8.69% 31.75% 25.37% 33.92% 38.53% 45.73% 47.63% 49.93% 51.97%
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority judgment ranking between two candidates in large electorate
Hollande Bayrou
Outstanding 12.48% 2.58%
Excellent 16.15% 9.77%
Very Good 16.42% 21.71%
Rida Laraki
Good 11.67% 25.24%
Accepable 14.79% 20.08%
Insufficient 14.25% 11.94%
Majority Judgment vs Majority Rule
Reject 14.24% 8.69%
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority judgment ranking between two candidates in large electorate
Hollande Bayrou
H B
Outstanding 12.48% 2.58%
Outstanding 12.48% 2.58%
Excellent 16.15% 9.77%
Excellent 16.15% 9.77%
Very Good 16.42% 21.71%
Very Good 16.42% 21.71%
Good 11.67% 25.24%
Accepable 14.79% 20.08%
Good
l
Good
4.95% 15.94%
l l
6.72% 9.30%
Rida Laraki
Insufficient 14.25% 11.94% Accepable 14.79% 20.08%
Reject 14.24% 8.69% Insufficient 14.25% 11.94%
Majority Judgment vs Majority Rule
Reject 14.24% 8.69%
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority judgment ranking between two candidates in large electorate
Hollande Bayrou
H B
Outstanding 12.48% 2.58%
Outstanding 12.48% 2.58%
Excellent 16.15% 9.77%
Excellent 16.15% 9.77%
Very Good 16.42% 21.71%
Very Good 16.42% 21.71%
Good 11.67% 25.24%
Accepable 14.79% 20.08%
Good
l
Good
4.95% 15.94%
l l
6.72% 9.30%
Insufficient 14.25% 11.94% Accepable 14.79% 20.08%
Reject 14.24% 8.69% Insufficient 14.25% 11.94%
Dropping 50% − (4.95% + %) of grades above and below yield :
Rida Laraki
Majority Judgment vs Majority Rule
Reject 14.24% 8.69%
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority judgment ranking between two candidates in large electorate
Hollande Bayrou
H B
Outstanding 12.48% 2.58%
Outstanding 12.48% 2.58%
Excellent 16.15% 9.77%
Excellent 16.15% 9.77%
Very Good 16.42% 21.71%
Very Good 16.42% 21.71%
Good 11.67% 25.24%
Accepable 14.79% 20.08%
Good
l
Good
4.95% 15.94%
l l
6.72% 9.30%
Insufficient 14.25% 11.94% Accepable 14.79% 20.08%
Reject 14.24% 8.69% Insufficient 14.25% 11.94%
Reject 14.24% 8.69%
Dropping 50% − (4.95% + %) of grades above and below yield : Outstanding H B
Excellent
Very Good %
Good
l
Good
4.95% 4.95+%
l l
4.95+% 4.95+%
Rida Laraki
Accepable
Insufficient
Majority Judgment vs Majority Rule
Reject
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority judgment ranking between two candidates in large electorate
Hollande Bayrou
H B
Outstanding 12.48% 2.58%
Outstanding 12.48% 2.58%
Excellent 16.15% 9.77%
Excellent 16.15% 9.77%
Very Good 16.42% 21.71%
Very Good 16.42% 21.71%
Good 11.67% 25.24%
Accepable 14.79% 20.08%
Good
l
Good
4.95% 15.94%
l l
6.72% 9.30%
Insufficient 14.25% 11.94% Accepable 14.79% 20.08%
Reject 14.24% 8.69% Insufficient 14.25% 11.94%
Reject 14.24% 8.69%
Dropping 50% − (4.95% + %) of grades above and below yield : Outstanding H B
Excellent
Very Good %
Good
l
Good
4.95% 4.95+%
l l
4.95+% 4.95+%
Accepable
Insufficient
Holland wins by first order dominance.
Rida Laraki
Majority Judgment vs Majority Rule
Reject
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority judgment ranking between two candidates in large electorate
Hollande Bayrou
H B
Outstanding 12.48% 2.58%
Outstanding 12.48% 2.58%
Excellent 16.15% 9.77%
Excellent 16.15% 9.77%
Very Good 16.42% 21.71%
Very Good 16.42% 21.71%
Good 11.67% 25.24%
Accepable 14.79% 20.08%
Good
l
Good
4.95% 15.94%
l l
6.72% 9.30%
Insufficient 14.25% 11.94% Accepable 14.79% 20.08%
Reject 14.24% 8.69% Insufficient 14.25% 11.94%
Reject 14.24% 8.69%
Dropping 50% − (4.95% + %) of grades above and below yield : Outstanding H B
Excellent
Very Good %
Good
l
Good
4.95% 4.95+%
l l
4.95+% 4.95+%
Accepable
Insufficient
Holland wins by first order dominance. In large electorate first order dominance applies almost surely.
Rida Laraki
Majority Judgment vs Majority Rule
Reject
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority Grade et Gauge
Hollande
Outstanding 12.48%
Excellent 16.15%
Very Good 16.42%
Rida Laraki
Good 11.67%
Accepable 14.79%
Insufficient 14.25%
Majority Judgment vs Majority Rule
Reject 14.24%
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority Grade et Gauge
Hollande
Outstanding 12.48%
Excellent 16.15%
Very Good 16.42%
Good 11.67%
Accepable 14.79%
Insufficient 14.25%
The Majority-Grade=median of Hollande is α=Good because :
Rida Laraki
Majority Judgment vs Majority Rule
Reject 14.24%
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority Grade et Gauge
Hollande
Outstanding 12.48%
Excellent 16.15%
Very Good 16.42%
Good 11.67%
Accepable 14.79%
Insufficient 14.25%
Reject 14.24%
The Majority-Grade=median of Hollande is α=Good because : 12.48 + 16.15 + 16.42 + 11.67 = 56.72% judge him Good or above.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority Grade et Gauge
Hollande
Outstanding 12.48%
Excellent 16.15%
Very Good 16.42%
Good 11.67%
Accepable 14.79%
Insufficient 14.25%
Reject 14.24%
The Majority-Grade=median of Hollande is α=Good because : 12.48 + 16.15 + 16.42 + 11.67 = 56.72% judge him Good or above. 11.67 + 14.79 + 14.25 + 14.24 = 54.95% judge him Good or below.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority Grade et Gauge
Hollande
Outstanding 12.48%
Excellent 16.15%
Very Good 16.42%
Good 11.67%
Accepable 14.79%
Insufficient 14.25%
Reject 14.24%
The Majority-Grade=median of Hollande is α=Good because : 12.48 + 16.15 + 16.42 + 11.67 = 56.72% judge him Good or above. 11.67 + 14.79 + 14.25 + 14.24 = 54.95% judge him Good or below. The Majority Gauge of Hollande is (p, α, q)=(45.05%, Good, 43.28%).
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority Grade et Gauge
Hollande
Outstanding 12.48%
Excellent 16.15%
Very Good 16.42%
Good 11.67%
Accepable 14.79%
Insufficient 14.25%
Reject 14.24%
The Majority-Grade=median of Hollande is α=Good because : 12.48 + 16.15 + 16.42 + 11.67 = 56.72% judge him Good or above. 11.67 + 14.79 + 14.25 + 14.24 = 54.95% judge him Good or below. The Majority Gauge of Hollande is (p, α, q)=(45.05%, Good, 43.28%). p = 45.05=12.48+16.15+16.42 = percentage of grade above Good.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority Grade et Gauge
Hollande
Outstanding 12.48%
Excellent 16.15%
Very Good 16.42%
Good 11.67%
Accepable 14.79%
Insufficient 14.25%
Reject 14.24%
The Majority-Grade=median of Hollande is α=Good because : 12.48 + 16.15 + 16.42 + 11.67 = 56.72% judge him Good or above. 11.67 + 14.79 + 14.25 + 14.24 = 54.95% judge him Good or below. The Majority Gauge of Hollande is (p, α, q)=(45.05%, Good, 43.28%). p = 45.05=12.48+16.15+16.42 = percentage of grade above Good. q = 43.25=14.79+14.25+14.24 = percentage of grades below Good.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority Grade et Gauge
Hollande
Outstanding 12.48%
Excellent 16.15%
Very Good 16.42%
Good 11.67%
Accepable 14.79%
Insufficient 14.25%
Reject 14.24%
The Majority-Grade=median of Hollande is α=Good because : 12.48 + 16.15 + 16.42 + 11.67 = 56.72% judge him Good or above. 11.67 + 14.79 + 14.25 + 14.24 = 54.95% judge him Good or below. The Majority Gauge of Hollande is (p, α, q)=(45.05%, Good, 43.28%). p = 45.05=12.48+16.15+16.42 = percentage of grade above Good. q = 43.25=14.79+14.25+14.24 = percentage of grades below Good. Because p=45.05 > q=43.28,
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Majority Grade et Gauge
Hollande
Outstanding 12.48%
Excellent 16.15%
Very Good 16.42%
Good 11.67%
Accepable 14.79%
Insufficient 14.25%
Reject 14.24%
The Majority-Grade=median of Hollande is α=Good because : 12.48 + 16.15 + 16.42 + 11.67 = 56.72% judge him Good or above. 11.67 + 14.79 + 14.25 + 14.24 = 54.95% judge him Good or below. The Majority Gauge of Hollande is (p, α, q)=(45.05%, Good, 43.28%). p = 45.05=12.48+16.15+16.42 = percentage of grade above Good. q = 43.25=14.79+14.25+14.24 = percentage of grades below Good. Because p=45.05 > q=43.28, Hollande Gauge is +45.05.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Result of the Pool OpinionWay-Terra Nova, April 12-16 2012
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Result of the Pool OpinionWay-Terra Nova, April 12-16 2012
Majority Judgment Ranking 1 Hollande 2 Bayrou
Majority Grade α Good Good
Rida Laraki
Gauge + or − p ou q +45.05% −40.71%
FirstPastthe-Post 1 5
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Result of the Pool OpinionWay-Terra Nova, April 12-16 2012
Majority Judgment Ranking 1 Hollande 2 Bayrou 3 Sarkozy 4 Mélenchon
Majority Grade α Good Good Acceptable Acceptable
Rida Laraki
Gauge + or − p ou q +45.05% −40.71% +49.25% +42.47%
FirstPastthe-Post 1 5 2 4
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Result of the Pool OpinionWay-Terra Nova, April 12-16 2012
Majority Judgment Ranking 1 Hollande 2 Bayrou 3 Sarkozy 4 Mélenchon 5 Dupont-Aignan 6 Joly 7 Poutou
Majority Grade α Good Good Acceptable Acceptable Insufficient Insufficient Insufficient
Rida Laraki
Gauge + or − p ou q +45.05% −40.71% +49.25% +42.47% +40.57% −38.53% −45.73%
FirstPastthe-Post 1 5 2 4 7 6 8
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Result of the Pool OpinionWay-Terra Nova, April 12-16 2012
Majority Judgment Ranking 1 Hollande 2 Bayrou 3 Sarkozy 4 Mélenchon 5 Dupont-Aignan 6 Joly 7 Poutou 8 Le Pen
Majority Grade α Good Good Acceptable Acceptable Insufficient Insufficient Insufficient Insuffisant
Rida Laraki
Gauge + or − p ou q +45.05% −40.71% +49.25% +42.47% +40.57% −38.53% −45.73% −47.63%
FirstPastthe-Post 1 5 2 4 7 6 8 3
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Result of the Pool OpinionWay-Terra Nova, April 12-16 2012
Majority Judgment Ranking 1 Hollande 2 Bayrou 3 Sarkozy 4 Mélenchon 5 Dupont-Aignan 6 Joly 7 Poutou 8 Le Pen 9 Arthaud 10 Cheminade
Majority Grade α Good Good Acceptable Acceptable Insufficient Insufficient Insufficient Insuffisant Insufficient To Rejetect
Gauge + or − p ou q +45.05% −40.71% +49.25% +42.47% +40.57% −38.53% −45.73% −47.63% −49.93% +48.03%
FirstPastthe-Post 1 5 2 4 7 6 8 3 9 10
Compared to first-past-the-post (plurality voting), majority judgment increases the ranking of moderates and decreases the ranking of the extremes.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Pew Research center poll results, March 17-27, 2016 Question asked : Regardless of who you currently support, I’d like to know what kind of president you think each of the following would be :
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Pew Research center poll results, March 17-27, 2016 Question asked : Regardless of who you currently support, I’d like to know what kind of president you think each of the following would be :
John Kasich Bernie Sanders Ted Cruz Hillary Clinton Donald Trump
Great 5% 10% 7% 11% 10%
Good 28% 26% 22% 22% 16%
Rida Laraki
Average 39% 26% 21% 20% 12%
Poor 13% 15% 17% 16% 15%
Terrible 7% 21% 19% 30% 44%
Majority Judgment vs Majority Rule
Never heard of 9% 3% 4% 1% 3%
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Pew Research center poll results, March 17-27, 2016 Question asked : Regardless of who you currently support, I’d like to know what kind of president you think each of the following would be :
John Kasich Bernie Sanders Ted Cruz Hillary Clinton Donald Trump
Great 5% 10% 7% 11% 10%
Good 28% 26% 22% 22% 16%
John Kasich Bernie Sanders Ted Cruz Hillary Clinton Donald Trump
Average 39% 26% 21% 20% 12%
p 33% 36% 29% 33% 38%
Rida Laraki
Poor 13% 15% 17% 16% 15%
α ± max{p, q} Average+ Average− Average− Average− Poor −
Terrible 7% 21% 19% 30% 44% q 29% 39% 40% 47% 47%
Majority Judgment vs Majority Rule
Never heard of 9% 3% 4% 1% 3%
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Could strategic voting have made Bayrou the winner in 2012 ?
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Could strategic voting have made Bayrou the winner in 2012 ? How could a voter who graded Bayrou above Hollande manipulate in 2012 with Majority Judgment ?
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Could strategic voting have made Bayrou the winner in 2012 ? How could a voter who graded Bayrou above Hollande manipulate in 2012 with Majority Judgment ? By lowering Hollande’s grade and raising Bayrou’s.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Could strategic voting have made Bayrou the winner in 2012 ? How could a voter who graded Bayrou above Hollande manipulate in 2012 with Majority Judgment ? By lowering Hollande’s grade and raising Bayrou’s. Definition : A method f is strategy-proof if whenever f ranks A above B, and a voter prefers B above A, he cannot lower A and cannot increase B.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Could strategic voting have made Bayrou the winner in 2012 ? How could a voter who graded Bayrou above Hollande manipulate in 2012 with Majority Judgment ? By lowering Hollande’s grade and raising Bayrou’s. Definition : A method f is strategy-proof if whenever f ranks A above B, and a voter prefers B above A, he cannot lower A and cannot increase B. We prove the following results : Theorem No method satisfying the basic axioms is strategy-proof in the entire domain.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Could strategic voting have made Bayrou the winner in 2012 ? How could a voter who graded Bayrou above Hollande manipulate in 2012 with Majority Judgment ? By lowering Hollande’s grade and raising Bayrou’s. Definition : A method f is strategy-proof if whenever f ranks A above B, and a voter prefers B above A, he cannot lower A and cannot increase B. We prove the following results : Theorem No method satisfying the basic axioms is strategy-proof in the entire domain. Majority-gauge method is partially strategy-proof in entire domain,
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Could strategic voting have made Bayrou the winner in 2012 ? How could a voter who graded Bayrou above Hollande manipulate in 2012 with Majority Judgment ? By lowering Hollande’s grade and raising Bayrou’s. Definition : A method f is strategy-proof if whenever f ranks A above B, and a voter prefers B above A, he cannot lower A and cannot increase B. We prove the following results : Theorem No method satisfying the basic axioms is strategy-proof in the entire domain. Majority-gauge method is partially strategy-proof in entire domain, and is the unique strategy proof on polarized domains.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Could strategic voting have made Bayrou the winner in 2012 ? How could a voter who graded Bayrou above Hollande manipulate in 2012 with Majority Judgment ? By lowering Hollande’s grade and raising Bayrou’s. Definition : A method f is strategy-proof if whenever f ranks A above B, and a voter prefers B above A, he cannot lower A and cannot increase B. We prove the following results : Theorem No method satisfying the basic axioms is strategy-proof in the entire domain. Majority-gauge method is partially strategy-proof in entire domain, and is the unique strategy proof on polarized domains. Partial strategy proofness : whenever majority-gauge ranks A higher than B, and a voter grades B above A :
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Could strategic voting have made Bayrou the winner in 2012 ? How could a voter who graded Bayrou above Hollande manipulate in 2012 with Majority Judgment ? By lowering Hollande’s grade and raising Bayrou’s. Definition : A method f is strategy-proof if whenever f ranks A above B, and a voter prefers B above A, he cannot lower A and cannot increase B. We prove the following results : Theorem No method satisfying the basic axioms is strategy-proof in the entire domain. Majority-gauge method is partially strategy-proof in entire domain, and is the unique strategy proof on polarized domains. Partial strategy proofness : whenever majority-gauge ranks A higher than B, and a voter grades B above A : if he can lower A’s majority-gauge, he cannot raise B.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Could strategic voting have made Bayrou the winner in 2012 ? How could a voter who graded Bayrou above Hollande manipulate in 2012 with Majority Judgment ? By lowering Hollande’s grade and raising Bayrou’s. Definition : A method f is strategy-proof if whenever f ranks A above B, and a voter prefers B above A, he cannot lower A and cannot increase B. We prove the following results : Theorem No method satisfying the basic axioms is strategy-proof in the entire domain. Majority-gauge method is partially strategy-proof in entire domain, and is the unique strategy proof on polarized domains. Partial strategy proofness : whenever majority-gauge ranks A higher than B, and a voter grades B above A : if he can lower A’s majority-gauge, he cannot raise B. if he can raise B’s majority-gauge, he cannot lower A.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Inspired May’s by practice Axioms Majority to n ≥ 3judgment [based onfor comparions] small jury Majority Extending J
Could strategic voting have made Bayrou the winner in 2012 ? How could a voter who graded Bayrou above Hollande manipulate in 2012 with Majority Judgment ? By lowering Hollande’s grade and raising Bayrou’s. Definition : A method f is strategy-proof if whenever f ranks A above B, and a voter prefers B above A, he cannot lower A and cannot increase B. We prove the following results : Theorem No method satisfying the basic axioms is strategy-proof in the entire domain. Majority-gauge method is partially strategy-proof in entire domain, and is the unique strategy proof on polarized domains. Partial strategy proofness : whenever majority-gauge ranks A higher than B, and a voter grades B above A : if he can lower A’s majority-gauge, he cannot raise B. if he can raise B’s majority-gauge, he cannot lower A. Conclusion : majority judgment best resists strategic manipulations among the methods satisfying the basic axioms. Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
1
Majority judgment method Inspired by practice Majority judgment for small jury Majority Judgment for a large electorate
2
May’s axioms for n = 2 candidates
3
Extending May’s Axioms to n ≥ 3 [based on comparions] Condorcet and Arrow Paradoxes Arrow’s Theorem
4
Extending May’s axioms to n ≥ 1 candidates [based on measures] Dahl’s intensity problem Ranking methods based on measures Strategy proofness and second characterization of MJ
5
Scale and language dependency
6
Conclusion and references
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
May’s (1952) axioms of majority rule : based on comparisons
For n = 2 candidates, majority rule is the unique method that satisfies the following axioms :
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
May’s (1952) axioms of majority rule : based on comparisons
For n = 2 candidates, majority rule is the unique method that satisfies the following axioms : A0 [Based on comparisons] A voter expresses her opinion by preferring one candidate or being indifferent.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
May’s (1952) axioms of majority rule : based on comparisons
For n = 2 candidates, majority rule is the unique method that satisfies the following axioms : A0 [Based on comparisons] A voter expresses her opinion by preferring one candidate or being indifferent. A1 [Universal domain] All opinions are admissible.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
May’s (1952) axioms of majority rule : based on comparisons
For n = 2 candidates, majority rule is the unique method that satisfies the following axioms : A0 [Based on comparisons] A voter expresses her opinion by preferring one candidate or being indifferent. A1 [Universal domain] All opinions are admissible. A2 [Anonymous] permuting names of voters does not change the outcome.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
May’s (1952) axioms of majority rule : based on comparisons
For n = 2 candidates, majority rule is the unique method that satisfies the following axioms : A0 [Based on comparisons] A voter expresses her opinion by preferring one candidate or being indifferent. A1 [Universal domain] All opinions are admissible. A2 [Anonymous] permuting names of voters does not change the outcome. A3 [Neutral] permuting names of candidates does not change the outcome.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
May’s (1952) axioms of majority rule : based on comparisons
For n = 2 candidates, majority rule is the unique method that satisfies the following axioms : A0 [Based on comparisons] A voter expresses her opinion by preferring one candidate or being indifferent. A1 [Universal domain] All opinions are admissible. A2 [Anonymous] permuting names of voters does not change the outcome. A3 [Neutral] permuting names of candidates does not change the outcome. A4 [Monotone] If candidate A wins or is in a tie and one or more voters change their preferences in favor of A then A wins.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
May’s (1952) axioms of majority rule : based on comparisons
For n = 2 candidates, majority rule is the unique method that satisfies the following axioms : A0 [Based on comparisons] A voter expresses her opinion by preferring one candidate or being indifferent. A1 [Universal domain] All opinions are admissible. A2 [Anonymous] permuting names of voters does not change the outcome. A3 [Neutral] permuting names of candidates does not change the outcome. A4 [Monotone] If candidate A wins or is in a tie and one or more voters change their preferences in favor of A then A wins. A5 [Complete] The rule guarantees an outcome : one of the two candidates wins or they are tied.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
May’s (1952) axioms of majority rule : based on comparisons
For n = 2 candidates, majority rule is the unique method that satisfies the following axioms : A0 [Based on comparisons] A voter expresses her opinion by preferring one candidate or being indifferent. A1 [Universal domain] All opinions are admissible. A2 [Anonymous] permuting names of voters does not change the outcome. A3 [Neutral] permuting names of candidates does not change the outcome. A4 [Monotone] If candidate A wins or is in a tie and one or more voters change their preferences in favor of A then A wins. A5 [Complete] The rule guarantees an outcome : one of the two candidates wins or they are tied. Proof : Simple.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
1
Majority judgment method Inspired by practice Majority judgment for small jury Majority Judgment for a large electorate
2
May’s axioms for n = 2 candidates
3
Extending May’s Axioms to n ≥ 3 [based on comparions] Condorcet and Arrow Paradoxes Arrow’s Theorem
4
Extending May’s axioms to n ≥ 1 candidates [based on measures] Dahl’s intensity problem Ranking methods based on measures Strategy proofness and second characterization of MJ
5
Scale and language dependency
6
Conclusion and references
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
The Condorcet Paradox (1786) : comparisons
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
The Condorcet Paradox (1786) : comparisons
The great hope—since Ramun Llull in 1299—has been to choose a Condorcet-winner : a candidate who beats every possible opponent according to majority rule.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
The Condorcet Paradox (1786) : comparisons
The great hope—since Ramun Llull in 1299—has been to choose a Condorcet-winner : a candidate who beats every possible opponent according to majority rule. There may be no Condorcet-winner : 30% A B C
32% B C A
38% C A B
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
The Condorcet Paradox (1786) : comparisons
The great hope—since Ramun Llull in 1299—has been to choose a Condorcet-winner : a candidate who beats every possible opponent according to majority rule. There may be no Condorcet-winner : 30% A B C
32% B C A
38% C A B
Rida Laraki
A B C
A – 32% 70%
B 68% – 38%
C 30% 62% –
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
The Condorcet Paradox (1786) : comparisons
The great hope—since Ramun Llull in 1299—has been to choose a Condorcet-winner : a candidate who beats every possible opponent according to majority rule. There may be no Condorcet-winner : 30% A B C
32% B C A
38% C A B
A B C
A – 32% 70%
B 68% – 38%
C 30% 62% –
because A(68%) B(62%) C (70%) A
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
The Condorcet Paradox (1786) : comparisons
The great hope—since Ramun Llull in 1299—has been to choose a Condorcet-winner : a candidate who beats every possible opponent according to majority rule. There may be no Condorcet-winner : 30% A B C
32% B C A
38% C A B
A B C
A – 32% 70%
B 68% – 38%
C 30% 62% –
because A(68%) B(62%) C (70%) A This is called the Condorcet paradox.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Arrow paradox in the French presidential election
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Arrow paradox in the French presidential election First round results 2002 (16 candidates, 72% participation) : Chirac 19,88%
Le Pen 16,86%
Mamère 5,25%
Besancenot 4,25%
(Pasqua) 0%
Jospin 16,18%
Bayrou 6,84%
Saint-Josse 4,23%
Taubira 2,32%
Lepage 1,88%
Rida Laraki
Laguiller 5,72% Madelin 3,91% Boutin 1,19%
Chévènement 5,33% Hue 3,37%
Mégret 2,34%
Gluckstein 0,47%
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Arrow paradox in the French presidential election First round results 2002 (16 candidates, 72% participation) : Chirac 19,88%
Le Pen 16,86%
Mamère 5,25%
Besancenot 4,25%
(Pasqua) 0%
Jospin 16,18%
Bayrou 6,84%
Saint-Josse 4,23%
Taubira 2,32%
Lepage 1,88%
Laguiller 5,72% Madelin 3,91% Boutin 1,19%
Chévènement 5,33% Hue 3,37%
Mégret 2,34%
Gluckstein 0,47%
Second round results 2002 (80% participation) : Chirac 82,21%
Le Pen 17,79%
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Arrow paradox in the French presidential election First round results 2002 (16 candidates, 72% participation) : Chirac 19,88%
Le Pen 16,86%
Mamère 5,25%
Besancenot 4,25%
(Pasqua) 0%
Jospin 16,18%
Bayrou 6,84%
Saint-Josse 4,23%
Taubira 2,32%
Lepage 1,88%
Laguiller 5,72% Madelin 3,91% Boutin 1,19%
Chévènement 5,33% Hue 3,37%
Mégret 2,34%
Gluckstein 0,47%
Second round results 2002 (80% participation) : Chirac 82,21%
Le Pen 17,79%
Chirac < 50% ?
Rida Laraki
Jospin > 50% ?
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Arrow paradox in the French presidential election First round results 2002 (16 candidates, 72% participation) : Chirac 19,88%
Le Pen 16,86%
Mamère 5,25%
Besancenot 4,25%
(Pasqua) 0%
Jospin 16,18%
Bayrou 6,84%
Saint-Josse 4,23%
Taubira 2,32%
Lepage 1,88%
Laguiller 5,72% Madelin 3,91% Boutin 1,19%
Chévènement 5,33% Hue 3,37%
Mégret 2,34%
Gluckstein 0,47%
Second round results 2002 (80% participation) : Chirac 82,21%
Le Pen 17,79%
Chirac < 50% ?
Rida Laraki
Jospin > 50% ?
Jospin > 75%
Le Pen < 25%
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Arrow paradox in the French presidential election First round results 2002 (16 candidates, 72% participation) : Chirac 19,88%
Le Pen 16,86%
Mamère 5,25%
Besancenot 4,25%
(Pasqua) 0%
Jospin 16,18%
Bayrou 6,84%
Saint-Josse 4,23%
Taubira 2,32%
Lepage 1,88%
Laguiller 5,72% Madelin 3,91% Boutin 1,19%
Chévènement 5,33% Hue 3,37%
Mégret 2,34%
Gluckstein 0,47%
Second round results 2002 (80% participation) : Chirac 82,21%
Le Pen 17,79%
Chirac < 50% ?
Jospin > 50% ?
Jospin > 75%
Le Pen < 25%
Arrow’s paradox : a candidate’s presence (having no chance of winning whatsoever) can change the winner.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Arrow paradox in the French presidential election First round results 2002 (16 candidates, 72% participation) : Chirac 19,88%
Le Pen 16,86%
Mamère 5,25%
Besancenot 4,25%
(Pasqua) 0%
Jospin 16,18%
Bayrou 6,84%
Saint-Josse 4,23%
Taubira 2,32%
Lepage 1,88%
Laguiller 5,72% Madelin 3,91% Boutin 1,19%
Chévènement 5,33% Hue 3,37%
Mégret 2,34%
Gluckstein 0,47%
Second round results 2002 (80% participation) : Chirac 82,21%
Le Pen 17,79%
Chirac < 50% ?
Jospin > 50% ?
Jospin > 75%
Le Pen < 25%
Arrow’s paradox : a candidate’s presence (having no chance of winning whatsoever) can change the winner. Without Ralph Nader in Florida, Albert Gore would have been the President of the US in 2000 instead of George W. Bush.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Traditional ranking methods based on comparisons
A method of ranking : a binary relation that compares any two candidates. It must satisfy : A0 [Based on comparisons] A voter expresses her opinion by ranking them. A1 [Unrestricted Domain] All voters opinions are admissible. A2 [Anonymous] Permuting names of voters does not change the outcome. A3 [Neutral] Permuting names of candidates does not change the outcome. A4 [Monotone] If A wins or is in a tie and one or more voters change their preferences in favor of A then A wins. A5 [Complete] The rule guarantees an outcome : or the two candidates are tie or one is the winner.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Traditional ranking methods based on comparisons
A method of ranking : a binary relation that compares any two candidates. It must satisfy : A0 [Based on comparisons] A voter expresses her opinion by ranking them. A1 [Unrestricted Domain] All voters opinions are admissible. A2 [Anonymous] Permuting names of voters does not change the outcome. A3 [Neutral] Permuting names of candidates does not change the outcome. A4 [Monotone] If A wins or is in a tie and one or more voters change their preferences in favor of A then A wins. A5 [Complete] The rule guarantees an outcome : or the two candidates are tie or one is the winner. A6 [Transitive] If A B and B C then A C .
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Traditional ranking methods based on comparisons
A method of ranking : a binary relation that compares any two candidates. It must satisfy : A0 [Based on comparisons] A voter expresses her opinion by ranking them. A1 [Unrestricted Domain] All voters opinions are admissible. A2 [Anonymous] Permuting names of voters does not change the outcome. A3 [Neutral] Permuting names of candidates does not change the outcome. A4 [Monotone] If A wins or is in a tie and one or more voters change their preferences in favor of A then A wins. A5 [Complete] The rule guarantees an outcome : or the two candidates are tie or one is the winner. A6 [Transitive] If A B and B C then A C . A7 [Independence of irrelevant alternatives (IIA)] If A B then whatever candidates are dropped or adjoined A B.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Impossibility of ranking methods based on comparisons
Theorem (Arrow’s Impossibility) No method of ranking based on comparisons satisfies axioms A1 to A7.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Impossibility of ranking methods based on comparisons
Theorem (Arrow’s Impossibility) No method of ranking based on comparisons satisfies axioms A1 to A7. It is not the usual formulation, but it is the one we will compare with.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Impossibility of ranking methods based on comparisons
Theorem (Arrow’s Impossibility) No method of ranking based on comparisons satisfies axioms A1 to A7. It is not the usual formulation, but it is the one we will compare with. Proof : Simple.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Arrow’s paradox in the 1997 European Championships, figure skating
Before the performance of Vlascenko, the order was : 1st Urmanov, 2nd Zagorodniuk, 3rd Candeloro.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Arrow’s paradox in the 1997 European Championships, figure skating
Before the performance of Vlascenko, the order was : 1st Urmanov, 2nd Zagorodniuk, 3rd Candeloro. After Vlascenko’s performance, the order was reversed : 1st Urmanov, 2nd Candeloro, 3rd Zagorodniuk.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Arrow’s paradox in the 1997 European Championships, figure skating
Before the performance of Vlascenko, the order was : 1st Urmanov, 2nd Zagorodniuk, 3rd Candeloro. After Vlascenko’s performance, the order was reversed : 1st Urmanov, 2nd Candeloro, 3rd Zagorodniuk. Why ?
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Arrow’s paradox in the 1997 European Championships, figure skating
Before the performance of Vlascenko, the order was : 1st Urmanov, 2nd Zagorodniuk, 3rd Candeloro. After Vlascenko’s performance, the order was reversed : 1st Urmanov, 2nd Candeloro, 3rd Zagorodniuk. Why ? Because the method is a function of : comparisons.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Arrow’s paradox in the 1997 European Championships, figure skating
Before the performance of Vlascenko, the order was : 1st Urmanov, 2nd Zagorodniuk, 3rd Candeloro. After Vlascenko’s performance, the order was reversed : 1st Urmanov, 2nd Candeloro, 3rd Zagorodniuk. Why ? Because the method is a function of : comparisons. Urmanov Candeloro Zagorodniuk Yagudin Kulik Vlascenko
J1 1 3 5 4 2 6
J2 1 2 5 3 4 6
J3 1 5 4 3 2 6
J4 1 2 4 6 3 5
J5 1 3 2 4 6 5
Rida Laraki
J6 2 3 4 6 5 1
J7 1 5 2 4 3 6
J8 1 6 2 3 4 5
J9 1 6 3 2 5 4
Mark 1/8 3/5 4/7 4/7 4/6 5/5
Majority Judgment vs Majority Rule
Place 1st 2nd 3rd 4th 5th 6th
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
Arrow’s paradox in the 1997 European Championships, figure skating
Before the performance of Vlascenko, the order was : 1st Urmanov, 2nd Zagorodniuk, 3rd Candeloro. After Vlascenko’s performance, the order was reversed : 1st Urmanov, 2nd Candeloro, 3rd Zagorodniuk. Why ? Because the method is a function of : comparisons. Urmanov Candeloro Zagorodniuk Yagudin Kulik Vlascenko
J1 1 3 5 4 2 6
J2 1 2 5 3 4 6
J3 1 5 4 3 2 6
J4 1 2 4 6 3 5
J5 1 3 2 4 6 5
J6 2 3 4 6 5 1
J7 1 5 2 4 3 6
J8 1 6 2 3 4 5
J9 1 6 3 2 5 4
Mark 1/8 3/5 4/7 4/7 4/6 5/5
Arrow’s paradox occurs because of Judge 6’s strategic voting !
Rida Laraki
Majority Judgment vs Majority Rule
Place 1st 2nd 3rd 4th 5th 6th
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
What happens in skating after the flip-flop ?
This flip-flop was so strident that the rules used for a century were changed.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
What happens in skating after the flip-flop ?
This flip-flop was so strident that the rules used for a century were changed. The ISU adopted the OBO rule (“one-by-one”) in 1998 :
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
What happens in skating after the flip-flop ?
This flip-flop was so strident that the rules used for a century were changed. The ISU adopted the OBO rule (“one-by-one”) in 1998 : rank the competitors by their number of wins (Condorcet’s) ;
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
What happens in skating after the flip-flop ?
This flip-flop was so strident that the rules used for a century were changed. The ISU adopted the OBO rule (“one-by-one”) in 1998 : rank the competitors by their number of wins (Condorcet’s) ; break any ties by using Borda’s rule.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
What happens in skating after the flip-flop ?
This flip-flop was so strident that the rules used for a century were changed. The ISU adopted the OBO rule (“one-by-one”) in 1998 : rank the competitors by their number of wins (Condorcet’s) ; break any ties by using Borda’s rule.
This rule is similar to the one proposed by Black 1958 and was proposed in (2004, 2008) by Dasgupta and Maskin.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
What happens in skating after the flip-flop ?
This flip-flop was so strident that the rules used for a century were changed. The ISU adopted the OBO rule (“one-by-one”) in 1998 : rank the competitors by their number of wins (Condorcet’s) ; break any ties by using Borda’s rule.
This rule is similar to the one proposed by Black 1958 and was proposed in (2004, 2008) by Dasgupta and Maskin. Just 4 years after adopting the OBO method, a “big scandal” of the 2002 Olympic games in Salt Lake City occurred because of strategic manipulation
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
What happens in skating after the flip-flop ?
This flip-flop was so strident that the rules used for a century were changed. The ISU adopted the OBO rule (“one-by-one”) in 1998 : rank the competitors by their number of wins (Condorcet’s) ; break any ties by using Borda’s rule.
This rule is similar to the one proposed by Black 1958 and was proposed in (2004, 2008) by Dasgupta and Maskin. Just 4 years after adopting the OBO method, a “big scandal” of the 2002 Olympic games in Salt Lake City occurred because of strategic manipulation Deep divisions in the skating world led to the formulation of another system (abandoning the newly adopted OBO rule).
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Condorcet May’s and Axioms Arrow to Paradoxes n ≥ 3 [based Arrow’s on comparions] Theorem Extending
What happens in skating after the flip-flop ?
This flip-flop was so strident that the rules used for a century were changed. The ISU adopted the OBO rule (“one-by-one”) in 1998 : rank the competitors by their number of wins (Condorcet’s) ; break any ties by using Borda’s rule.
This rule is similar to the one proposed by Black 1958 and was proposed in (2004, 2008) by Dasgupta and Maskin. Just 4 years after adopting the OBO method, a “big scandal” of the 2002 Olympic games in Salt Lake City occurred because of strategic manipulation Deep divisions in the skating world led to the formulation of another system (abandoning the newly adopted OBO rule). The new system is based on measures and is similar to one used in gymnastic, in diving and other sport competitions.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
1
Majority judgment method Inspired by practice Majority judgment for small jury Majority Judgment for a large electorate
2
May’s axioms for n = 2 candidates
3
Extending May’s Axioms to n ≥ 3 [based on comparions] Condorcet and Arrow Paradoxes Arrow’s Theorem
4
Extending May’s axioms to n ≥ 1 candidates [based on measures] Dahl’s intensity problem Ranking methods based on measures Strategy proofness and second characterization of MJ
5
Scale and language dependency
6
Conclusion and references
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
The domination paradox
National poll, 10 days before first-round, French presidential election, 2012.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
The domination paradox
National poll, 10 days before first-round, French presidential election, 2012. Merit profile :
Hollande : Sarkozy :
Outstanding 12.5% 9.6%
Excellent 16.2% 12.3%
Very Good 16.4% 16.3%
Rida Laraki
Good 11.7% 11.0%
Acceptable 14.8% 11.1%
Poor 14.2% 7.9%
To Reject 14.2% 31.8%
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
The domination paradox
National poll, 10 days before first-round, French presidential election, 2012. Merit profile :
Hollande : Sarkozy :
Outstanding 12.5% 9.6%
Excellent 16.2% 12.3%
Very Good 16.4% 16.3%
Good 11.7% 11.0%
Acceptable 14.8% 11.1%
Poor 14.2% 7.9%
To Reject 14.2% 31.8%
Possible opinion profile : Hollande : Sarkozy :
9.6% Exc. Outs.
12.3% V.Good Exc.
11.7% Good V.Good
4.6% Accept. V.Good
10.2% Accept. Good
5.9% Poor Accept.
14.2% Rej. Rej.
Hollande : Sarkozy :
0.8% Outs. Good
5.2% Outs. Accept.
6.5% Outs. Poor
1.4% Exc. Poor
5.2% Exc. Rej.
4.1% V.Good Rej.
8.3% Poor Rej.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
The domination paradox
National poll, 10 days before first-round, French presidential election, 2012. Merit profile :
Hollande : Sarkozy :
Outstanding 12.5% 9.6%
Excellent 16.2% 12.3%
Very Good 16.4% 16.3%
Good 11.7% 11.0%
Acceptable 14.8% 11.1%
Poor 14.2% 7.9%
To Reject 14.2% 31.8%
Possible opinion profile : Hollande : Sarkozy :
9.6% Exc. Outs.
12.3% V.Good Exc.
11.7% Good V.Good
4.6% Accept. V.Good
10.2% Accept. Good
5.9% Poor Accept.
14.2% Rej. Rej.
Hollande : Sarkozy :
0.8% Outs. Good
5.2% Outs. Accept.
6.5% Outs. Poor
1.4% Exc. Poor
5.2% Exc. Rej.
4.1% V.Good Rej.
8.3% Poor Rej.
Majority Rule :
Sarkozy : 54.3%
Hollande : 31.5%
Rida Laraki
Indifferent : 14.2%
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
The intensity problem
Dahl in his A Preface to Democratic Theory (1956) formally recognized the intensity problem :
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
The intensity problem
Dahl in his A Preface to Democratic Theory (1956) formally recognized the intensity problem : “What if the minority prefers its alternative much more passionately than the majority prefers a contrary alternative ? Does the majority principle still make sense ?”
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
The intensity problem
Dahl in his A Preface to Democratic Theory (1956) formally recognized the intensity problem : “What if the minority prefers its alternative much more passionately than the majority prefers a contrary alternative ? Does the majority principle still make sense ?” “If there is any case that might be considered the modern analogue to Madison’s implicit concept of tyranny, I suppose it is this one.”
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
The intensity problem
Dahl in his A Preface to Democratic Theory (1956) formally recognized the intensity problem : “What if the minority prefers its alternative much more passionately than the majority prefers a contrary alternative ? Does the majority principle still make sense ?” “If there is any case that might be considered the modern analogue to Madison’s implicit concept of tyranny, I suppose it is this one.” To solve the problem, Dahl proposes using an ordinal “intensity scale” obtained “simply by reference to some observable response, such as a statement of one’s feelings.”
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
May’s and Arrow’s axioms
A method of ranking is an asymmetric binary relation that compares any two candidates. It must satisfy the following axioms : A0∗ [Based on measures] A voter’s opinion is expressed by evaluating each candidate in an ordinal scale of grades Γ.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
May’s and Arrow’s axioms
A method of ranking is an asymmetric binary relation that compares any two candidates. It must satisfy the following axioms : A0∗ [Based on measures] A voter’s opinion is expressed by evaluating each candidate in an ordinal scale of grades Γ. A1 [Unrestricted Domain] All voter’s opinions are admissible. A2 [Anonymous] Permuting names of voters does not change the outcome. A3 [Neutral] Permuting names of candidates does not change the outcome. A4 [Monotone] If A B and one or more of A’s grades are raised then A B. A5 [Complete] For any two candidates either A B or A B (or both, implying A ≈ B). A6 [Transitive] If A B and B C then A C . A7 [Independence of irrelevant alternatives (IIA)] If A B then whatever candidates are dropped or adjoined A B.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Ranking methods based on measures
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Ranking methods based on measures
Theorem For variable n ≥ 1, infinitely many methods, based on measures, satisfy axioms A1 to A7. All depend only on the merit profile, and all respect domination.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Ranking methods based on measures
Theorem For variable n ≥ 1, infinitely many methods, based on measures, satisfy axioms A1 to A7. All depend only on the merit profile, and all respect domination.
Corollary : in cases where majority rule does not respect domination, all rules satisfying the basic axioms disagree with it.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Ranking methods based on measures
Theorem For variable n ≥ 1, infinitely many methods, based on measures, satisfy axioms A1 to A7. All depend only on the merit profile, and all respect domination.
Corollary : in cases where majority rule does not respect domination, all rules satisfying the basic axioms disagree with it. Why is it a bad property to disagree with majority rule in that case ?
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Ranking methods based on measures
Theorem For variable n ≥ 1, infinitely many methods, based on measures, satisfy axioms A1 to A7. All depend only on the merit profile, and all respect domination.
Corollary : in cases where majority rule does not respect domination, all rules satisfying the basic axioms disagree with it. Why is it a bad property to disagree with majority rule in that case ? When does majority rule works well ? (does not have the domination paradox)
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Polarization Definition : an opinion profil is polarized between a pair of candidates A and B if for every two voters i and j, if they disagree, they do in opposite directions : if i evaluates A higher than j, then i evaluates B lower than j.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Polarization Definition : an opinion profil is polarized between a pair of candidates A and B if for every two voters i and j, if they disagree, they do in opposite directions : if i evaluates A higher than j, then i evaluates B lower than j. Merit profile :
Hollande : Sarkozy :
Outstanding 12.5% 9.6%
Excellent 16.2% 12.3%
Very Good 16.4% 16.3%
Rida Laraki
Good 11.7% 11.0%
Acceptable 14.8% 11.1%
Poor 14.2% 7.9%
To Reject 14.2% 31.8%
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Polarization Definition : an opinion profil is polarized between a pair of candidates A and B if for every two voters i and j, if they disagree, they do in opposite directions : if i evaluates A higher than j, then i evaluates B lower than j. Merit profile :
Hollande : Sarkozy :
Outstanding 12.5% 9.6%
Excellent 16.2% 12.3%
Very Good 16.4% 16.3%
Good 11.7% 11.0%
Acceptable 14.8% 11.1%
Poor 14.2% 7.9%
To Reject 14.2% 31.8%
Polarized opinion profile : Hollande : Sarkozy :
12.5% Outs. Rej.
Hollande : Sarkozy :
5.0% Accept. Good
16.2% Exc. Rej. 9.8% Accept. V.Good
3.1% V.Good Rej. 6.5% Poor V.Good
Rida Laraki
7.9% V.Good Poor 7.7% Poor Exc.
5.4% V.Good Accept. 4.6% Rej. Exc.
5.7% Good Accept. 9.6% Rej. Outs.
Majority Judgment vs Majority Rule
6.0% Good. Good
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Polarization Definition : an opinion profil is polarized between a pair of candidates A and B if for every two voters i and j, if they disagree, they do in opposite directions : if i evaluates A higher than j, then i evaluates B lower than j. Merit profile :
Hollande : Sarkozy :
Outstanding 12.5% 9.6%
Excellent 16.2% 12.3%
Very Good 16.4% 16.3%
Good 11.7% 11.0%
Acceptable 14.8% 11.1%
Poor 14.2% 7.9%
To Reject 14.2% 31.8%
Polarized opinion profile : Hollande : Sarkozy :
12.5% Outs. Rej.
Hollande : Sarkozy :
5.0% Accept. Good
Holland : 50.8%
16.2% Exc. Rej. 9.8% Accept. V.Good
3.1% V.Good Rej. 6.5% Poor V.Good
7.9% V.Good Poor 7.7% Poor Exc.
Sarkozy : 43.2% Rida Laraki
5.4% V.Good Accept. 4.6% Rej. Exc.
5.7% Good Accept. 9.6% Rej. Outs.
Indifferent : 6.0%
Majority Judgment vs Majority Rule
6.0% Good. Good
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Statistical Polarization True opinion profile, Hollande-Sarkozy, 2012 French presidential poll :
S a r k o z y
Outs. Exc. V.G. Good Fair Poor Rej. Total
Outs. 0.14% 0.27% 0.27% 1.22% 1.63% 1.75% 7.19% 12.48%
Exc. 0.00% 1.09% 1.22% 1.09% 2.44% 2.58% 7.73% 16.15%
V.G. 0.41% 0.95% 2.04% 1.76% 2.58% 1.09% 7.60% 16.42%
Rida Laraki
Hollande Good 1.09% 2.17% 3.12% 1.76% 1.09% 0.27% 2.17% 11.67%
Fair 2.04% 2.71% 2.99% 2.85% 2.31% 0.54% 1.36% 14.79%
Poor 2.99% 2.71% 3.93% 1.63% 0.68% 0.81% 1.49% 14.25%
Majority Judgment vs Majority Rule
Rej. 2.99% 2.44% 2.71% 0.68% 0.41% 0.81% 4.21% 14.25%
Total 09.63% 12.35% 16.28% 10.99% 11.13% 07.87% 31.75%
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Statistical Polarization True opinion profile, Hollande-Sarkozy, 2012 French presidential poll :
S a r k o z y
Outs. Exc. V.G. Good Fair Poor Rej. Total
Outs. 0.14% 0.27% 0.27% 1.22% 1.63% 1.75% 7.19% 12.48%
Exc. 0.00% 1.09% 1.22% 1.09% 2.44% 2.58% 7.73% 16.15%
V.G. 0.41% 0.95% 2.04% 1.76% 2.58% 1.09% 7.60% 16.42%
Hollande Good 1.09% 2.17% 3.12% 1.76% 1.09% 0.27% 2.17% 11.67%
Fair 2.04% 2.71% 2.99% 2.85% 2.31% 0.54% 1.36% 14.79%
Poor 2.99% 2.71% 3.93% 1.63% 0.68% 0.81% 1.49% 14.25%
Rej. 2.99% 2.44% 2.71% 0.68% 0.41% 0.81% 4.21% 14.25%
Cumulative distibutions of Hollande’s grades for each of Sarkozy’s grades
S a r k o z y
Outs. Exc. V.Good Good Fair Poor Rej.
Outs. 01.41% 02.20% 01.67% 11.11% 14.63% 22.41% 22.65%
Exc. 01.41% 10.99% 09.17% 20.99% 36.58% 55.17% 47.01%
V.Good 05.64% 18.68% 21.67% 37.04% 59.75% 68.96% 70.94%
Rida Laraki
Hollande Good 16.91% 36.26% 40.84% 53.09% 69.51% 72.41% 77.78%
Fair 38.04% 58.24% 59.17% 79.02% 90.24% 79.31% 82.05%
Poor 69.03% 80.23% 83.34% 93.83% 96.34% 89.65% 86.75%
Majority Judgment vs Majority Rule
Rej. 100% 100% 100% 100% 100% 100% 100%
Total 09.63% 12.35% 16.28% 10.99% 11.13% 07.87% 31.75%
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Consistency with majority rule on polarized domains Theorem When the language of grades is sufficiently rich, and if the opinion profile is polarized,
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Consistency with majority rule on polarized domains Theorem When the language of grades is sufficiently rich, and if the opinion profile is polarized, majority rule respects domination (no domination paradox !).
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Consistency with majority rule on polarized domains Theorem When the language of grades is sufficiently rich, and if the opinion profile is polarized, majority rule respects domination (no domination paradox !). Definition : a method of ranking is consistent with majority rule on polarized pairs if it gives the same ranking as majority rule between every polarized pair of candidates, whenever majority rule is decisive.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Consistency with majority rule on polarized domains Theorem When the language of grades is sufficiently rich, and if the opinion profile is polarized, majority rule respects domination (no domination paradox !). Definition : a method of ranking is consistent with majority rule on polarized pairs if it gives the same ranking as majority rule between every polarized pair of candidates, whenever majority rule is decisive. Theorem When the language of grades is sufficiently rich, a method of ranking based on measures satisfying basic axioms A1 to A7 is consistent with the majority rule on polarized pairs if and only if it coincide with the majority-gauge.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Consistency with majority rule on polarized domains Theorem When the language of grades is sufficiently rich, and if the opinion profile is polarized, majority rule respects domination (no domination paradox !). Definition : a method of ranking is consistent with majority rule on polarized pairs if it gives the same ranking as majority rule between every polarized pair of candidates, whenever majority rule is decisive. Theorem When the language of grades is sufficiently rich, a method of ranking based on measures satisfying basic axioms A1 to A7 is consistent with the majority rule on polarized pairs if and only if it coincide with the majority-gauge. Putting aside Dahl’s desiderata, why is it a good axiom to coincide with majority rule on polarized pairs ?
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Consistency with majority rule on polarized domains Theorem When the language of grades is sufficiently rich, and if the opinion profile is polarized, majority rule respects domination (no domination paradox !). Definition : a method of ranking is consistent with majority rule on polarized pairs if it gives the same ranking as majority rule between every polarized pair of candidates, whenever majority rule is decisive. Theorem When the language of grades is sufficiently rich, a method of ranking based on measures satisfying basic axioms A1 to A7 is consistent with the majority rule on polarized pairs if and only if it coincide with the majority-gauge. Putting aside Dahl’s desiderata, why is it a good axiom to coincide with majority rule on polarized pairs ? Because it is in this situation that voters have the greatest temptation to vote strategically and MR is stable to strategic voting. Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Strategy proofness for methods based on measures A0∗ [Based on measures] A voter’s opinion is expressed by evaluating each candidate in a scale of grades Γ. A1 [Unrestricted Domain] All voters opinions are admissible. A2 [Anonymous] Interchanging the names of voters does not change the outcome. A3 [Neutral] Interchanging the names of candidates does not change the outcome. A4 [Monotone] If A B and one or more of A’s grades are raised then A B. A5 [Complete] For any two candidates either A B or A B (or both, implying A ≈ B).
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Strategy proofness for methods based on measures A0∗ [Based on measures] A voter’s opinion is expressed by evaluating each candidate in a scale of grades Γ. A1 [Unrestricted Domain] All voters opinions are admissible. A2 [Anonymous] Interchanging the names of voters does not change the outcome. A3 [Neutral] Interchanging the names of candidates does not change the outcome. A4 [Monotone] If A B and one or more of A’s grades are raised then A B. A5 [Complete] For any two candidates either A B or A B (or both, implying A ≈ B). Theorem When there are n = 2 candidates, majority rule is the unique strategy proof method based on measures that satisfies axioms A1 to A5.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Strategy proofness for methods based on measures A0∗ [Based on measures] A voter’s opinion is expressed by evaluating each candidate in a scale of grades Γ. A1 [Unrestricted Domain] All voters opinions are admissible. A2 [Anonymous] Interchanging the names of voters does not change the outcome. A3 [Neutral] Interchanging the names of candidates does not change the outcome. A4 [Monotone] If A B and one or more of A’s grades are raised then A B. A5 [Complete] For any two candidates either A B or A B (or both, implying A ≈ B). Theorem When there are n = 2 candidates, majority rule is the unique strategy proof method based on measures that satisfies axioms A1 to A5. Theorem For variable n ≥ 1, no method based on measures satisfy axioms A1 to A7 is strategy proof.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Strategy proofness for methods based on measures A0∗ [Based on measures] A voter’s opinion is expressed by evaluating each candidate in a scale of grades Γ. A1 [Unrestricted Domain] All voters opinions are admissible. A2 [Anonymous] Interchanging the names of voters does not change the outcome. A3 [Neutral] Interchanging the names of candidates does not change the outcome. A4 [Monotone] If A B and one or more of A’s grades are raised then A B. A5 [Complete] For any two candidates either A B or A B (or both, implying A ≈ B). Theorem When there are n = 2 candidates, majority rule is the unique strategy proof method based on measures that satisfies axioms A1 to A5. Theorem For variable n ≥ 1, no method based on measures satisfy axioms A1 to A7 is strategy proof. When the language of grades is rich, Majority-Gauge is partially strategy proof on the entire domain, and Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending Dahl’s May’s intensity Axioms problem to n Ranking ≥ 3 [based methods on comparions] based on measures ExtendingS
Strategy proofness for methods based on measures A0∗ [Based on measures] A voter’s opinion is expressed by evaluating each candidate in a scale of grades Γ. A1 [Unrestricted Domain] All voters opinions are admissible. A2 [Anonymous] Interchanging the names of voters does not change the outcome. A3 [Neutral] Interchanging the names of candidates does not change the outcome. A4 [Monotone] If A B and one or more of A’s grades are raised then A B. A5 [Complete] For any two candidates either A B or A B (or both, implying A ≈ B). Theorem When there are n = 2 candidates, majority rule is the unique strategy proof method based on measures that satisfies axioms A1 to A5. Theorem For variable n ≥ 1, no method based on measures satisfy axioms A1 to A7 is strategy proof. When the language of grades is rich, Majority-Gauge is partially strategy proof on the entire domain, and is the unique strategy proof on the domain of polarized pairs. Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
1
Majority judgment method Inspired by practice Majority judgment for small jury Majority Judgment for a large electorate
2
May’s axioms for n = 2 candidates
3
Extending May’s Axioms to n ≥ 3 [based on comparions] Condorcet and Arrow Paradoxes Arrow’s Theorem
4
Extending May’s axioms to n ≥ 1 candidates [based on measures] Dahl’s intensity problem Ranking methods based on measures Strategy proofness and second characterization of MJ
5
Scale and language dependency
6
Conclusion and references
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Scale dependency Definition : A method based on measures is scale-stable if when it ranks A above B, it does the same when two or more neighboring grades are merged.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Scale dependency Definition : A method based on measures is scale-stable if when it ranks A above B, it does the same when two or more neighboring grades are merged. Theorem No good method is scale-stable on the entire domain.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Scale dependency Definition : A method based on measures is scale-stable if when it ranks A above B, it does the same when two or more neighboring grades are merged. Theorem No good method is scale-stable on the entire domain. Every good method is scale-stable when A dominates B.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Scale dependency Definition : A method based on measures is scale-stable if when it ranks A above B, it does the same when two or more neighboring grades are merged. Theorem No good method is scale-stable on the entire domain. Every good method is scale-stable when A dominates B. Conclusion : since richer scale means more information and so better decision, a scale must be as rich as possible.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Scale dependency Definition : A method based on measures is scale-stable if when it ranks A above B, it does the same when two or more neighboring grades are merged. Theorem No good method is scale-stable on the entire domain. Every good method is scale-stable when A dominates B. Conclusion : since richer scale means more information and so better decision, a scale must be as rich as possible. In a famous paper, George Miller in (Psychological Review, 1956) proved that 7 ± 2 grades is an optimal number in a human’s capacity for judgement.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Scale dependency Definition : A method based on measures is scale-stable if when it ranks A above B, it does the same when two or more neighboring grades are merged. Theorem No good method is scale-stable on the entire domain. Every good method is scale-stable when A dominates B. Conclusion : since richer scale means more information and so better decision, a scale must be as rich as possible. In a famous paper, George Miller in (Psychological Review, 1956) proved that 7 ± 2 grades is an optimal number in a human’s capacity for judgement. In our field experiments, 4 grades were few, 6 grades were sufficient No. of grades : 2007 : 2012 :
1 1% 1%
2 2% 6%
3 10% 13%
Rida Laraki
4 31% 31%
5 42% 36%
6 14% 13%
7 – 1%
Majority Judgment vs Majority Rule
Total 100% 100%
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Poll Opinion Way/Terra Nova, French presidential, April 12-16, 2012
Condorcetranking 1 Hollande 2 Bayrou 3 Sarkozy 4 Mélenchon 5 Le Pen
Hollande – 48.4% 46.1% 31.5% 35.9%
Bayrou 51.6% – 43.5% 40.6% 29.5%
Sarkozy 53.9% 56.5% – 49.5% 34.3%
Rida Laraki
Mélenchon 68.5% 59.4% 50.5% – 40.3%
Le Pen 64.1% 70.5% 65.7% 59.7% –
Majority Judgment vs Majority Rule
Bordaranking 1) 59.5% 2) 58.7% 3) 51.4% 4) 45.3% 5) 35.0%
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Poll Opinion Way-Terra Nova, French presidential, April 12-16, 2012.
Majority judgment 1 Hollande 2 Bayrou 3 Sarkozy 4 Mélenchon 5 Dupont-Aignan 6 Joly 7 Poutou 8 Le Pen 9 Arthaud 10 Cheminade
Majority grade Good Good Accept Accept Poor Poor Poor Poor Poor to Reject
Gauge +45.1% −40.7% +49.3% +42.5% +40.6% −38.5% −45.7% −47.6% −49.9% +48.0%
Rida Laraki
First-pastthe-post 1 28.6% 5 9.1% 2 27.3% 4 11.0% 7 1.5% 6 2.3% 8 1.2% 3 17.9% 9 0.7% 10 0.4%
Approval Voting 1 49.4% 3 39.2% 2 40.5% 4 39.1% 8 10.7% 6 26.7% 7 13.3% 5 27.4% 9 8.4% 10 3.2%
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Poll Opinion Way-Terra Nova, French presidential, April 12-16, 2012.
Majority judgment 1 Hollande 2 Bayrou 3 Sarkozy 4 Mélenchon 5 Dupont-Aignan 6 Joly 7 Poutou 8 Le Pen 9 Arthaud 10 Cheminade
Majority grade Good Good Accept Accept Poor Poor Poor Poor Poor to Reject
Gauge +45.1% −40.7% +49.3% +42.5% +40.6% −38.5% −45.7% −47.6% −49.9% +48.0%
First-pastthe-post 1 28.6% 5 9.1% 2 27.3% 4 11.0% 7 1.5% 6 2.3% 8 1.2% 3 17.9% 9 0.7% 10 0.4%
Approval Voting 1 49.4% 3 39.2% 2 40.5% 4 39.1% 8 10.7% 6 26.7% 7 13.3% 5 27.4% 9 8.4% 10 3.2%
Methods that ask more information (MJ-, Condorcet and Borda) have identical rankings and put Bayrou comfortably ahead of Sarkozy.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Poll Opinion Way-Terra Nova, French presidential, April 12-16, 2012.
Majority judgment 1 Hollande 2 Bayrou 3 Sarkozy 4 Mélenchon 5 Dupont-Aignan 6 Joly 7 Poutou 8 Le Pen 9 Arthaud 10 Cheminade
Majority grade Good Good Accept Accept Poor Poor Poor Poor Poor to Reject
Gauge +45.1% −40.7% +49.3% +42.5% +40.6% −38.5% −45.7% −47.6% −49.9% +48.0%
First-pastthe-post 1 28.6% 5 9.1% 2 27.3% 4 11.0% 7 1.5% 6 2.3% 8 1.2% 3 17.9% 9 0.7% 10 0.4%
Approval Voting 1 49.4% 3 39.2% 2 40.5% 4 39.1% 8 10.7% 6 26.7% 7 13.3% 5 27.4% 9 8.4% 10 3.2%
Methods that ask more information (MJ-, Condorcet and Borda) have identical rankings and put Bayrou comfortably ahead of Sarkozy. Methods that ask less information (first-past-the-post and AV) fail.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
1
Majority judgment method Inspired by practice Majority judgment for small jury Majority Judgment for a large electorate
2
May’s axioms for n = 2 candidates
3
Extending May’s Axioms to n ≥ 3 [based on comparions] Condorcet and Arrow Paradoxes Arrow’s Theorem
4
Extending May’s axioms to n ≥ 1 candidates [based on measures] Dahl’s intensity problem Ranking methods based on measures Strategy proofness and second characterization of MJ
5
Scale and language dependency
6
Conclusion and references
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Conclusion Majority rule says nothing when there is n = 1 candidate ; it may fail for n = 2 ; and fails for n ≥ 3 candidates.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Conclusion Majority rule says nothing when there is n = 1 candidate ; it may fail for n = 2 ; and fails for n ≥ 3 candidates. There is a method —based on ordinal measures—that meets May’s axioms and responds to Dahl’s requirements : majority judgment.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Conclusion Majority rule says nothing when there is n = 1 candidate ; it may fail for n = 2 ; and fails for n ≥ 3 candidates. There is a method —based on ordinal measures—that meets May’s axioms and responds to Dahl’s requirements : majority judgment. MJ is the unique that avoids Arrow and Condorcet paradoxes and best resists strategic manipulation.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Conclusion Majority rule says nothing when there is n = 1 candidate ; it may fail for n = 2 ; and fails for n ≥ 3 candidates. There is a method —based on ordinal measures—that meets May’s axioms and responds to Dahl’s requirements : majority judgment. MJ is the unique that avoids Arrow and Condorcet paradoxes and best resists strategic manipulation. MJ is closed to method used in practice (diving, skating, gymnastic).
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Conclusion Majority rule says nothing when there is n = 1 candidate ; it may fail for n = 2 ; and fails for n ≥ 3 candidates. There is a method —based on ordinal measures—that meets May’s axioms and responds to Dahl’s requirements : majority judgment. MJ is the unique that avoids Arrow and Condorcet paradoxes and best resists strategic manipulation. MJ is closed to method used in practice (diving, skating, gymnastic). It has been used to higher professors in several universities (Santiago, Ecole Polytechnique, Montpellier, Paris Dauphine), and associations (Eco-Festival, Nieman Fellows at Harvard University).
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Conclusion Majority rule says nothing when there is n = 1 candidate ; it may fail for n = 2 ; and fails for n ≥ 3 candidates. There is a method —based on ordinal measures—that meets May’s axioms and responds to Dahl’s requirements : majority judgment. MJ is the unique that avoids Arrow and Condorcet paradoxes and best resists strategic manipulation. MJ is closed to method used in practice (diving, skating, gymnastic). It has been used to higher professors in several universities (Santiago, Ecole Polytechnique, Montpellier, Paris Dauphine), and associations (Eco-Festival, Nieman Fellows at Harvard University). Terra Nova (a left progressist think thank) and Nouvelle Donne (a centrist political party) have included MJ in their recommendations for reforming the French electoral system.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Conclusion Majority rule says nothing when there is n = 1 candidate ; it may fail for n = 2 ; and fails for n ≥ 3 candidates. There is a method —based on ordinal measures—that meets May’s axioms and responds to Dahl’s requirements : majority judgment. MJ is the unique that avoids Arrow and Condorcet paradoxes and best resists strategic manipulation. MJ is closed to method used in practice (diving, skating, gymnastic). It has been used to higher professors in several universities (Santiago, Ecole Polytechnique, Montpellier, Paris Dauphine), and associations (Eco-Festival, Nieman Fellows at Harvard University). Terra Nova (a left progressist think thank) and Nouvelle Donne (a centrist political party) have included MJ in their recommendations for reforming the French electoral system. LaPrimaire.org used (October-November 2016) MJ to select the "candidat citoyen" for the 2017 French presidential election.
Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
Conclusion Majority rule says nothing when there is n = 1 candidate ; it may fail for n = 2 ; and fails for n ≥ 3 candidates. There is a method —based on ordinal measures—that meets May’s axioms and responds to Dahl’s requirements : majority judgment. MJ is the unique that avoids Arrow and Condorcet paradoxes and best resists strategic manipulation. MJ is closed to method used in practice (diving, skating, gymnastic). It has been used to higher professors in several universities (Santiago, Ecole Polytechnique, Montpellier, Paris Dauphine), and associations (Eco-Festival, Nieman Fellows at Harvard University). Terra Nova (a left progressist think thank) and Nouvelle Donne (a centrist political party) have included MJ in their recommendations for reforming the French electoral system. LaPrimaire.org used (October-November 2016) MJ to select the "candidat citoyen" for the 2017 French presidential election. It has been proposed to the Special Committee on Electoral Reform in Quebec City by the deputy Raymond Côté, in September 22, 2016. Rida Laraki
Majority Judgment vs Majority Rule
Majority judgment method May’s axioms for n = 2 candidates Extending May’s Axioms to n ≥ 3 [based on comparions] Extending
References
Michel Balinski and Rida Laraki 2016. Majority Judgment vs Majority Rule. Preprint — and —. 2014. “Judge : Don’t vote !” Operations Research. — and —. 2011. Majority Judgment : Measuring, Ranking, and Electing. MIT Press. — and —. 2007. A Theory of Measuring, Electing, and Ranking. PNAS USA. Terra Nova. 2011. “Rendre les élections aux lecteurs : le jugement majoritaire,” http ://www.tnova.fr/note/rendre-les- lections-aux-lecteurs-le-jugement-majoritaire
Rida Laraki
Majority Judgment vs Majority Rule
