t In a large electorate it is natural to consider voters’ preference profiles as frequency distributions over the set of all possible preferences. We assume coherence in voters’ preferences resulting in accumulation of voters preferences. We show that such distributions can be studied via superpositions of simpler so called unimodal distributions. At these, it is shown that all well-known rules choose the mode as the outcome. We provide a set of sufficient conditions for a rule to have this trait of choosing the mode under unimodal distributions. Further we show that Condorcet consistent rules, Borda rule, plurality rule are robust under tail-perturbations of unimodal distributions.

1. Frequency Based Analysis of Voting Rules
S. Chatterjee and T. Storcken
March 15, 2017
1

2. Outline
1. Introduction
Motivation
Literature review
2. Model
Unimodal Frequency Distributions
3. Analysis
Part-I
Part-II
4. Conclusion
2

3. Introduction

4. Motivation
• Think about voting with a large electorate: nation wide
elections.
• Voters’ preferences are diverse.
• But there is some coherence between these voters’
preferences.
• Coherence is expressed in one or several accumulations of
voters’ preferences.
3

5. Motivation
Preferences p q p+q
abc 100 80 180
acb 80 60 140
bac 80 100 180
cab 60 20 80
bca 60 80 140
cba 20 60 80
Visual
• Preference profiles are represented by frequency
distributions. Each accumulation in the distribution can
be seen as an agglomeration of preferences around a
local mode.
• The whole profile: addition or superposition of such
(local) unimodal distributions yielding a multimodal
distribution.
• Unimodal distributions =⇒ multimodal distributions.
4

6. Motivation
• Admissible set of preferences: the set of linear orders over
all candidates. A profile is a frequency distribution on this
set of linear orders.
• This set is structured by the Kemeny distance.
• Kemeny distance: minimal path-length needed to convert
one order into another by swapping consecutively ordered
candidates.
eg.
5

7. Motivation
• A unimodal distribution: mode has highest frequency and
further frequency decreases with the distance from the
mode.
• Multimodal distributions: additions of such unimodal
distributions.
• Choosing the mode as the collective order at unimodal
distribution.
• Multimodal distributions: natural to infer the outcome
from the intersection of the local modes.
6

8. Connection with Existing Literature
• Some empirical papers: impartial culture or uniform
distribution.
• Gehrlein (2006), Riker (1982, p. 122): calculates the
probability of a majority cycle occurring. Cycle
• Criticism of the impartial culture.
• Grofman et al. (2003): impartial culture is a worst-case
scenario.
7

9. Connection with Existing Literature
• Other probabilistic models: Multinomial likelihood models
(Gillett, 1976, 1978), Dual Culture (Gehrlein; 1978), Maximal
Culture Condition (Fishburn and Gehrlein, 1977).
• Message: when voters’ preferences are homogeneous,
there is an increased likelihood that a Pairwise Majority
Rule Winner exists meaning that Condorcet cycles are
avoided.
• Our results: at unimodal distributions not only pairwise
majority cycles are absent, but also all well-known rules
have the same collective order: the mode.
8

10. Connection with Existing Literature
• Merlin et. al. (2004): probability of conflicts in a U.S.
presidential type of election.
• Main difference: distributions are structured by the
Kemeny distance – A structure that therewith stems from
the preferences.
9

11. Our Approach
• Starting with observing the behaviour of several
well-known rules like Condorcet consistent rules, Scoring
rules, Elimination rules, under unimodal distributions.
• Common trait: choose the mode at unimodal distributions.
• We provide sufficient conditions for choosing the mode at
unimodal distribution.
C B E
10

12. Our Approach
• Investigate the robustness of this result under
perturbations.
• We show that even under this tail-perturbed distributions
Condorcet consistent rules, Borda rule and plurality rule
choose the mode as the outcome.
P
11

13. Model

14. Model
• N: finite but large set of voters, and A: finite set of
candidates, |A| ≥ 3.
• R ⊆ A × A: complete, anti-symmetric and transitive
relations on A.
• complete: for any a, b ∈ A either a ≽ b or b ≽ a.
• anti-symmetric: a ≽ b and b ≽ a implies a ≈ b.
• transitive: a ≽ b and b ≽ c implies a ≽ c.
• L: set of these linear orders on A.
• τxyR: the preference relation where the positions of x and
y are swapped.
• Ex: τadabcd = dbca.
12

15. Model
• p: voter i’s preference (linear order) p(i) ∈ L. p ∈ LN.
• Ties in the collective orders → collective orders are weak
orders, i.e. complete and transitive orders on A.
• W: the set of all these weak orders.
13

16. Model
• Lxy: set of linear orders R at which x is strictly preferred to
y, that is Lxy = {R ∈ L|(x, y) ∈ R}.
• Ex: For 3 candidates a, b, c, Lab = {abc, acb, cab} and
Lba = {cba, bca, bac}.
• The collective decision is formalized by preference rule.
• F: a function that assigns to every profile p in LN a
collective preference F(p) in W.
14

17. Frequency Distribution
• A frequency distribution: a function representing the
number of times each linear order appears in a profile.
• Given a profile p and a preference R in L, then f(R, p)
denotes the number of voters with preference R at profile
p, that is
f(R, p) = |{i ∈ N | p(i) = R}|.
Preferences p
abc 100
acb 80
bac 80
cab 60
bca 60
cba 20
• f(bac, p) = 80.
15

18. Frequency Distribution
• There is a metric space over L induced by the Kemeny
distance function d. For two preferences R1 and R2,
d(R1, R2) = 1
2|(R1 R2) ∪ (R2 R1)|. If R1 and R2 are linear
orders, then d(R1, R2) = |(R1 R2)|.
• Ex: d(abc, bca)
= |({(a, b), (a, c), (b, c)} {(b, a), (c, a), (b, c)})|
= |{(a, b), (a, c)}|
= 2
back
16

19. • A profile p is called unimodal if there exists a preference
R, the mode, such that for every two preferences R1 and R2
in L, if d(R, R1) < d(R, R2), then we have f(R1, p) > f(R2, p).
• Symmetric unimodal: if in addition f(R1, p) = f(R2, p)
whenever d(R, R1) = d(R, R2).
17

20. Unimodal (symmetric) Frequency Distributions
adbc
bcad
dacb
cbda
abdc
dcab
bacd
cdba
dbac
acdb
bdca
cabd
bdac
acbd
dbca
cadb
badc
dcbaabcd
cdab
dabc
bcda
adcb
cbad
18

21. Unimodal (symmetric) Frequency Distributions
adbc
bcad
dacb
cbda
abdc
dcab
bacd
cdba
dbac
acdb
bdca
cabd
bdac
acbd
dbca
cadb
badc
dcbaabcd
cdab
dabc
bcda
adcb
cbad
|
19

22. Unimodal (symmetric) Frequency Distributions
adbc
bcad
dacb
cbda
abdc
dcab
bacd
cdba
dbac
acdb
bdca
cabd
bdac
acbd
dbca
cadb
badc
dcbaabcd
cdab
dabc
bcda
adcb
cbad
||
|
|
20

23. Unimodal (symmetric) Frequency Distributions
adbc
bcad
dacb
cbda
abdc
dcab
bacd
cdba
dbac
acdb
bdca
cabd
bdac
acbd
dbca
cadb
badc
dcbaabcd
cdab
dabc
bcda
adcb
cbad
||
|
|
|
|
||
|
21

24. Unimodal (symmetric) Frequency Distributions
adbc
bcad
dacb
cbda
abdc
dcab
bacd
cdba
dbac
acdb
bdca
cabd
bdac
acbd
dbca
cadb
badc
dcbaabcd
cdab
dabc
bcda
adcb
cbad
||
|
|
|
|
||
|
|
|
|
| |
|
22

25. Unimodal (symmetric) Frequency Distributions
adbc
bcad
dacb
cbda
abdc
dcab
bacd
cdba
dbac
acdb
bdca
cabd
bdac
acbd
dbca
cadb
badc
dcbaabcd
cdab
dabc
bcda
adcb
cbad
||
|
|
|
|
||
|
|
|
|
| |
|
|
|
|
|
|
23

26. Unimodal (symmetric) Frequency Distributions
adbc
bcad
dacb
cbda
abdc
dcab
bacd
cdba
dbac
acdb
bdca
cabd
bdac
acbd
dbca
cadb
badc
dcbaabcd
cdab
dabc
bcda
adcb
cbad
||
|
|
|
|
||
|
|
|
|
| |
|
|
|
|
|
|
|
|
|
24

27. Unimodal (symmetric) Frequency Distributions
adbc
bcad
dacb
cbda
abdc
dcab
bacd
cdba
dbac
acdb
bdca
cabd
bdac
acbd
dbca
cadb
badc
dcbaabcd
cdab
dabc
bcda
adcb
cbad
||
|
|
|
|
||
|
|
|
|
| |
|
|
|
|
|
|
|
|
|
|
25

28. Unimodal (symmetric) Frequency Distributions
26

29. Analysis

30. Intuition Behind the Analysis
R R τabR
. . .
. a b
. . .
a . .
. . .
. . .
b . .
. b a
. . .
. . .
• R ↔ τabR. R ∈ Lab and τabR ∈ Lba.
• In pairwise comparison a will beat b.
• The mode consists of all pairs which in pairwise
comparisons beat each other: the Condorcet order.
27

31. Intuition Behind the Analysis
• The same argument =⇒ at any score rule a gets a higher
score than b and that b is eliminated before a.
• Mode: “positional” order + order of “elimination”.
28

32. Sufficient Conditions
Formalizing this intuition:
• Anonymity, neutrality, discrimination and monotonicity.
• Not the usual monotonicity.
29

33. Monotonicity
p
a b d c a …
b c b d c …
c d c b d …
d a a a b …
q
a d d c a …
d c b d c …
c b c b d …
b a a a b …
• Monotonicity: (d, b) ∈ F(p) implies (d, b) ∈ F(q).
• In this way monotonicity is defined pairwise but it is
sensitive to the positions of candidates. It is therefore
satisfied by many rules.
30

34. Monotonicity
p
a b d c a …
b c b d c …
c d c b d …
d a a a b …
q
d d d c a …
a c b d c …
c b c b d …
b a a a b …
• Monotonicity: (d, b) ∈ F(p) does not imply (d, b) ∈ F(q).
• In this way monotonicity is defined pairwise but it is
sensitive to the positions of candidates. It is therefore
satisfied by many rules.
30

35. Discrimination
p
a a b c b c
b c a a c b
c b c b a a
10 8 8 6 6 2
• Discrimination: (a ≈ b) /∈ F(p)
31

36. Discrimination
p
a a b c b c
b c a a c b
c b c b a a
10 8 8 6 6 2
• Discrimination: (a ≈ b) /∈ F(p), (b ≈ c) /∈ F(p),
31

37. Discrimination
p
a a b c b c
b c a a c b
c b c b a a
10 8 8 6 6 2
• Discrimination: (a ≈ b) /∈ F(p), (b ≈ c) /∈ F(p),
(a ≈ c) /∈ F(p).
31

38. Discrimination
p
a a b c b c
b c a a c b
c b c b a a
10 8 8 6 6 2
• Discrimination: (a ≈ b) /∈ F(p), (b ≈ c) /∈ F(p),
(a ≈ c) /∈ F(p).
• positive discrimination: (a, b) ∈ F(p)
31

39. Discrimination
p
a a b c b c
b c a a c b
c b c b a a
10 8 8 6 6 2
• Discrimination: (a ≈ b) /∈ F(p), (b ≈ c) /∈ F(p),
(a ≈ c) /∈ F(p).
• positive discrimination: (a, b) ∈ F(p), (b, c) ∈ F(p),
31

40. Discrimination
p
a a b c b c
b c a a c b
c b c b a a
10 8 8 6 6 2
• Discrimination: (a ≈ b) /∈ F(p), (b ≈ c) /∈ F(p),
(a ≈ c) /∈ F(p).
• positive discrimination: (a, b) ∈ F(p), (b, c) ∈ F(p),
(a, c) ∈ F(p).
31

41. First Result
Theorem
Let p be a unimodal profile with mode R. Then F(p) = R for a
rule F from LN to W in each of the following two cases:
1. F is positively discriminating;
2. F is anonymous, neutral, monotone and discriminating.
32

42. Tail-perturbed Distributions
33

43. Tail-perturbed Distributions
There is a linear order, say R∗, and a real number ν such that
33

44. Tail-perturbed Distributions
There is a linear order, say R∗, and a real number ν such that
1. Frequencies are constant at any given distance from R∗.
So, f(R1, p) = f(R2, p) whenever d(R1, R∗) = d(R2, R∗) for all
linear orders R1 and R2.
33

45. Tail-perturbed Distributions
There is a linear order, say R∗, and a real number ν such that
1. Frequencies are constant at any given distance from R∗.
So, f(R1, p) = f(R2, p) whenever d(R1, R∗) = d(R2, R∗) for all
linear orders R1 and R2.
2. For linear orders R1, R2 and R3, with
d(R1, R∗) < d(R2, R∗) ν < d(R3, R∗) we have
f(R1, p) > f(R2, p) > f(R3, p).
33

46. Tail-perturbed Distributions
• δ =
(m
2
)
: diameter distance of L.
• ρ = 1
2δ: radius distance.
• ρ ν: perturbations only occur in the second half of the
set of linear orders.
• Ex: For an election with 4 (i.e.m = 4) candidates,
δ =
(4
2
)
= 6 and ρ = 3.
34

47. Condorcet Consistent Rules
• Condorcet consistent rules depend on pairwise majority
comparisons of the candidates.
• may yield cycles and break cycles differently.
• If, however, at a certain profile pairwise majority
comparisons yield a complete, strict and transitive order
from overall winner (the Condorcet winner) to overall loser
(the Condorcet loser), then this is the outcome of all these
rules at that profile.
C
35

48. Perturbation and Condorcet Consistent Rules
36

49. Perturbation and Condorcet Consistent Rules
• Frequency doesn’t only depend on the distance to R∗.
• First half: frequency declines with the distance to R∗.
Second half: frequency at distance, say δ − k, is bounded
by the minimum frequency at the ’opposite’ distance k.
36

50. Perturbation and Condorcet Consistent Rules
We prove that R∗ is the Condorcet order if frequencies satisfy
min
R∈Lk
xy
f(R) + min
R∈Lδ−k
xy
f(R) > max
R∈Lk
yx
f(R) + max
R∈Lδ−k
yx
f(R)
for all 0 ≤ k < ρ.
37

51. Borda Rule
• Borda rule is based on Borda score.
• Borda score of a candidate x is the number of candidates
below x in a preference R, summed over all preferences.
• In the collective order we rank the candidates according to
their Borda score.
B
38

52. Perturbation and Borda Rule
39

53. Perturbation and Borda Rule
• Frequency depends on the distance to R∗.
• At distances larger than radius distance, say δ − k,
frequency is lower than at the ’opposite’ distance k. 39

54. Perturbation and Borda Rule
We prove that R∗ is the chosen order by Borda rule if
frequencies satisfy
f(k) > f(δ − k) for all 0 ≤ k < ρ.
40

55. Plurality Rule
• Plurality rule is based on Plurality score.
• That is the number of times a candidate is at the top of
voter’s preferences.
• In the collective order we rank the candidates according to
their Plurality score.
P
41

56. Perturbation and Plurality Rule
42

57. Perturbation and Plurality Rule
• Frequency depends on the distance to R∗.
• First half: ν = 1
2
(m−1
2
)
+ m − 3
2 > ρ = 1
2
(m
2
)
.
42

58. Perturbation and Plurality Rule
We prove that R∗ is the chosen order by Plurality rule if the
distribution is ν-tail perturbed unimodal, with
ν = 1
2
(m−1
2
)
+ m − 3
2
43

59. Multimodal Distribution
+ ⇒
• Unimodal constituents =⇒ multimodal distribution.
back
44

60. Multimodal Distribution
• For discriminating collective decision rules: the outcome
at the union of two unimodal distributions is in the
intersection of the two modes of these unimodal
distributions.
• Ex: Intersection of abc and bac: set of concordant pairs:
{(a, c), (b, c)}.
• Generalizes to more than two unimodal distributions.
• Difficult situations: empty intersection.
45

61. Conclusion

62. Conclusion
• Recognized a common trait in all well-known collective
decision rules.
• Extended this result outside the domain of unimodal
distribution.
• As Borda rule and Plurality rule are “opposite extremes” in
the class of score rules, we expect that a large subclass of
score rules also choose the mode at such ν−tail
perturbed unimodal distribution.
• Predictability of the outcome of multimodal distributions.
46

63. Thank you

64. Pairwise Majority Cycle
p
a b c
b c a
c a b
1 1 1
Candidate
Candidate
a b c
a - 2 1
b 1 - 2
c 2 1 -
Table 1: Occurrence of a Cycle
(a, b) : (b, c) : (c, a): Cycle
q
a a c
b c a
c b b
1 1 1
Candidate
Candidate
a b c
a - 3 2
b 0 - 1
c 1 2 -
Table 2: Non-occurrence of a Cycle
(a, b) : (a, c) : (c, b) =⇒ acb back
48

65. Condorcet Consistent Rules
Copeland Rule:
p
a a c
b c a
c b b
1 1 1
Candidate
Candidate
a b c
a - 3 2
b 0 - 1
c 1 2 -
Candidate Score
a 2
b 0
c 1
F(p) = acb.
(a, b)(a, c)(c, b) =⇒ acb.
back back to CR
49

66. Borda Rule
Borda Rule:
p
a a c
c c b
b d d
d b a
1 1 1
Candidate
Candidate
a b c d Sum
a - 2 2 2 6
b 1 - 0 2 3
c 1 3 - 3 7
d 1 1 0 - 2
Candidate Score Rank
a 6 2
b 3 3
c 7 1
d 2 4
(a, b) : (a, c) : (a, d) : (b, d) : (c, b) : (c, d) =⇒ acbd.
F(p) = cabd.
back back to BR
50

67. Plurality Rule
Plurality Rule:
p
b c d
a a a
c b b
d d c
1 1 1
Candidate
Candidate
a b c d
a - 2 2 2
b 1 - 2 2
c 1 1 - 2
d 1 1 1 -
Candidate Score Rank
a 0 2
b 1 1
c 1 1
d 1 1
(a, b) : (a, c) : (a, d) : (b, c) : (b, d) : (c, d) =⇒ abcd.
F(p) = (b − c − d)a.
back back to PR
51

68. Elimination Rules
Coombs Rule:
p
b c d
a a a
c b b
d d c
1 1 1
Round 1 Scores
a 3
b 3
c 2
d 1
d eliminated.
p
b c a
a a b
c b c
1 1 1
Round 2 Scores
a 3
b 2
c 1
c eliminated.
p
b a a
a b b
1 1 1
Round 3 Scores
a 2
b 1
b eliminated.
F(p) = abcd.
(a, b) : (a, c) : (a, d) : (b, c) : (b, d) : (c, d) =⇒ abcd. back 52