Elections and Strategic Voting:
Condorcet and Borda
E. Maskin
Harvard University
The 2016 NSF/NBER/CEME Conference on
Mathematical Economics
John Hopkins University
October 30, 2016
• voting rule
2
• voting rule
method for choosing winning candidate on basis of
voters’ preferences (rankings, utility functions)
3
• voting rule
method for choosing winning candidate on basis of
voters’ preferences (rankings, utility functions)
• prominent examples
4
• voting rule
method for choosing winning candidate on basis of
voters’ preferences (rankings, utility functions)
• prominent examples
– Plurality Rule (MPs in Britain, members of Congress in
U.S.)
5
• voting rule
method for choosing winning candidate on basis of
voters’ preferences (rankings, utility functions)
• prominent examples
– Plurality Rule (MPs in Britain, members of Congress in
U.S.)
choose candidate ranked first by more voters than any
other
6
• voting rule
method for choosing winning candidate on basis of
voters’ preferences (rankings, utility functions)
• prominent examples
– Plurality Rule (MPs in Britain, members of Congress in
U.S.)
choose candidate ranked first by more voters than any
other
– Majority Rule (Condorcet Method)
7
• voting rule
method for choosing winning candidate on basis of
voters’ preferences (rankings, utility functions)
• prominent examples
– Plurality Rule (MPs in Britain, members of Congress in
U.S.)
choose candidate ranked first by more voters than any
other
– Majority Rule (Condorcet Method)
choose candidate preferred by majority to each other
candidate
8
− Run-off Voting (presidential elections in France)
9
− Run-off Voting (presidential elections in France)
• choose candidate ranked first by more voters than any
other, unless number of first-place rankings
less than majority
10
− Run-off Voting (presidential elections in France)
• choose candidate ranked first by more voters than any
other, unless number of first-place rankings
less than majority
among top 2 candidates, choose alternative preferred
by majority
11
− Run-off Voting (presidential elections in France)
• choose candidate ranked first by more voters than any
other, unless number of first-place rankings
less than majority
among top 2 candidates, choose alternative preferred
by majority
− Rank-Order Voting (Borda Count)
12
− Run-off Voting (presidential elections in France)
• choose candidate ranked first by more voters than any
other, unless number of first-place rankings
less than majority
among top 2 candidates, choose alternative preferred
by majority
− Rank-Order Voting (Borda Count)
• candidate assigned 1 point every time some voter ranks
her first, 2 points every time ranked second, etc.
13
− Run-off Voting (presidential elections in France)
• choose candidate ranked first by more voters than any
other, unless number of first-place rankings
less than majority
among top 2 candidates, choose alternative preferred
by majority
− Rank-Order Voting (Borda Count)
• candidate assigned 1 point every time some voter ranks
her first, 2 points every time ranked second, etc.
• choose candidate with lowest point total
14
− Run-off Voting (presidential elections in France)
• choose candidate ranked first by more voters than any
other, unless number of first-place rankings
less than majority
among top 2 candidates, choose alternative preferred
by majority
− Rank-Order Voting (Borda Count)
• candidate assigned 1 point every time some voter ranks
her first, 2 points every time ranked second, etc.
• choose candidate with lowest point total
− Utilitarian Principle
15
− Run-off Voting (presidential elections in France)
• choose candidate ranked first by more voters than any
other, unless number of first-place rankings
less than majority
among top 2 candidates, choose alternative preferred
by majority
− Rank-Order Voting (Borda Count)
• candidate assigned 1 point every time some voter ranks
her first, 2 points every time ranked second, etc.
• choose candidate with lowest point total
− Utilitarian Principle
• choose candidate who maximizes sum of voters’
utilities
16
• Which voting rule to adopt?
17
• Which voting rule to adopt?
• Answer depends on what one wants in voting rule
18
• Which voting rule to adopt?
• Answer depends on what one wants in voting rule
– can specify criteria (axioms) voting rule should satisfy
19
• Which voting rule to adopt?
• Answer depends on what one wants in voting rule
– can specify criteria (axioms) voting rule should satisfy
– see which rules best satisfy them
20
• Which voting rule to adopt?
• Answer depends on what one wants in voting rule
– can specify criteria (axioms) voting rule should satisfy
– see which rules best satisfy them
• One important criterion: nonmanipulability
21
• Which voting rule to adopt?
• Answer depends on what one wants in voting rule
– can specify criteria (axioms) voting rule should satisfy
– see which rules best satisfy them
• One important criterion: nonmanipulability
– voters shouldn’t have incentive to misrepresent
preferences, i.e., vote strategically
22
• Which voting rule to adopt?
• Answer depends on what one wants in voting rule
– can specify criteria (axioms) voting rule should satisfy
– see which rules best satisfy them
• One important criterion: nonmanipulability
– voters shouldn’t have incentive to misrepresent
preferences, i.e., vote strategically
– otherwise
23
• Which voting rule to adopt?
• Answer depends on what one wants in voting rule
– can specify criteria (axioms) voting rule should satisfy
– see which rules best satisfy them
• One important criterion: nonmanipulability
– voters shouldn’t have incentive to misrepresent
preferences, i.e., vote strategically
– otherwise
not implementing intended voting rule
24
• Which voting rule to adopt?
• Answer depends on what one wants in voting rule
– can specify criteria (axioms) voting rule should satisfy
– see which rules best satisfy them
• One important criterion: nonmanipulability
– voters shouldn’t have incentive to misrepresent
preferences, i.e., vote strategically
– otherwise
not implementing intended voting rule
decision problem for voters may be hard
25
• But basic negative result
Gibbard-Satterthwaite (GS) theorem
26
• But basic negative result
Gibbard-Satterthwaite (GS) theorem
– if 3 or more candidates, no voting rule is always
nonmanipulable
(except for dictatorial rules - - where one voter has all
the power)
27
• But basic negative result
Gibbard-Satterthwaite (GS) theorem
– if 3 or more candidates, no voting rule is always
nonmanipulable
(except for dictatorial rules - - where one voter has all
the power)
• Still, GS overly pessimistic
28
• But basic negative result
Gibbard-Satterthwaite (GS) theorem
– if 3 or more candidates, no voting rule is always
nonmanipulable
(except for dictatorial rules - - where one voter has all
the power)
• Still, GS overly pessimistic
– requires that voting rule never be manipulable
29
• But basic negative result
Gibbard-Satterthwaite (GS) theorem
– if 3 or more candidates, no voting rule is always
nonmanipulable
(except for dictatorial rules - - where one voter has all
the power)
• Still, GS overly pessimistic
– requires that voting rule never be manipulable
– but some circumstances where manipulation can occur
may be unlikely
30
• But basic negative result
Gibbard-Satterthwaite (GS) theorem
– if 3 or more candidates, no voting rule is always
nonmanipulable
(except for dictatorial rules - - where one voter has all
the power)
• Still, GS overly pessimistic
– requires that voting rule never be manipulable
– but some circumstances where manipulation can occur
may be unlikely
• In any case, natural question:
31
• But basic negative result
Gibbard-Satterthwaite (GS) theorem
– if 3 or more candidates, no voting rule is always
nonmanipulable
(except for dictatorial rules - - where one voter has all
the power)
• Still, GS overly pessimistic
– requires that voting rule never be manipulable
– but some circumstances where manipulation can occur
may be unlikely
• In any case, natural question:
Which (reasonable) voting rule(s) nonmanipulable most
often?
32
• But basic negative result
Gibbard-Satterthwaite (GS) theorem
– if 3 or more candidates, no voting rule is always
nonmanipulable
(except for dictatorial rules - - where one voter has all
the power)
• Still, GS overly pessimistic
– requires that voting rule never be manipulable
– but some circumstances where manipulation can occur
may be unlikely
• In any case, natural question:
Which (reasonable) voting rule(s) nonmanipulable most
often?
• Paper tries to answer question
33
• X = finite set of candidates
34
• X = finite set of candidates
• society consists of a continuum of voters [0,1]
35
• X = finite set of candidates
• society consists of a continuum of voters [0,1]
– typical voter i ∈ [ 0,1]
36
• X = finite set of candidates
• society consists of a continuum of voters [0,1]
– typical voter i ∈ [ 0,1]
– reason for continuum clear soon
37
• X = finite set of candidates
• society consists of a continuum of voters [0,1]
– typical voter i ∈ [ 0,1]
– reason for continuum clear soon
• utility function for voter i U i : X → 
38
• X = finite set of candidates
• society consists of a continuum of voters [0,1]
– typical voter i ∈ [ 0,1]
– reason for continuum clear soon
• utility function for voter i U i : X → 
– restrict attention to strict utility functions
if x ≠ y, then U i ( x ) ≠ U i ( y )
U X = set of strict utility functions
39
• X = finite set of candidates
• society consists of a continuum of voters [0,1]
– typical voter i ∈ [ 0,1]
– reason for continuum clear soon
• utility function for voter i U i : X → 
– restrict attention to strict utility functions
if x ≠ y, then U i ( x ) ≠ U i ( y )
U X = set of strict utility functions
• profile U  - - specification of each individual's utility
function
40
• voting rule F
for all profiles U  and all Y ⊆ X ,
F (U  , Y ) ∈ Y
41
• voting rule F
for all profiles U  and all Y ⊆ X ,
F (U  , Y ) ∈ Y
− Y is ballot
42
• voting rule F
for all profiles U  and all Y ⊆ X ,
F (U  , Y ) ∈ Y
− Y is ballot
− F (U  , Y ) =
optimal candidate in Y if profile
is U 
43
• voting rule F
for all profiles U  and all Y ⊆ X ,
F (U  , Y ) ∈ Y
− Y is ballot
− F (U  , Y ) =
optimal candidate in Y if profile
is U 
• definition isn’t quite right - - ignores ties
44
• voting rule F
for all profiles U  and all Y ⊆ X ,
F (U  , Y ) ∈ Y
− Y is ballot
− F (U  , Y ) =
optimal candidate in Y if profile
is U 
• definition isn’t quite right - - ignores ties
– with plurality rule, might be two candidates who are both ranked
first the most
45
• voting rule F
for all profiles U  and all Y ⊆ X ,
F (U  , Y ) ∈ Y
− Y is ballot
− F (U  , Y ) =
optimal candidate in Y if profile
is U 
• definition isn’t quite right - - ignores ties
– with plurality rule, might be two candidates who are both ranked
first the most
– with rank-order voting, might be two candidates who each get
lowest number of points
46
• voting rule F
for all profiles U  and all Y ⊆ X ,
F (U  , Y ) ∈ Y
− Y is ballot
− F (U  , Y ) =
optimal candidate in Y if profile
is U 
• definition isn’t quite right - - ignores ties
– with plurality rule, might be two candidates who are both ranked
first the most
– with rank-order voting, might be two candidates who each get
lowest number of points
• But exact ties unlikely with many voters
47
• voting rule F
for all profiles U  and all Y ⊆ X ,
F (U  , Y ) ∈ Y
− Y is ballot
− F (U  , Y ) =
optimal candidate in Y if profile
is U 
• definition isn’t quite right - - ignores ties
– with plurality rule, might be two candidates who are both ranked
first the most
– with rank-order voting, might be two candidates who each get
lowest number of points
• But exact ties unlikely with many voters
– with continuum, ties are nongeneric
48
• voting rule F
for all profiles U  and all Y ⊆ X ,
F (U  , Y ) ∈ Y
− Y is ballot
− F (U  , Y ) =
optimal candidate in Y if profile
is U 
• definition isn’t quite right - - ignores ties
– with plurality rule, might be two candidates who are both ranked
first the most
– with rank-order voting, might be two candidates who each get
lowest number of points
• But exact ties unlikely with many voters
– with continuum, ties are nongeneric
• so, correct definition:
for generic profile U  and all Y ⊆ X
F (U  , Y ) ∈ Y
49
plurality rule:
50
plurality rule:
=
F P (U  , Y )
{a µ {i U ( a ) ≥ U ( b ) for all b}
i
i
}
≥ µ {i U i ( a′) ≥ U i ( b ) for all b} for all a′
51
plurality rule:
=
F P (U  , Y )
{a µ {i U ( a ) ≥ U ( b ) for all b}
i
i
}
≥ µ {i U i ( a′) ≥ U i ( b ) for all b} for all a′
majority rule:
52
plurality rule:
=
F P (U  , Y )
{a µ {i U ( a ) ≥ U ( b ) for all b}
i
i
}
≥ µ {i U i ( a′) ≥ U i ( b ) for all b} for all a′
majority rule:
F C (U=
 ,Y )
{a µ {i U ( a ) ≥ U ( b )} ≥
i
i
1
2
}
for all b
53
plurality rule:
=
F P (U  , Y )
{a µ {i U ( a ) ≥ U ( b ) for all b}
i
i
}
≥ µ {i U i ( a′) ≥ U i ( b ) for all b} for all a′
majority rule:
F C (U=
 ,Y )
{a µ {i U ( a ) ≥ U ( b )} ≥
i
i
1
2
}
for all b
rank-order voting:
54
plurality rule:
=
F P (U  , Y )
{a µ {i U ( a ) ≥ U ( b ) for all b}
i
i
}
≥ µ {i U i ( a′) ≥ U i ( b ) for all b} for all a′
majority rule:
F C (U=
 ,Y )
{a µ {i U ( a ) ≥ U ( b )} ≥
i
i
1
2
}
for all b
rank-order voting:
=
F B (U  , Y )
{a ∫ r
Ui
( a ) d µ ( i ) ≤ ∫ rU ( b ) d µ ( i ) for all b},
i
where
rUi ( a ) # {b U i ( b ) ≥ U i ( a )}
=
55
plurality rule:
=
F P (U  , Y )
{a µ {i U ( a ) ≥ U ( b ) for all b}
i
i
}
≥ µ {i U i ( a′) ≥ U i ( b ) for all b} for all a′
majority rule:
F C (U=
 ,Y )
{a µ {i U ( a ) ≥ U ( b )} ≥
i
i
1
2
}
for all b
rank-order voting:
=
F B (U  , Y )
{a ∫ r
Ui
( a ) d µ ( i ) ≤ ∫ rU ( b ) d µ ( i ) for all b},
i
where
rUi ( a ) # {b U i ( b ) ≥ U i ( a )}
=
utilitarian principle:
56
plurality rule:
=
F P (U  , Y )
{a µ {i U ( a ) ≥ U ( b ) for all b}
i
i
}
≥ µ {i U i ( a′) ≥ U i ( b ) for all b} for all a′
majority rule:
F C (U=
 ,Y )
{a µ {i U ( a ) ≥ U ( b )} ≥
i
i
1
2
}
for all b
rank-order voting:
=
F B (U  , Y )
{a ∫ r
Ui
( a ) d µ ( i ) ≤ ∫ rU ( b ) d µ ( i ) for all b},
i
where
rUi ( a ) # {b U i ( b ) ≥ U i ( a )}
=
utilitarian principle:
=
F U (U  , Y )
{a ∫ U ( a ) d µ (i ) ≥∫ U ( b ) d µ (i ) for all b}
i
i
57
What properties should reasonable voting rule satisfy?
58
What properties should reasonable voting rule satisfy?
• Pareto Property (P): if U i ( x ) > U i ( y ) for all i
and x ∈ Y , then y ≠ F (U  , Y )
59
What properties should reasonable voting rule satisfy?
• Pareto Property (P): if U i ( x ) > U i ( y ) for all i
and x ∈ Y , then y ≠ F (U  , Y )
– if everybody prefers x to y, y should not be chosen
60
What properties should reasonable voting rule satisfy?
• Pareto Property (P): if U i ( x ) > U i ( y ) for all i
and x ∈ Y , then y ≠ F (U  , Y )
– if everybody prefers x to y, y should not be chosen
• Anonymity (A): suppose π : [ 0,1] → [ 0,1] measure-preserving
permutation. If U iπ = U π (i ) for all i, then
61
What properties should reasonable voting rule satisfy?
• Pareto Property (P): if U i ( x ) > U i ( y ) for all i
and x ∈ Y , then y ≠ F (U  , Y )
– if everybody prefers x to y, y should not be chosen
• Anonymity (A): suppose π : [ 0,1] → [ 0,1] measure-preserving
permutation. If U iπ = U π (i ) for all i, then
F (U π , Y ) = F (U  , Y ) for all Y
62
What properties should reasonable voting rule satisfy?
• Pareto Property (P): if U i ( x ) > U i ( y ) for all i
and x ∈ Y , then y ≠ F (U  , Y )
– if everybody prefers x to y, y should not be chosen
• Anonymity (A): suppose π : [ 0,1] → [ 0,1] measure-preserving
permutation. If U iπ = U π (i ) for all i, then
F (U π , Y ) = F (U  , Y ) for all Y
– candidate chosen depends only on voters’ preferences and not who
has those preferences
63
What properties should reasonable voting rule satisfy?
• Pareto Property (P): if U i ( x ) > U i ( y ) for all i
and x ∈ Y , then y ≠ F (U  , Y )
– if everybody prefers x to y, y should not be chosen
• Anonymity (A): suppose π : [ 0,1] → [ 0,1] measure-preserving
permutation. If U iπ = U π (i ) for all i, then
F (U π , Y ) = F (U  , Y ) for all Y
– candidate chosen depends only on voters’ preferences and not who
has those preferences
– voters treated symmetrically
64
• Neutrality (N): Suppose ρ : Y → Y permutation.
65
• Neutrality (N): Suppose ρ : Y → Y permutation.
If U iρ ,Y ( ρ ( x ) ) >U iρ ,Y ( ρ ( y ) ) ⇔ U i ( x ) > U i ( y ) for all x, y, i,
66
• Neutrality (N): Suppose ρ : Y → Y permutation.
If U iρ ,Y ( ρ ( x ) ) >U iρ ,Y ( ρ ( y ) ) ⇔ U i ( x ) > U i ( y ) for all x, y, i,
then
67
• Neutrality (N): Suppose ρ : Y → Y permutation.
If U iρ ,Y ( ρ ( x ) ) >U iρ ,Y ( ρ ( y ) ) ⇔ U i ( x ) > U i ( y ) for all x, y, i,
then
F (U ρ ,Y , Y ) = ρ ( F (U  , Y ) ) .
68
• Neutrality (N): Suppose ρ : Y → Y permutation.
If U iρ ,Y ( ρ ( x ) ) >U iρ ,Y ( ρ ( y ) ) ⇔ U i ( x ) > U i ( y ) for all x, y, i,
then
F (U ρ ,Y , Y ) = ρ ( F (U  , Y ) ) .
– candidates treated symmetrically
69
• Neutrality (N): Suppose ρ : Y → Y permutation.
If U iρ ,Y ( ρ ( x ) ) >U iρ ,Y ( ρ ( y ) ) ⇔ U i ( x ) > U i ( y ) for all x, y, i,
then
F (U ρ ,Y , Y ) = ρ ( F (U  , Y ) ) .
– candidates treated symmetrically
• All four voting rules – plurality, majority, rank-order,
utilitarian – satisfy P, A, N
70
• Neutrality (N): Suppose ρ : Y → Y permutation.
If U iρ ,Y ( ρ ( x ) ) >U iρ ,Y ( ρ ( y ) ) ⇔ U i ( x ) > U i ( y ) for all x, y, i,
then
F (U ρ ,Y , Y ) = ρ ( F (U  , Y ) ) .
– candidates treated symmetrically
• All four voting rules – plurality, majority, rank-order,
utilitarian – satisfy P, A, N
• Next axiom most controversial
still
71
• Neutrality (N): Suppose ρ : Y → Y permutation.
If U iρ ,Y ( ρ ( x ) ) >U iρ ,Y ( ρ ( y ) ) ⇔ U i ( x ) > U i ( y ) for all x, y, i,
then
F (U ρ ,Y , Y ) = ρ ( F (U  , Y ) ) .
– candidates treated symmetrically
• All four voting rules – plurality, majority, rank-order,
utilitarian – satisfy P, A, N
• Next axiom most controversial
still
• has quite compelling justification
72
• Neutrality (N): Suppose ρ : Y → Y permutation.
If U iρ ,Y ( ρ ( x ) ) >U iρ ,Y ( ρ ( y ) ) ⇔ U i ( x ) > U i ( y ) for all x, y, i,
then
F (U ρ ,Y , Y ) = ρ ( F (U  , Y ) ) .
– candidates treated symmetrically
• All four voting rules – plurality, majority, rank-order,
utilitarian – satisfy P, A, N
• Next axiom most controversial
still
• has quite compelling justification
• invoked by both Arrow (1951) and Nash (1950)
73
• Independence of Irrelevant Candidates (I):
74
• Independence of Irrelevant Candidates (I):
if x F (U  , Y ) and x ∈ Y ′ ⊆ Y
=
75
• Independence of Irrelevant Candidates (I):
if x F (U  , Y ) and x ∈ Y ′ ⊆ Y
=
then
76
• Independence of Irrelevant Candidates (I):
if x F (U  , Y ) and x ∈ Y ′ ⊆ Y
=
then
x = F (U  , Y ′ )
77
• Independence of Irrelevant Candidates (I):
if x F (U  , Y ) and x ∈ Y ′ ⊆ Y
=
then
x = F (U  , Y ′ )
– if x chosen and some non-chosen candidates removed, x still
chosen
78
• Independence of Irrelevant Candidates (I):
if x F (U  , Y ) and x ∈ Y ′ ⊆ Y
=
then
x = F (U  , Y ′ )
– if x chosen and some non-chosen candidates removed, x still
chosen
– Nash formulation (rather than Arrow)
79
• Independence of Irrelevant Candidates (I):
if x F (U  , Y ) and x ∈ Y ′ ⊆ Y
=
then
x = F (U  , Y ′ )
– if x chosen and some non-chosen candidates removed, x still
chosen
– Nash formulation (rather than Arrow)
– no “spoilers” (e.g. Nader in 2000 U.S. presidential election, Le Pen
in 2002 French presidential election)
80
• Majority rule and utilitarianism satisfy I, but
others don’t:
81
• Majority rule and utilitarianism satisfy I, but
others don’t:
– plurality rule
.35
x
y
z
.33
y
z
x
.32
z
y
x
F P (U  , {x, y , z}) = x
F P (U  , {x, y}) = y
82
• Majority rule and utilitarianism satisfy I, but
others don’t:
– plurality rule
.35
x
y
z
.33
y
z
x
.32
z
y
x
F P (U  , {x, y , z}) = x
F P (U  , {x, y}) = y
– rank-order voting
.55
x
y
z
.45
y
z
x
F B (U  , {x, y , z}) = y
F B (U  , {x, y}) = x
83
Final Axiom:
84
Final Axiom:
• Nonmanipulability (NM):
85
Final Axiom:
• Nonmanipulability (NM):
if x
F=
(U  , Y ) and x′ F (U ′, Y ) ,
where
=
U ′j U j for all j ∉ C ⊆ [ 0,1]
86
Final Axiom:
• Nonmanipulability (NM):
if x
F=
(U  , Y ) and x′ F (U ′, Y ) ,
where
=
U ′j U j for all j ∉ C ⊆ [ 0,1]
then
87
Final Axiom:
• Nonmanipulability (NM):
if x
F=
(U  , Y ) and x′ F (U ′, Y ) ,
where
=
U ′j U j for all j ∉ C ⊆ [ 0,1]
then
U i ( x ) > U i ( x′ ) for some i ∈ C
88
Final Axiom:
• Nonmanipulability (NM):
F=
(U  , Y ) and x′ F (U ′, Y ) ,
if x
where
=
U ′j U j for all j ∉ C ⊆ [ 0,1]
then
U i ( x ) > U i ( x′ ) for some i ∈ C
– the members of coalition C can’t all gain from misrepresenting
utility functions as U i′
89
• NM implies voting rule must be ordinal (no cardinal
information used)
90
• NM implies voting rule must be ordinal (no cardinal
information used)
• F is ordinal if whenever, for profiles U  and U ′ ,
91
• NM implies voting rule must be ordinal (no cardinal
information used)
• F is ordinal if whenever, for profiles U  and U ′ ,
U i ( x ) > U i ( y ) ⇔ U i′ ( x ) > U i′ ( y ) for all i, x, y
92
• NM implies voting rule must be ordinal (no cardinal
information used)
• F is ordinal if whenever, for profiles U  and U ′ ,
U i ( x ) > U i ( y ) ⇔ U i′ ( x ) > U i′ ( y ) for all i, x, y
(*)
F (U  , Y ) = F (U ′, Y ) for all Y
93
• NM implies voting rule must be ordinal (no cardinal
information used)
• F is ordinal if whenever, for profiles U  and U ′ ,
U i ( x ) > U i ( y ) ⇔ U i′ ( x ) > U i′ ( y ) for all i, x, y
(*)
F (U  , Y ) = F (U ′, Y ) for all Y
• Lemma: If F satisfies NM, F ordinal
94
• NM implies voting rule must be ordinal (no cardinal
information used)
• F is ordinal if whenever, for profiles U  and U ′ ,
U i ( x ) > U i ( y ) ⇔ U i′ ( x ) > U i′ ( y ) for all i, x, y
(*)
F (U  , Y ) = F (U ′, Y ) for all Y
• Lemma: If F satisfies NM, F ordinal
• NM rules out utilitarianism
95
But majority rule also violates NM
96
But majority rule also violates NM
• F C not even always defined
.35
x
y
z
.33
y
z
x
.32
z
x
y
F C (U  , { x, y, z}) = ∅
97
But majority rule also violates NM
• F C not even always defined
.35
x
y
z
.33
y
z
x
.32
z
x
y
F C (U  , { x, y, z}) = ∅
– example of Condorcet cycle
98
But majority rule also violates NM
• F C not even always defined
.35
x
y
z
.33
y
z
x
.32
z
x
y
F C (U  , { x, y, z}) = ∅
– example of Condorcet cycle
– F C must be extended to Condorcet cycles
99
But majority rule also violates NM
• F C not even always defined
.35
x
y
z
.33
y
z
x
.32
z
x
y
F C (U  , { x, y, z}) = ∅
– example of Condorcet cycle
– F C must be extended to Condorcet cycles
– one possibility
 F C (U  , Y ) , if nonempty

F C / B (U  , Y ) = 
 F B (U  , Y ) , otherwise
(Black's method)
100
But majority rule also violates NM
• F C not even always defined
.35
x
y
z
.33
y
z
x
.32
z
x
y
F C (U  , { x, y, z}) = ∅
– example of Condorcet cycle
– F C must be extended to Condorcet cycles
– one possibility
 F C (U  , Y ) , if nonempty

F C / B (U  , Y ) = 
 F B (U  , Y ) , otherwise
–
(Black's method)
extensions make F C vulnerable to manipulation
101
But majority rule also violates NM
• F C not even always defined
.35
x
y
z
.33
y
z
x
.32
z
x
y
F C (U  , { x, y, z}) = ∅
– example of Condorcet cycle
– F C must be extended to Condorcet cycles
– one possibility
 F C (U  , Y ) , if nonempty

F C / B (U  , Y ) = 
 F B (U  , Y ) , otherwise
–
–
(Black's method)
extensions make F C vulnerable to manipulation
.35
x
y
z
.33
y
z
x
z
y
x
.32
z
x
y
F C / B (U  , { x, y, z}) = x
F C / B (U ′, { x, y, z}) = z
102
Theorem: There exists no voting rule satisfying
P,A,N,I and NM
103
Theorem: There exists no voting rule satisfying
P,A,N,I and NM
Proof: similar to that of GS
104
Theorem: There exists no voting rule satisfying
P,A,N,I and NM
Proof: similar to that of GS
overly pessimistic - - many cases in which some rankings
unlikely
105
Lemma: Majority rule satisfies all 5 properties if and only if
preferences restricted to domain with no Condorcet cycles
106
Lemma: Majority rule satisfies all 5 properties if and only if
preferences restricted to domain with no Condorcet cycles
When can we rule out Condorcet cycles?
107
Lemma: Majority rule satisfies all 5 properties if and only if
preferences restricted to domain with no Condorcet cycles
When can we rule out Condorcet cycles?
• preferences single-peaked
2000 US election
•
•
•
Nader
Gore
Bush
108
Lemma: Majority rule satisfies all 5 properties if and only if
preferences restricted to domain with no Condorcet cycles
When can we rule out Condorcet cycles?
• preferences single-peaked
2000 US election
•
•
•
Nader
Gore
Bush
unlikely that many had ranking
Bush
Nader
Gore
Nader
or
Bush
Gore
109
Lemma: Majority rule satisfies all 5 properties if and only if
preferences restricted to domain with no Condorcet cycles
When can we rule out Condorcet cycles?
• preferences single-peaked
2000 US election
•
•
•
Nader
Gore
Bush
unlikely that many had ranking
Bush
Nader
Gore
Nader
or
Bush
Gore
• strongly-felt candidate
110
Lemma: Majority rule satisfies all 5 properties if and only if
preferences restricted to domain with no Condorcet cycles
When can we rule out Condorcet cycles?
• preferences single-peaked
2000 US election
•
•
•
Nader
Gore
Bush
unlikely that many had ranking
Bush
Nader
Gore
Nader
or
Bush
Gore
• strongly-felt candidate
– in 2002 French election, 3 main candidates: Chirac, Jospin, Le Pen
111
Lemma: Majority rule satisfies all 5 properties if and only if
preferences restricted to domain with no Condorcet cycles
When can we rule out Condorcet cycles?
• preferences single-peaked
2000 US election
•
•
•
Nader
Gore
Bush
unlikely that many had ranking
Bush
Nader
Gore
Nader
or
Bush
Gore
• strongly-felt candidate
– in 2002 French election, 3 main candidates: Chirac, Jospin, Le Pen
– voters didn’t feel strongly about Chirac and Jospin
112
Lemma: Majority rule satisfies all 5 properties if and only if
preferences restricted to domain with no Condorcet cycles
When can we rule out Condorcet cycles?
• preferences single-peaked
2000 US election
•
•
•
Nader
Gore
Bush
unlikely that many had ranking
Bush
Nader
Gore
Nader
or
Bush
Gore
• strongly-felt candidate
– in 2002 French election, 3 main candidates: Chirac, Jospin, Le Pen
– voters didn’t feel strongly about Chirac and Jospin
– felt strongly about Le Pen (ranked him first or last)
113
• Voting rule F works well on domain U if satisfies P,A,N,I,NM
when utility functions restricted to U
114
• Voting rule F works well on domain U if satisfies P,A,N,I,NM
when utility functions restricted to U
– e.g., F C works well when preferences single-peaked
115
• Theorem 1: Suppose F works well on domain U , then F C works well on U too.
116
• Theorem 1: Suppose F works well on domain U , then F C works well on U too.
C
C
• Conversely, suppose that F works well on U .
117
• Theorem 1: Suppose F works well on domain U , then F C works well on U too.
C
C
• Conversely, suppose that F works well on U .
Then if there exisits profile U  on U
C
such that
118
• Theorem 1: Suppose F works well on domain U , then F C works well on U too.
C
C
• Conversely, suppose that F works well on U .
Then if there exisits profile U  on U
C
such that
F (U  , Y ) ≠ F C (U  , Y ) for some Y ,
119
• Theorem 1: Suppose F works well on domain U , then F C works well on U too.
C
C
• Conversely, suppose that F works well on U .
Then if there exisits profile U  on U
C
such that
F (U  , Y ) ≠ F C (U  , Y ) for some Y ,
there exists domain U ′ on which F C works well but F does not
120
• Theorem 1: Suppose F works well on domain U , then F C works well on U too.
C
C
• Conversely, suppose that F works well on U .
Then if there exisits profile U  on U
C
such that
F (U  , Y ) ≠ F C (U  , Y ) for some Y ,
there exists domain U ′ on which F C works well but F does not
Proof: From NM and I, if F works well on U , F must be ordinal
121
• Theorem 1: Suppose F works well on domain U , then F C works well on U too.
C
C
• Conversely, suppose that F works well on U .
Then if there exisits profile U  on U
C
such that
F (U  , Y ) ≠ F C (U  , Y ) for some Y ,
there exists domain U ′ on which F C works well but F does not
Proof: From NM and I, if F works well on U , F must be ordinal
• Hence result follows from
Dasgupta-Maskin (2008), JEEA
122
• Theorem 1: Suppose F works well on domain U , then F C works well on U too.
C
C
• Conversely, suppose that F works well on U .
Then if there exisits profile U  on U
C
such that
F (U  , Y ) ≠ F C (U  , Y ) for some Y ,
there exists domain U ′ on which F C works well but F does not
Proof: From NM and I, if F works well on U , F must be ordinal
• Hence result follows from
Dasgupta-Maskin (2008), JEEA
– shows that Theorem 1 holds when NM replaced by ordinality
123
To show this D-M uses
124
To show this D-M uses
Lemma: F C works well on U if and only if U has no Condorcet cycles
125
To show this D-M uses
Lemma: F C works well on U if and only if U has no Condorcet cycles
• Suppose F works well on U
126
To show this D-M uses
Lemma: F C works well on U if and only if U has no Condorcet cycles
• Suppose F works well on U
• If F C doesn't work well on U , Lemma impliesU must contain
Condorcet cycle x y z
y z x
z x y
127
•
Consider
128
•
Consider
1 2 n
U 1 = x z z
z x x
129
•
(*)
Consider
1 2 n
U 1 = x z z
z x x
(
)
Suppose F U 1 , { x, z} = z
130
•
(*)
•
Consider
1 2 n
U 1 = x z z
z x x
(
)
Suppose F U 1 , { x, z} = z
1 2 3
U = x y z
y z x
z x y
2

n
z
x
y
131
•
(*)
•
Consider
1 2 n
U 1 = x z z
z x x
(
)
Suppose F U 1 , { x, z} = z
1 2 3
U = x y z
y z x
z x y
2

(
n
z
x
y
)
(
)
F U 2 , { x, y, z} =
x ⇒ (from I) F U 2 , { x, z} =
x, contradicts (*)
132
•
(*)
•
Consider
1 2 n
U 1 = x z z
z x x
(
)
Suppose F U 1 , { x, z} = z
1 2 3
U = x y z
y z x
z x y
2

n
z
x
y
(
)
(
)
F (U , { x, y, z}) =
y ⇒ (from I) F (U , { x, y}) =
y, contradicts (*) (A,N)
F U 2 , { x, y, z} =
x ⇒ (from I) F U 2 , { x, z} =
x, contradicts (*)
2

2

133
•
(*)
•
Consider
1 2 n
U 1 = x z z
z x x
(
)
Suppose F U 1 , { x, z} = z
1 2 3
U = x y z
y z x
z x y
2

n
z
x
y
(
)
(
)
F (U , { x, y, z}) =
y ⇒ (from I) F (U , { x, y}) =
y, contradicts (*) (A,N)
F U 2 , { x, y, z} =
x ⇒ (from I) F U 2 , { x, z} =
x, contradicts (*)
so
2

2

134
•
(*)
•
Consider
1 2 n
U 1 = x z z
z x x
(
)
Suppose F U 1 , { x, z} = z
1 2 3
U = x y z
y z x
z x y
2

n
z
x
y
(
)
(
)
F (U , { x, y, z}) =
y ⇒ (from I) F (U , { x, y}) =
y, contradicts (*) (A,N)
F (U , { x, y, z}) = z
F U 2 , { x, y, z} =
x ⇒ (from I) F U 2 , { x, z} =
x, contradicts (*)
so
2

2

2

135
•
(*)
•
Consider
1 2 n
U 1 = x z z
z x x
(
)
Suppose F U 1 , { x, z} = z
1 2 3
U = x y z
y z x
z x y
n
z
x
y
2

(
)
(
)
F (U , { x, y, z}) =
y ⇒ (from I) F (U , { x, y}) =
y, contradicts (*) (A,N)
F (U , { x, y, z}) = z
F U 2 , { x, y, z} =
x ⇒ (from I) F U 2 , { x, z} =
x, contradicts (*)
so
•
2

(
2

2

)
so F U 2 , { y, z} = z (I)
136
•
(*)
•
Consider
1 2 n
U 1 = x z z
z x x
(
)
Suppose F U 1 , { x, z} = z
1 2 3
U = x y z
y z x
z x y
n
z
x
y
2

(
)
(
)
F (U , { x, y, z}) =
y ⇒ (from I) F (U , { x, y}) =
y, contradicts (*) (A,N)
F (U , { x, y, z}) = z
F U 2 , { x, y, z} =
x ⇒ (from I) F U 2 , { x, z} =
x, contradicts (*)
so
•
•
2

(
2

2

)
so F U 2 , { y, z} = z (I)
so for
137
•
(*)
•
Consider
1 2 n
U 1 = x z z
z x x
(
)
Suppose F U 1 , { x, z} = z
1 2 3
U = x y z
y z x
z x y
n
z
x
y
2

(
)
(
)
F (U , { x, y, z}) =
y ⇒ (from I) F (U , { x, y}) =
y, contradicts (*) (A,N)
F (U , { x, y, z}) = z
F U 2 , { x, y, z} =
x ⇒ (from I) F U 2 , { x, z} =
x, contradicts (*)
so
•
•
2

(
2

2

)
so F U 2 , { y, z} = z (I)
so for
1 2 3  n
z
U =x x z
z z x
x
3

138
•
(*)
•
Consider
1 2 n
U 1 = x z z
z x x
(
)
Suppose F U 1 , { x, z} = z
1 2 3
U = x y z
y z x
z x y
n
z
x
y
2

(
)
(
)
F (U , { x, y, z}) =
y ⇒ (from I) F (U , { x, y}) =
y, contradicts (*) (A,N)
F (U , { x, y, z}) = z
F U 2 , { x, y, z} =
x ⇒ (from I) F U 2 , { x, z} =
x, contradicts (*)
so
•
•
2

2

(
2

)
so F U 2 , { y, z} = z (I)
so for
1 2 3  n
z
U =x x z
z z x
x
3

(
)
F U 3 , { x, z} = z (N)
139
•
(*)
•
Consider
1 2 n
U 1 = x z z
z x x
(
)
Suppose F U 1 , { x, z} = z
1 2 3
U = x y z
y z x
z x y
n
z
x
y
2

(
)
(
)
F (U , { x, y, z}) =
y ⇒ (from I) F (U , { x, y}) =
y, contradicts (*) (A,N)
F (U , { x, y, z}) = z
F U 2 , { x, y, z} =
x ⇒ (from I) F U 2 , { x, z} =
x, contradicts (*)
so
•
•
2

2

(
2

)
so F U 2 , { y, z} = z (I)
so for
1 2 3  n
z
U =x x z
z z x
x
3

(
)
F U 3 , { x, z} = z (N)
•
1 n −1
Continuing in the same way, let U = x
x
z
z
4

n
z
x
140
•
(*)
•
Consider
1 2 n
U 1 = x z z
z x x
(
)
Suppose F U 1 , { x, z} = z
1 2 3
U = x y z
y z x
z x y
n
z
x
y
2

(
)
(
)
F (U , { x, y, z}) =
y ⇒ (from I) F (U , { x, y}) =
y, contradicts (*) (A,N)
F (U , { x, y, z}) = z
F U 2 , { x, y, z} =
x ⇒ (from I) F U 2 , { x, z} =
x, contradicts (*)
so
•
•
2

2

(
2

)
so F U 2 , { y, z} = z (I)
so for
1 2 3  n
z
U =x x z
z z x
x
3

(
)
F U 3 , { x, z} = z (N)
•
1 n −1
Continuing in the same way, let U = x
x
z
z
4

(
)
n
z
x
F U 4 , { x, z} = z , contradicts (*)
141
• So F can’t work well on U with Condorcet cycle
142
• So F can’t work well on U with Condorcet cycle
C
• Conversely, suppose that F works well on U
C
and
143
• So F can’t work well on U with Condorcet cycle
C
• Conversely, suppose that F works well on U
(
)
(
C
and
)
F U  , Y ≠ F C U  , Y for some U  and Y
144
• So F can’t work well on U with Condorcet cycle
C
• Conversely, suppose that F works well on U
(
)
(
C
and
)
F U  , Y ≠ F C U  , Y for some U  and Y
• Then there exist α with 1 − α > α and
145
• So F can’t work well on U with Condorcet cycle
C
• Conversely, suppose that F works well on U
(
)
(
C
and
)
F U  , Y ≠ F C U  , Y for some U  and Y
• Then there exist α with 1 − α > α and
U =


1−α
x
y
α
y
x
146
• So F can’t work well on U with Condorcet cycle
C
• Conversely, suppose that F works well on U
(
)
(
C
and
)
F U  , Y ≠ F C U  , Y for some U  and Y
• Then there exist α with 1 − α > α and
U =


1−α
x
y
α
y
x
such that
147
• So F can’t work well on U with Condorcet cycle
C
• Conversely, suppose that F works well on U
(
)
(
C
and
)
F U  , Y ≠ F C U  , Y for some U  and Y
• Then there exist α with 1 − α > α and
U =


1−α
x
y
α
y
x
such that
x
(
)
(
C
F=
U  , { x, y} and y F U  , { x, y}
)
148
• So F can’t work well on U with Condorcet cycle
C
• Conversely, suppose that F works well on U
(
)
(
C
and
)
F U  , Y ≠ F C U  , Y for some U  and Y
• Then there exist α with 1 − α > α and
U =


1−α
x
y
α
y
x
such that
x
•
(
)
(
C
F=
U  , { x, y} and y F U  , { x, y}
)
But not hard to show that F C unique voting rule satisfying P,A,N, and NM
when X = 2 - - contradiction
149
• Let’s drop I
150
• Let’s drop I
– most controversial
151
• Let’s drop I
– most controversial
• no voting rule satisfies P,A,N,NM on U X
152
• Let’s drop I
– most controversial
• no voting rule satisfies P,A,N,NM on U X
– GS again
153
• Let’s drop I
– most controversial
• no voting rule satisfies P,A,N,NM on U X
– GS again
• F works nicely on U if satisfies P,A,N,NM on U
154
Theorem 2:
155
Theorem 2:
• Suppose F works nicely on U , then F C or F B works nicely on U too.
156
Theorem 2:
• Suppose F works nicely on U , then F C or F B works nicely on U too.
• Conversely suppose F ∗ works nicely onU ∗ , where F ∗ = F C or F B .
157
Theorem 2:
• Suppose F works nicely on U , then F C or F B works nicely on U too.
• Conversely suppose F ∗ works nicely onU ∗ , where F ∗ = F C or F B .
Then, if there exisits profile U  on U
∗
such that
158
Theorem 2:
• Suppose F works nicely on U , then F C or F B works nicely on U too.
• Conversely suppose F ∗ works nicely onU ∗ , where F ∗ = F C or F B .
Then, if there exisits profile U  on U
∗
such that
F (U  , Y ) ≠ F ∗ (U  , Y ) for some Y ,
159
Theorem 2:
• Suppose F works nicely on U , then F C or F B works nicely on U too.
• Conversely suppose F ∗ works nicely onU ∗ , where F ∗ = F C or F B .
Then, if there exisits profile U  on U
∗
such that
F (U  , Y ) ≠ F ∗ (U  , Y ) for some Y ,
there exists domain U ′ on which F * works nicely but F does not
160
Theorem 2:
• Suppose F works nicely on U , then F C or F B works nicely on U too.
• Conversely suppose F ∗ works nicely onU ∗ , where F ∗ = F C or F B .
Then, if there exisits profile U  on U
∗
such that
F (U  , Y ) ≠ F ∗ (U  , Y ) for some Y ,
there exists domain U ′ on which F * works nicely but F does not
Proof:
161
Theorem 2:
• Suppose F works nicely on U , then F C or F B works nicely on U too.
• Conversely suppose F ∗ works nicely onU ∗ , where F ∗ = F C or F B .
Then, if there exisits profile U  on U
∗
such that
F (U  , Y ) ≠ F ∗ (U  , Y ) for some Y ,
there exists domain U ′ on which F * works nicely but F does not
Proof:
• F C works nicely on any Condorcet-cycle-free domain
162
Theorem 2:
• Suppose F works nicely on U , then F C or F B works nicely on U too.
• Conversely suppose F ∗ works nicely onU ∗ , where F ∗ = F C or F B .
Then, if there exisits profile U  on U
∗
such that
F (U  , Y ) ≠ F ∗ (U  , Y ) for some Y ,
there exists domain U ′ on which F * works nicely but F does not
Proof:
• F C works nicely on any Condorcet-cycle-free domain
• F B works nicely only when U is subset of Condorcet cycle
163
Theorem 2:
• Suppose F works nicely on U , then F C or F B works nicely on U too.
• Conversely suppose F ∗ works nicely onU ∗ , where F ∗ = F C or F B .
Then, if there exisits profile U  on U
∗
such that
F (U  , Y ) ≠ F ∗ (U  , Y ) for some Y ,
there exists domain U ′ on which F * works nicely but F does not
Proof:
• F C works nicely on any Condorcet-cycle-free domain
• F B works nicely only when U is subset of Condorcet cycle
C
B
• so F and F complement each other
164
Theorem 2:
• Suppose F works nicely on U , then F C or F B works nicely on U too.
• Conversely suppose F ∗ works nicely onU ∗ , where F ∗ = F C or F B .
Then, if there exisits profile U  on U
∗
such that
F (U  , Y ) ≠ F ∗ (U  , Y ) for some Y ,
there exists domain U ′ on which F * works nicely but F does not
Proof:
• F C works nicely on any Condorcet-cycle-free domain
• F B works nicely only when U is subset of Condorcet cycle
C
B
• so F and F complement each other
– if F works nicely on U and U doesn't contain Condorcet cycle, F C works nicely too
165
Theorem 2:
• Suppose F works nicely on U , then F C or F B works nicely on U too.
• Conversely suppose F ∗ works nicely onU ∗ , where F ∗ = F C or F B .
Then, if there exisits profile U  on U
∗
such that
F (U  , Y ) ≠ F ∗ (U  , Y ) for some Y ,
there exists domain U ′ on which F * works nicely but F does not
Proof:
• F C works nicely on any Condorcet-cycle-free domain
• F B works nicely only when U is subset of Condorcet cycle
C
B
• so F and F complement each other
– if F works nicely on U and U doesn't contain Condorcet cycle, F C works nicely too
– if F works nicely on U and U contains Condorcet cycle, then U can't contain any
other ranking (otherwise no voting rule works nicely)
166
Theorem 2:
• Suppose F works nicely on U , then F C or F B works nicely on U too.
• Conversely suppose F ∗ works nicely onU ∗ , where F ∗ = F C or F B .
Then, if there exisits profile U  on U
∗
such that
F (U  , Y ) ≠ F ∗ (U  , Y ) for some Y ,
there exists domain U ′ on which F * works nicely but F does not
Proof:
• F C works nicely on any Condorcet-cycle-free domain
• F B works nicely only when U is subset of Condorcet cycle
C
B
• so F and F complement each other
– if F works nicely on U and U doesn't contain Condorcet cycle, F C works nicely too
– if F works nicely on U and U contains Condorcet cycle, then U can't contain any
other ranking (otherwise no voting rule works nicely)
– so F B works nicely on U .
167
Striking that the 2 longest-studied voting rules
(Condorcet and Borda) are also
168
Striking that the 2 longest-studied voting rules
(Condorcet and Borda) are also
• only two that work nicely on maximal domains
169