This talk explores possibilities for exploiting structure in voting profiles to obtain efficient algorithms for problems that are computationally intractable in general.

1. Efficient Algorithms for Hard Problems on
Structured Electorates
Neeldhara Misra
Aspects of Computation 2017

2. and some problems that we will encounter.
The standard Voting Setup

3. definition, recognition, strategy-proofness, elicitation
Single Peaked Preferences
and some problems that we will encounter.
The standard Voting Setup

4. definition, recognition, strategy-proofness, elicitation
definition, recognition and Condorcet winners
Single Crossing preferences
Single Peaked Preferences
and some problems that we will encounter.
The standard Voting Setup

5. definition, recognition, strategy-proofness, elicitation
definition, recognition and Condorcet winners
nearly structured preferences, dichotomous preferences
Single Crossing preferences
Concluding remarks
Single Peaked Preferences
and some problems that we will encounter.
The standard Voting Setup

6. The standard Voting Setup
and some problems that we will encounter.
The Handbook of Computational Social Choice,
Brandt, Conitzer, Endriss, Lang and Procaccia; 2016

8. A typical voting scenario involves
a set of alternatives, and
a set of voters.
Voters have preferences over the alternatives,
which can be modelled in various ways.
In this talk, we’ll think of preferences as
linear orders over the rankings.

9. A typical voting scenario involves
a set of alternatives, and
a set of voters.
Voters have preferences over the alternatives,
which can be modelled in various ways.
In this talk, we’ll think of preferences as
linear orders over the rankings.

10. A typical voting scenario involves
a set of alternatives, and
a set of voters.
Voters have preferences over the alternatives,
which can be modelled in various ways.
In this talk, we’ll think of preferences as
linear orders over the rankings.

11. A group of friends have
an evening to spend.

14. (Plurality)

15. (Plurality)

17. We say that a voter (or a group of voters)
can manipulate if they can obtain a more desirable
outcome by misreporting their preferences.
We’ll see that plurality is vulnerable to this behaviour.

18. We say that a voter (or a group of voters)
can manipulate if they can obtain a more desirable
outcome by misreporting their preferences.
We’ll see that plurality is vulnerable to this behaviour.

19. (Plurality)

20. (Plurality)

21. (Plurality)

22. (Plurality)

23. (Plurality)
Manipulation

24. This scheme is intended only for honest men.
Borda, 1770

25. Elimination over multiple rounds

29. (STV)

30. Pairwise Elections

36. An alternative that beats all the others
in pairwise comparisons.

37. (Condorcet)
An alternative that beats all the others
in pairwise comparisons.

39. An alternative that beats all the others
in pairwise comparisons.

41. may not exist!
An alternative that beats all the others
in pairwise comparisons.

42. Desirable properties of voting rules.

44. We’ll now pause to formally define
social choice functions and
social welfare functions.

46. Social Welfare Functions (SWF)

48. Social Choice Functions (SCF)

50. Unanimity

51. Unanimity

52. Unanimity
If everyone prefers A to B then
the consensus ranking also prefers
A over B.

54. Independence of Irrelevant Alternatives

55. Independence of Irrelevant Alternatives

56. Independence of Irrelevant Alternatives

57. Independence of Irrelevant Alternatives
Suppose a SWF prefers A over B in
the consensus ranking for a profile P.
Let Q be a profile that is the same as P when
projected on the candidates A and B.
Then the SWF must prefer A over B in the
consensus ranking that it determines for Q as well.

58. Arrow (1949)

59. When voters have three or more alternatives,
any social welfare function which respects
unanimity and
independence of irrelevant alternatives
is a dictatorship.
Arrow (1949)

60. Gibbard-Satterthwaite (1973/75)

61. Any strategy- proof SCF where at least three
candidates have some chance of being selected
must be a dictatorship.
Gibbard-Satterthwaite (1973/75)

62. Some Popular Workarounds

63. Domain Restrictions
Some Popular Workarounds

64. Domain Restrictions
Some Popular Workarounds
Randomised Voting Rules

65. Domain Restrictions
Some Popular Workarounds
Computational Hardness
Randomised Voting Rules

66. Domain Restrictions
Some Popular Workarounds
Computational Hardness
Randomised Voting Rules

67. Single Peaked Preferences
definition, recognition, strategy-proofness, elicitation

68. Definition
The Theory of Committees and Elections.
Black, D.,NewYork: Cambridge University Press, 1958
Single Peaked Preferences

69. Left RightCenter
A B C D E F G

70. Left RightCenter
A B C D E F G

71. Left RightCenter
A B C D E F G
E D C F G B A

72. Left RightCenter
A B C D E F G
E D C F G B A

73. Left RightCenter
A B C D E F G
E D C F G B A

74. Left RightCenter
A B C D E F G
E D C F G B A

75. Left RightCenter
A B C D E F G
E D C F G B A
E D CF GB A

76. Left RightCenter
A B C D E F G

77. Left RightCenter
A B C D E F G

78. Left RightCenter
A B C D E F G
If an agent with single-peaked preferences prefers x to y,
one of the following must be true:
- x is the agent’s peak,
- x and y are on opposite sides of the agent’s peak, or
- x is closer to the peak than y.

79. Left RightCenter
A B C D E F G
The notion is popular for several reasons:
- No Condorcet Cycles.
- No incentive for an agent to misreport its preferences.
- Identifiable in polynomial time.
- Reasonable (?) model of actual elections.

80. Single Peaked Preferences
Recognition
Stable Matching with Preferences Derived from a Psychological Model
Bartholdi andTrick, Operations Research Letters, 1986

82. Recognising if a given profile is single-peaked with
respect to some preference ordering.
Checking if a 0/1 matrix has the
consecutive-ones property.

83. A B C D E F G

84. A B C D E F G
ABCDE F G

85. A B C D E F G
ABCDE F G
A fixed
permutation

86. A B C D E F G
01000 0 0
ABCDE F G
A fixed
permutation

87. A B C D E F G
01000 0 0
ABCDE F G
A fixed
permutation

88. A B C D E F G
01000 0 0
01100 0 0
ABCDE F G
A fixed
permutation

89. A B C D E F G
01000 0 0
01100 0 0
ABCDE F G
A fixed
permutation

90. A B C D E F G
01000 0 0
01100 0 0
ABCDE F G
11100 0 0
A fixed
permutation

91. A B C D E F G
01000 0 0
01100 0 0
ABCDE F G
11100 0 0
A fixed
permutation

92. A B C D E F G
01000 0 0
01100 0 0
ABCDE F G
11100 0 0
11100 1 0
A fixed
permutation

93. A B C D E F G
01000 0 0
01100 0 0
ABCDE F G
11100 0 0
11100 1 0
A fixed
permutation

94. A B C D E F G
01000 0 0
01100 0 0
ABCDE F G
11100 0 0
11100 1 0
11110 1 0
A fixed
permutation

95. A B C D E F G
01000 0 0
01100 0 0
ABCDE F G
11100 0 0
11100 1 0
11110 1 0
A fixed
permutation

96. A B C D E F G
01000 0 0
01100 0 0
ABCDE F G
11100 0 0
11100 1 0
11110 1 0
11111 1 0
A fixed
permutation

97. A B C D E F G
01000 0 0
01100 0 0
ABCDE F G
11100 0 0
11100 1 0
11110 1 0
11111 1 0
A fixed
permutation

98. A B C D E F G
01000 0 0
01100 0 0
ABCDE F G
11100 0 0
11100 1 0
11110 1 0
11111 1 0
11111 1 1
A fixed
permutation

99. Single Peaked Preferences
Strategyproofness
The Theory of Committees and Elections.
Black, D.,NewYork: Cambridge University Press, 1958

100. A B C D E F G

101. A B C D E F G

102. A B C D E F G

103. A B C D E F G
Claim: D beats all other candidates in pairwise elections.

104. A B C D E F G
Claim: D beats all other candidates in pairwise elections.

105. A B C D E F G
Claim: D beats all other candidates in pairwise elections.

106. A B C D E F G
Claim: D beats all other candidates in pairwise elections.
(Peak to the left of D,
F further than D)

107. A B C D E F G
Claim: D beats all other candidates in pairwise elections.

108. A B C D E F G
Claim: D beats all other candidates in pairwise elections.

109. A B C D E F G
Claim: D beats all other candidates in pairwise elections.

110. A B C D E F G
Claim: D beats all other candidates in pairwise elections.
(Peak to the right of D,
B further than D)

111. A B C D E F G
Claim: Choosing D also leaves nobody
with any incentive to manipulate.

112. A B C D E F G
Claim: Choosing D also leaves nobody
with any incentive to manipulate.

113. A B C D E F G
Claim: Choosing D also leaves nobody
with any incentive to manipulate.

114. A B C D E F G
Claim: Choosing D also leaves nobody
with any incentive to manipulate.

115. A B C D E F G
Claim: Choosing D also leaves nobody
with any incentive to manipulate.

116. A B C D E F G
Claim: Choosing D also leaves nobody
with any incentive to manipulate.

117. Single Peaked Preferences
Preference Elicitation
Eliciting single-peaked preferences using comparison queries,
Conitzer, J.Artif. Intell. Res.; 2016

118. When the number of candidates is large,
soliciting a full ranking can be a little unmanageable.

119. When the number of candidates is large,
soliciting a full ranking can be a little unmanageable.
Voters typically find it easier to answer
“comparison queries”:
Would you rather hang out over coffee
or join me for a concert?

120. How many such queries to do we need to make
to be able to reconstruct the full preference?

121. How many such queries to do we need to make
to be able to reconstruct the full preference?
Just like the weighing scale puzzles, except you can
only compare two single options at a time.

122. Using a “merge sort” like idea,
O(m log m) comparisons are enough.

123. Using a “merge sort” like idea,
O(m log m) comparisons are enough.
Recursively order half the alternatives.

124. Using a “merge sort” like idea,
O(m log m) comparisons are enough.
Recursively order half the alternatives.

125. Using a “merge sort” like idea,
O(m log m) comparisons are enough.
Merge the two lists with a linear number of queries.

126. Left RightCenter
A B C D E F G

127. Left RightCenter
A B C D E F G
…and we can do better if the preferences are single-peaked!

128. Left RightCenter
A B C D E F G
…and we can do better if the preferences are single-peaked!
(i) Identify the peak: use binary search.

129. Left RightCenter
A B C D E F G
(i) Identify the peak: use binary search.
[Better query complexities for elicitation.]

130. Left RightCenter
A B C D
E
F G
(i) Identify the peak: use binary search.
[Better query complexities for elicitation.]

131. Left RightCenter
A B C D
E
F G
(i) Identify the peak: use binary search.
this signals that the
peak is to the right.
[Better query complexities for elicitation.]

132. Left RightCenter
A B C D E F G
(i) Identify the peak: use binary search.
[Better query complexities for elicitation.]

133. Left RightCenter
A B C
D
E F G
(i) Identify the peak: use binary search.
[Better query complexities for elicitation.]

134. Left RightCenter
A B C
D
E F G
(i) Identify the peak: use binary search.
this signals that the
peak is to the left.
[Better query complexities for elicitation.]

135. Left RightCenter
A B C D E F G
(ii) Once we know the peak, the rest is O(m) queries.
[Better query complexities for elicitation.]

136. Left Right
A B C D E F G
(ii) Once we know the peak, the rest is O(m) queries.
[Better query complexities for elicitation.]

137. Left Right
A B C D E F G
(ii) Once we know the peak, the rest is O(m) queries.
C
[Better query complexities for elicitation.]

138. Left Right
A B C D E F G
(ii) Once we know the peak, the rest is O(m) queries.
C
[Better query complexities for elicitation.]

139. Left Right
A B C D E F G
(ii) Once we know the peak, the rest is O(m) queries.
C
[Better query complexities for elicitation.]

140. Left Right
A
B
C D E F G
(ii) Once we know the peak, the rest is O(m) queries.
C
[Better query complexities for elicitation.]

141. Left Right
A
B
C D E F G
(ii) Once we know the peak, the rest is O(m) queries.
C
[Better query complexities for elicitation.]

142. Left Right
A B C D E F G
(ii) Once we know the peak, the rest is O(m) queries.
C
[Better query complexities for elicitation.]

143. Left Right
A B C
D
E F G
(ii) Once we know the peak, the rest is O(m) queries.
C
[Better query complexities for elicitation.]

144. Left Right
A B C
D
E F G
(ii) Once we know the peak, the rest is O(m) queries.
C
[Better query complexities for elicitation.]

145. Left Right
A B C D E F G
Center
[Better query complexities for elicitation.]

146. Left Right
A B C D E F G
Center
(ii) Once we know the peak, the rest is O(m) queries.
(i) Identify the peak: use binary search (O(log m) queries).
[Better query complexities for elicitation.]

147. Single Crossing Preferences
definition, recognition and Condorcet winners

148. Definition
Single Crossing Preferences

150. A profile is single-crossing if it admits an ordering of the
voters such that for every pair of candidates (a,b), either:
a) all voters who prefer a over b appear before all voters
who prefer b over a, or,
b) all voters who prefer a over b appear after all voters
who prefer b over a, or,

152. The notion is popular for several reasons:
- No Condorcet Cycles.
- Identifiable in polynomial time.
- Reasonable (?) model of actual elections.

153. Recognition
Single Crossing Preferences
A Characterization of the Single-Crossing Domain
Bredereck, Chen andWoeginger, Social Choice and Welfare, 2013

155. Recognising if a given profile is single-crossing
with respect to some preference ordering.
Checking if a 0/1 matrix has the
consecutive-ones property.

156. 1 0 1
1 0
0
10
1
0

157. No Condorcet Cycles
Single Crossing Preferences

160. For any pair of candidates,
the opinion of the middle
voter is also the majority opinion.

161. Chamberlin-Courant
Single Crossing Preferences
The Complexity of Fully Proportional Representation for Single-Crossing Electorates
Skowron, SAGT, 2013

162. Voters
Candidates

163. Voters
Candidates

164. Voters
Candidates

165. Voters
Candidates

166. dissatisfaction of voter v =
rank of best candidate in the committee in his vote
Voters
Candidates

167. dissatisfaction of voter v =
rank of best candidate in the committee in his vote
Voters
Candidates

168. dissatisfaction of voter v =
rank of best candidate in the committee in his vote
On single-crossing profiles, optimal CC solutions exhibit a
“contiguous blocks property”.
Voters
Candidates

169. dissatisfaction of voter v =
rank of best candidate in the committee in his vote
On single-crossing profiles, optimal CC solutions exhibit a
“contiguous blocks property”.
Voters
Candidates

170. dissatisfaction of voter v =
rank of best candidate in the committee in his vote
On single-crossing profiles, optimal CC solutions exhibit a
“contiguous blocks property”.
Voters
Candidates

171. Concluding Remarks
nearly structured preferences, dichotomous preferences

172. The single-peaked and single-crossing domains have been
generalised to notions of single-peaked and single-
crossing on trees. The generalised domains continue to
exhibit many of the nice properties we saw today.
Generalizing the Single-Crossing Property on Lines and Trees to Intermediate
Preferences on Median Graphs, Clearwater, Puppe, and Slinko, IJCAI 2015
Single-peaked orders on a tree, Gabrielle Demange,
Math. Soc. Sci, 3(4), 1982.

173. The single-peaked and single-crossing domains have been
generalised to notions of single-peaked-width and single-
crossing-width.
Here, it is common that algorithms that work in the single-
peaked or single-crossing settings can be generalised to
profiles of width w at an expense that is exponential in w.
Kemeny Elections with Bounded Single-peaked or Single-crossing Width,
Cornaz, Galand, and Spanjaard, IJCAI 2013

174. Profiles that are “close” to being single-peaked or single-
crossing (closeness measured usually in terms of candidate
or voter deletion) have also been studied.
It’s typically NP-complete to determine the optimal
distance, but FPT and approximation algorithms are
known.
Are There Any Nicely Structured Preference Profiles Nearby?
Bredereck, Chen, andWoeginger, AAAI 2013
On Detecting Nearly Structured Preference Profiles
Elkind and Lackner, AAAI 2014
Computational aspects of nearly single-peaked electorates,
Erdélyi, Lackner, and Pfandler, AAAI 2013

175. Summary from:
On Detecting Nearly Structured Preference Profiles Elkind and Lackner, AAAI 2014

176. While the domain restrictions we saw today apply to votes
given as linear orders, similar restrictions have also been
studied in the context of votes that are given by approval
ballots.
As was the case here, there are close connections with
COP and many hard problems become easy on these
structured (dichotomous) profiles.
Structure in Dichotomous Preferences,
Elikind and Lackner; IJCAI 2015.

177. Euclidean preferences capture settings where voters and
alternatives can be identified with points in the Euclidean
space, with voters’ preferences driven by distances to
alternatives.
Psychological scaling without a unit of measurement,
Coombs; Psychological review, 1950

178. The Dark Side: Domain restrictions also have some side-
effects: problems like manipulation, bribery, and so forth
also become easy!
Bypassing Combinatorial Protections: Polynomial-Time Algorithms for Single-Peaked Electorates,
Brandt et al; AAAI 2010
The Shield that Never Was: Societies with Single-Peaked Preferences are More Open to
Manipulation and Control, Faliszewski et al; TARK 2009

179. Thank you!
Preference Restrictions in Computational Social Choice: Recent Progress, Elikind,
Lackner, and Peters; IJCAI 2016.
The Handbook of Computational Social Choice,
Brandt, Conitzer, Endriss, Lang and Procaccia; 2016