Elections and Strategic Voting:
Condorcet and Borda
E. Maskin
Harvard University
Lecture II
Gorman Lectures
University College, London
February, 2016
• Today (as yesterday) will discuss voting rules
methods for choosing winning candidate on basis of
voters’ preferences (rankings, utility functions)
2
• Today (as yesterday) will discuss voting rules
methods for choosing winning candidate on basis of
voters’ preferences (rankings, utility functions)
• prominent examples
3
• Today (as yesterday) will discuss voting rules
methods 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.)
4
• Today (as yesterday) will discuss voting rules
methods 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
5
• Today (as yesterday) will discuss voting rules
methods 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)
6
• Today (as yesterday) will discuss voting rules
methods 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
7
− Run-off Voting (presidential elections in France)
8
− 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
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
among top 2 candidates, choose alternative preferred
by 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
− Rank-Order Voting (Borda Count)
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)
• candidate assigned 1 point every time some voter ranks
her first, 2 points every time ranked second, etc.
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.
• choose candidate with lowest point total
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
− Utilitarian Principle
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
• choose candidate who maximizes sum of voters’
utilities
15
• Which voting rule to adopt?
16
• Which voting rule to adopt?
• As discussed yesterday, answer depends on what
one wants in voting rule
17
• Which voting rule to adopt?
• As discussed yesterday, answer depends on what
one wants in voting rule
– can specify criteria (axioms) voting rule should satisfy
18
• Which voting rule to adopt?
• As discussed yesterday, answer depends on what
one wants in voting rule
– can specify criteria (axioms) voting rule should satisfy
– see which rules best satisfy them
19
• Which voting rule to adopt?
• As discussed yesterday, 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
20
• Which voting rule to adopt?
• As discussed yesterday, 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
21
• Which voting rule to adopt?
• As discussed yesterday, 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
22
• Which voting rule to adopt?
• As discussed yesterday, 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
23
• Which voting rule to adopt?
• As discussed yesterday, 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
24
• But basic negative result
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?
• Lecture 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
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 
40
• 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
41
• 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
42
• voting rule F
43
• voting rule F
for all profiles U and all Y  X ,
F U , Y   Y
44
• voting rule F
for all profiles U and all Y  X ,
F U , Y   Y
 Y is ballot
45
• voting rule F
for all profiles U and all Y  X ,
F U , Y   Y
 Y is ballot
 F U , Y   optimal candidate(s) in Y if profile
is U
46
• voting rule F
for all profiles U and all Y  X ,
F U , Y   Y
 Y is ballot
 F U , Y   optimal candidate(s) in Y if profile
is U
• will require F to be decisive (D)
47
• voting rule F
for all profiles U and all Y  X ,
F U , Y   Y
 Y is ballot
 F U , Y   optimal candidate(s) in Y if profile
is U
• will require F to be decisive (D)
 F U , Y   single candidate
48
• voting rule F
for all profiles U and all Y  X ,
F U , Y   Y
 Y is ballot
 F U , Y   optimal candidate(s) in Y if profile
is U
• will require F to be decisive (D)
 F U , Y   single candidate
– reflects fact that there is single office to be filled
49
• voting rule F
for all profiles U and all Y  X ,
F U , Y   Y
 Y is ballot
 F U , Y   optimal candidate(s) in Y if profile
is U
• will require F to be decisive (D)
 F U , Y   single candidate
– reflects fact that there is single office to be filled
• definition isn’t quite right - - ignores ties
50
• voting rule F
for all profiles U and all Y  X ,
F U , Y   Y
 Y is ballot
 F U , Y   optimal candidate(s) in Y if profile
is U
• will require F to be decisive (D)
 F U , Y   single candidate
– reflects fact that there is single office to be filled
• definition isn’t quite right - - ignores ties
– with any voting rule, there could be exact tie
51
• voting rule F
for all profiles U and all Y  X ,
F U , Y   Y
 Y is ballot
 F U , Y   optimal candidate(s) in Y if profile
is U
• will require F to be decisive (D)
 F U , Y   single candidate
– reflects fact that there is single office to be filled
• definition isn’t quite right - - ignores ties
– with any voting rule, there could be exact tie
– with plurality rule, might be two candidates who are both ranked first the
most
52
• voting rule F
for all profiles U and all Y  X ,
F U , Y   Y
 Y is ballot
 F U , Y   optimal candidate(s) in Y if profile
is U
• will require F to be decisive (D)
 F U , Y   single candidate
– reflects fact that there is single office to be filled
• definition isn’t quite right - - ignores ties
– with any voting rule, there could be exact tie
– 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
53
• voting rule F
for all profiles U and all Y  X ,
F U , Y   Y
 Y is ballot
 F U , Y   optimal candidate(s) in Y if profile
is U
• will require F to be decisive (D)
 F U , Y   single candidate
– reflects fact that there is single office to be filled
• definition isn’t quite right - - ignores ties
– with any voting rule, there could be exact tie
– 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
54
• voting rule F
for all profiles U and all Y  X ,
F U , Y   Y
 Y is ballot
 F U , Y   optimal candidate(s) in Y if profile
is U
• will require F to be decisive (D)
 F U , Y   single candidate
– reflects fact that there is single office to be filled
• definition isn’t quite right - - ignores ties
– with any voting rule, there could be exact tie
– 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
55
• voting rule F
for all profiles U and all Y  X ,
F U , Y   Y
 Y is ballot
 F U , Y   optimal candidate(s) in Y if profile
is U
• will require F to be decisive (D)
 F U , Y   single candidate
– reflects fact that there is single office to be filled
• definition isn’t quite right - - ignores ties
– with any voting rule, there could be exact tie
– 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 of decisiveness:
56
• voting rule F
for all profiles U and all Y  X ,
F U , Y   Y
 Y is ballot
 F U , Y   optimal candidate(s) in Y if profile
is U
• will require F to be decisive (D)
 F U , Y   single candidate
– reflects fact that there is single office to be filled
• definition isn’t quite right - - ignores ties
– with any voting rule, there could be exact tie
– 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 of decisiveness:
for generic profile U and all Y  X
F U , Y  is singleton
57
plurality rule:
58
plurality rule:

F P U , Y   a  i U i  a   U i  b  for all b

  i U i  a  U i  b  for all b for all a 
59
plurality rule:

F P U , Y   a  i U i  a   U i  b  for all b

  i U i  a  U i  b  for all b for all a 
majority rule:
60
plurality rule:

F P U , Y   a  i U i  a   U i  b  for all b

  i U i  a  U i  b  for all b for all a 
majority rule:

F C U , Y   a  i U i  a   U i  b  
1
2

for all b
61
plurality rule:

F P U , Y   a  i U i  a   U i  b  for all b

  i U i  a  U i  b  for all b for all a 
majority rule:

F C U , Y   a  i U i  a   U i  b  
1
2

for all b
rank-order voting:
62
plurality rule:

F P U , Y   a  i U i  a   U i  b  for all b

  i U i  a  U i  b  for all b for all a 
majority rule:

F C U , Y   a  i U i  a   U i  b  
1
2

for all b
rank-order voting:


F B U , Y   a  rUi  a  d   i    rUi  b  d   i  for all b ,
where rUi  a   # b U i  b   U i  a 
63
plurality rule:

F P U , Y   a  i U i  a   U i  b  for all b

  i U i  a  U i  b  for all b for all a 
majority rule:

F C U , Y   a  i U i  a   U i  b  
1
2

for all b
rank-order voting:


F B U , Y   a  rUi  a  d   i    rUi  b  d   i  for all b ,
where rUi  a   # b U i  b   U i  a 
utilitarian principle:


F U U , Y   a  U i  a  d   i    U i  b  d   i  for all b
64
What properties should reasonable voting rule satisfy?
65
What properties should reasonable voting rule satisfy?
• Consensus / Pareto Property (P): if Ui  x   U i  y  for all i
and x  Y , then y  F U , Y 
66
What properties should reasonable voting rule satisfy?
• Consensus / Pareto Property (P): if Ui  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
67
What properties should reasonable voting rule satisfy?
• Consensus / Pareto Property (P): if Ui  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  : 0,1  0,1 measure-preserving
permutation. If Ui  U (i ) for all i, then
68
What properties should reasonable voting rule satisfy?
• Consensus / Pareto Property (P): if Ui  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  : 0,1  0,1 measure-preserving
permutation. If Ui  U (i ) for all i, then
F U  , Y   F U , Y  for all Y
69
What properties should reasonable voting rule satisfy?
• Consensus / Pareto Property (P): if Ui  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  : 0,1  0,1 measure-preserving
permutation. If Ui  U (i ) for all i, then
F U  , Y   F U , Y  for all Y
– candidate chosen depends only on voters’ preferences and not who
has those preferences
70
What properties should reasonable voting rule satisfy?
• Consensus / Pareto Property (P): if Ui  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  : 0,1  0,1 measure-preserving
permutation. If Ui  U (i ) for all i, then
F U  , Y   F U , Y  for all Y
– candidate chosen depends only on voters’ preferences and not who
has those preferences
– voters treated symmetrically
71
• Neutrality (N): Suppose  : Y  Y permutation.
72
• Neutrality (N): Suppose  : Y  Y permutation.
If U i ,Y    x   >U i ,Y    y    U i  x   U i  y  for all x, y, i,
73
• Neutrality (N): Suppose  : Y  Y permutation.
If U i ,Y    x   >U i ,Y    y    U i  x   U i  y  for all x, y, i,
then
74
• Neutrality (N): Suppose  : Y  Y permutation.
If U i ,Y    x   >U i ,Y    y    U i  x   U i  y  for all x, y, i,
then
F U  ,Y , Y     F U , Y   .
75
• Neutrality (N): Suppose  : Y  Y permutation.
If U i ,Y    x   >U i ,Y    y    U i  x   U i  y  for all x, y, i,
then
F U  ,Y , Y     F U , Y   .
– candidates treated symmetrically
76
• Neutrality (N): Suppose  : Y  Y permutation.
If U i ,Y    x   >U i ,Y    y    U i  x   U i  y  for all x, y, i,
then
F U  ,Y , Y     F U , Y   .
– candidates treated symmetrically
• All four voting rules – plurality, majority, rank-order,
utilitarian – satisfy P, A, N
77
• Neutrality (N): Suppose  : Y  Y permutation.
If U i ,Y    x   >U i ,Y    y    U i  x   U i  y  for all x, y, i,
then
F U  ,Y , Y     F U , Y   .
– candidates treated symmetrically
• All four voting rules – plurality, majority, rank-order,
utilitarian – satisfy P, A, N
• Next axiom most controversial
78
• Neutrality (N): Suppose  : Y  Y permutation.
If U i ,Y    x   >U i ,Y    y    U i  x   U i  y  for all x, y, i,
then
F U  ,Y , Y     F U , Y   .
– candidates treated symmetrically
• All four voting rules – plurality, majority, rank-order,
utilitarian – satisfy P, A, N
• Next axiom most controversial
still
79
• Neutrality (N): Suppose  : Y  Y permutation.
If U i ,Y    x   >U i ,Y    y    U i  x   U i  y  for all x, y, i,
then
F U  ,Y , Y     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
80
• Neutrality (N): Suppose  : Y  Y permutation.
If U i ,Y    x   >U i ,Y    y    U i  x   U i  y  for all x, y, i,
then
F U  ,Y , Y     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)
81
• No Spoilers / Independence of Irrelevant Candidates (I):
if x  F U , Y  and x  Y   Y
82
• No Spoilers / Independence of Irrelevant Candidates (I):
if x  F U , Y  and x  Y   Y
then
83
• No Spoilers / Independence of Irrelevant Candidates (I):
if x  F U , Y  and x  Y   Y
then
x  F U , Y  
84
• No Spoilers / 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
85
• No Spoilers / 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)
86
• No Spoilers / 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)
87
• Majority rule and utilitarianism satisfy I, but
others don’t:
88
• Majority rule and utilitarianism satisfy I, but
others don’t:
– plurality rule
89
• Majority rule and utilitarianism satisfy I, but
others don’t:
– plurality rule
.35
x
y
z
.33
y
z
x
.32
z
y
x
90
• 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
91
• 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
92
• 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
93
• 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
94
• 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
95
• 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
96
Final Axiom:
97
Final Axiom:
• Nonmanipulability (NM):
98
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
99
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
100
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
101
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
– the members of coalition C can’t all gain from misrepresenting
utility functions as U i
102
• NM implies voting rule must be ordinal (no cardinal
information used)
103
• 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
104
• 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
105
• 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
106
• 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
107
But majority rule also violates NM
108
But majority rule also violates NM
•
F C violates decisiveness
109
But majority rule also violates NM
•
F C violates decisiveness
.35
x
y
z
.33
y
z
x
.32
z
x
y
110
But majority rule also violates NM
•
F C violates decisiveness
.35
x
y
z
.33
y
z
x
.32
z
x
y
F C U ,  x, y, z  
111
But majority rule also violates NM
•
F C violates decisiveness
.35
x
y
z
.33
y
z
x
.32
z
x
y
F C U ,  x, y, z  
– example of Condorcet cycle
112
But majority rule also violates NM
•
F C violates decisiveness
.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 when no majority winner
113
But majority rule also violates NM
•
F C violates decisiveness
.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 when no majority winner
– one possibility
114
But majority rule also violates NM
•
F C violates decisiveness
.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 when no majority winner
– one possibility
 F C U , Y  , if nonempty

F C / B U , Y   
 F B U , Y  , otherwise
115
But majority rule also violates NM
•
F C violates decisiveness
.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 when no majority winner
– one possibility
 F C U , Y  , if nonempty

F C / B U , Y   
 F B U , Y  , otherwise
(Black's method)
116
But majority rule also violates NM
•
F C violates decisiveness
.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 when no majority winner
– 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
117
But majority rule also violates NM
•
F C violates decisiveness
.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 when no majority winner
– 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
.32
z
x
y
118
But majority rule also violates NM
•
F C violates decisiveness
.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 when no majority winner
– 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
.32
z
x
y
F C / B U ,  x, y, z  x
119
But majority rule also violates NM
•
F C violates decisiveness
.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 when no majority winner
– 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
.32
z
x
y
F C / B U ,  x, y, z  x
z
y
x
120
But majority rule also violates NM
•
F C violates decisiveness
.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 when no majority winner
– 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
121
Theorem: There exists no voting rule satisfying
D,P,A,N,I, and NM
122
Theorem: There exists no voting rule satisfying
D,P,A,N,I, and NM
Proof: similar to that of GS
123
Theorem: There exists no voting rule satisfying
D,P,A,N,I, and NM
Proof: similar to that of GS
overly pessimistic - - many cases in which some rankings
unlikely
124
Lemma: Majority rule satisfies all 6 properties if and only if
preferences restricted to domain with no Condorcet cycles
125
Lemma: Majority rule satisfies all 6 properties if and only if
preferences restricted to domain with no Condorcet cycles
When can we rule out Condorcet cycles
126
Lemma: Majority rule satisfies all 6 properties if and only if
preferences restricted to domain with no Condorcet cycles
When can we rule out Condorcet cycles?
• preferences single-peaked
127
Lemma: Majority rule satisfies all 6 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
128
Lemma: Majority rule satisfies all 6 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
129
Lemma: Majority rule satisfies all 6 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
• if you have ranking order
Nader
or
Bush
Gore
Nader
Gore
Bush
130
Lemma: Majority rule satisfies all 6 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
• if you have ranking order
– would it pay to say Gore ?
Nader
or
Bush
Gore
Nader
Gore
Bush
Nader
Bush
131
Lemma: Majority rule satisfies all 6 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
• if you have ranking order
– would it pay to say Gore ?
Nader
or
Bush
Gore
Nader
Gore
Bush
Nader
Bush
– no - just makes it more likely that Gore beats Nader
132
Lemma: Majority rule satisfies all 6 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
• if you have ranking order
– would it pay to say Gore ?
Nader
or
Bush
Gore
Nader
Gore
Bush
Nader
Bush
– no - just makes it more likely that Gore beats Nader
– but prefer Nader
133
• strongly-felt candidate
134
• strongly-felt candidate
– in 2002 French election, 3 main candidates: Chirac, Jospin, Le Pen
135
• strongly-felt candidate
– in 2002 French election, 3 main candidates: Chirac, Jospin, Le Pen
– voters didn’t feel strongly about Chirac and Jospin
136
• 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)
137
Voting rule F works very well on domain U if satisfies D,P,A,N,I,NM
when utility functions restricted to U
138
Voting rule F works very well on domain U if satisfies D,P,A,N,I,NM
when utility functions restricted to U
 e.g., F C works very well when preferences single-peaked
139
• Theorem 1: Suppose F works very well on domainU , then F C works well on U too.
140
• Theorem 1: Suppose F works very well on domainU , then F C works well on U too.
C
C
• Conversely, suppose that F works well on U .
141
• Theorem 1: Suppose F works very well on domainU , 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 ,
142
• Theorem 1: Suppose F works very well on domainU , 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
143
• Theorem 1: Suppose F works very well on domainU , 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, if F works very well on U , F must be ordinal
144
• Theorem 1: Suppose F works very well on domainU , 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, if F works very well on U , F must be ordinal
• Hence first half follows from
Dasgupta-Maskin (2008)
145
• Theorem 1: Suppose F works very well on domainU , 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, if F works very well on U , F must be ordinal
• Hence first half follows from
Dasgupta-Maskin (2008)
– shows that Theorem 1 holds when NM replaced by ordinality
and “works very well” replaced by “works well”
146
• Suppose F works very well on U
147
• Suppose F works very well on U
• If F C doesn't work very well on U , Lemma impliesU must contain
Condorcet cycle x y z
y z x
z x y
148
•
Consider
149
•
Consider
1 2
U1  x z
z x
n
z
x
150
•
Consider
1 2
U1  x z
z x
(*)

n
z
x

Suppose F U 1 ,  x, z  z
151
•
Consider
1 2
U1  x z
z x
(*)
•
n
z
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
152
•
Consider
1 2
U1  x z
z x
(*)
•
n
z
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 (*)
153
•
Consider
1 2
U1  x z
z x
(*)
•
n
z
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
154
•
Consider
1 2
U1  x z
z x
(*)
•
n
z
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
so
2
155
•
Consider
1 2
U1  x z
z x
(*)
•
n
z
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
F U ,  x, y, z  z


 (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
so
2
2
156
•
Consider
1 2
U1  x z
z x
(*)
•
n
z
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
F U ,  x, y, z  z


 (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
so
2
2
•


so F U 2 ,  y, z  z (I)
157
•
Consider
1 2
U1  x z
z x
(*)
•
n
z
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
F U ,  x, y, z  z


 (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
so
2
2
•
•


so F U 2 ,  y, z  z (I)
so for
158
•
Consider
1 2
U1  x z
z x
(*)
•
n
z
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
F U ,  x, y, z  z


 (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
so
2
2
•
•


so F U 2 ,  y, z  z (I)
so for
1 2 3
U x x z
z z x
3
n
z
x
159
•
Consider
1 2
U1  x z
z x
(*)
•
n
z
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
F U ,  x, y, z  z


 (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
so
2
2
•
•


so F U 2 ,  y, z  z (I)
so for
1 2 3
U x x z
z z x
3

n
z
x

F U 3 ,  x, z  z (N)
160
•
Consider
1 2
U1  x z
z x
(*)
•
n
z
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
F U ,  x, y, z  z


 (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
so
2
2
•
•


so F U 2 ,  y, z  z (I)
so for
1 2 3
U x x z
z z x
3

n
z
x

F U 3 ,  x, z  z (N)
•
1
Continuing in the same way, let U  x
z
4
n 1
x
z
n
z
x
161
•
Consider
1 2
U1  x z
z x
(*)
•
n
z
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
F U ,  x, y, z  z


 (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
so
2
2
•
•


so F U 2 ,  y, z  z (I)
so for
1 2 3
U x x z
z z x
3

n
z
x

F U 3 ,  x, z  z (N)
•
1
Continuing in the same way, let U  x
z
4


n 1
x
z
n
z
x
F U 4 ,  x, z  z , contradicts (*)
162
• So F can’t work very well on U with Condorcet cycle
163
• So F can’t work very 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
164
• So F can’t work very 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  with 1     and
165
• So F can’t work very 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  with 1     and
U 
1
x
y

y
x
166
• So F can’t work very 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  with 1     and
U 
1
x
y

y
x
such that
167
• So F can’t work very 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  with 1     and
U 
1
x
y

y
x
such that



x  F C U ,  x, y and y  F U , x, y

168
• So F can’t work very 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  with 1     and
U 
1
x
y

y
x
such that



x  F C U ,  x, y and y  F U , x, y
•

But not hard to show that F C unique voting rule satisfying D,P,A,N, and NM
when X  2 - - contradiction
169
• Let’s drop I
170
• Let’s drop I
– most controversial
171
• Let’s drop I
– most controversial
• no voting rule satisfies D,P,A,N,NM on U X
172
• Let’s drop I
– most controversial
• no voting rule satisfies D,P,A,N,NM on U X
– GS again
173
• Let’s drop I
– most controversial
• no voting rule satisfies D,P,A,N,NM on U X
– GS again
• F works nicely on U if satisfies D,P,A,N,NM on
U
174
Theorem 2:
175
Theorem 2:
• Suppose F works nicely on U , then F C or F B works nicely on U too.
176
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 .
177
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
178
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 ,
179
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
180
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:
181
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
182
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
183
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
184
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
185
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)
186
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 .
187
• Striking that the 2 longest-studied voting rules (Condorcet
and Borda) are also
188
• Striking that the 2 longest-studied voting rules (Condorcet
and Borda) are also
– only two that work nicely on maximal domains
189
• Striking that the 2 longest-studied voting rules (Condorcet
and Borda) are also
– only two that work nicely on maximal domains
• sense in which Condorcet is better than Borda
190
• Striking that the 2 longest-studied voting rules (Condorcet
and Borda) are also
– only two that work nicely on maximal domains
• sense in which Condorcet is better than Borda
– if there are 3 alternatives, then 6 strict orderings, including 2
Condorcet cycles x y z
x z y
y z
z x
x
y
z y x
y x z
191
• Striking that the 2 longest-studied voting rules (Condorcet
and Borda) are also
– only two that work nicely on maximal domains
• sense in which Condorcet is better than Borda
– if there are 3 alternatives, then 6 strict orderings, including 2
Condorcet cycles x y z
x z y
y z x
z y x
z x y
y x z
– Borda works nicely on either cycle (domain of 3 orderings)
192
• Striking that the 2 longest-studied voting rules (Condorcet
and Borda) are also
– only two that work nicely on maximal domains
• sense in which Condorcet is better than Borda
– if there are 3 alternatives, then 6 strict orderings, including 2
Condorcet cycles x y z
x z y
y z x
z y x
z x y
y x z
– Borda works nicely on either cycle (domain of 3 orderings)
– Condorcet works nicely as long as one ordering from each cycle
missing (domain of 4 orderings)
193