The document discusses eliciting single-peaked preferences over trees, focusing on the query model for preference elicitation and complexities involved. It outlines the advantages of single-peaked preferences, such as having no Condorcet cycles and being easily identifiable, while describing improved querying strategies to reconstruct full preferences through comparison queries. The main strategy involves splitting the preference tree into paths, processing each path, and merging results, achieving efficient query complexities.

1. Elicitation for Preferences Single
Peaked on Trees
Palash Dey and Neeldhara Misra
*IJCAI 2016

2. and the query model for eliciting preferences.
The standard Voting Setup

3. and better query complexities for elicitation.
Single-peaked Preferences
and the query model for eliciting preferences.
The standard Voting Setup

4. and better query complexities for elicitation.
so we consider broader domains but still have nice properties.
GENERALIZING “SINGLE PEAKEDNESS”
Single-peaked Preferences
and the query model for eliciting preferences.
The standard Voting Setup

5. and better query complexities for elicitation.
so we consider broader domains but still have nice properties.
for preferences single-peaked on trees.
GENERALIZING “SINGLE PEAKEDNESS”
Back to QUERY COMPLEXITY
Single-peaked Preferences
and the query model for eliciting preferences.
The standard Voting Setup

6. The standard Voting Setup
and the query model for eliciting preferences.

7. Candidates/Alternatives

8. Voters express their preferences over alternatives
(here, as rankings).

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

10. 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?

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

12. 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.

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

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

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

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

17. 21 25 32 41 55 63 72
10 22 45 53 62 64 90

18. 21 25 32 41 55 63 72
10 22 45 53 62 64 90

19. 21 25 32 41 55 63 72
10
22 45 53 62 64 90

20. 21
25 32 41 55 63 72
10
22 45 53 62 64 90

21. 21
25 32 41 55 63 72
10 22
45 53 62 64 90

22. 21 25
32 41 55 63 72
10 22
45 53 62 64 90

23. 21 25 32
41 55 63 72
10 22
45 53 62 64 90

24. 21 25 32 41
55 63 72
10 22
45 53 62 64 90

25. 21 25 32 41
55 63 72
10 22 45
53 62 64 90

26. 21 25 32 41
55 63 72
10 22 45
53
62 64 90

27. 21 25 32 41
55 63 72
10 22 45
53 62 64 90

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

29. Single Peaked Preferences
and better query complexities for elicitation. [1]

30. Left RightCenter
A B C D E F G

31. Left RightCenter
A B C D E F G

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

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

34. Left RightCenter
A B C D E F G

35. Left RightCenter
A B C D E F G

36. 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.

37. 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.

38. Left RightCenter
A B C D E F G

39. Left RightCenter
A B C D E F G
…and better query complexities for elicitation.

40. Left RightCenter
A B C D E F G
…and better query complexities for elicitation.
(i) Identify the peak: use binary search.

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

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

43. 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.]

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

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

46. 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.]

47. 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.]

48. 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.]

49. 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.]

50. 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.]

51. 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.]

52. 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.]

53. 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.]

54. 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.]

55. 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.]

56. 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.]

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

58. 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.]

59. GeneraliZing “single-Peakedness”
so we consider broader domains but still have nice properties. [2]

60. Profiles single-peaked on trees

61. A preference profile is single-peaked with respect to a
tree T (labeled with the alternatives)
if it is single peaked when restricted to the candidates
involved in any path on the tree.
Profiles single-peaked on trees

65. A preference profile is single-peaked with respect to a
tree T (labeled with the alternatives)
if it is single peaked when restricted to the candidates
involved in any path on the tree.
Profiles single-peaked on trees

66. A preference profile is single-peaked with respect to a
tree T (labeled with the alternatives)
if it is single peaked when restricted to the candidates
involved in any path on the tree.
Profiles single-peaked on trees

67. A preference profile is single-peaked with respect to a
tree T (labeled with the alternatives)
if it is single peaked when restricted to the candidates
involved in any path on the tree.
The special case when T is a path corresponds to the
usual notion of single-peakedness.
Profiles single-peaked on trees

68. A preference profile is single-peaked with respect to a
tree T (labeled with the alternatives)
if it is single peaked when restricted to the candidates
involved in any path on the tree.
Profiles single-peaked on trees
Are these structures still “nice enough”, like single-peaked profiles?

69. Profiles single-peaked on trees
Are these structures still “nice enough”, like single-peaked profiles?
- No Condorcet Cycles.
- No incentive for an agent to misreport its preferences.
- Identifiable in polynomial time.
- Reasonable (?) model of actual elections.

70. Back to elicitation
for preferences single-peaked on trees.

71. The main idea:
- Split off the tree into paths.
- Elicit along each path.
- Merge the information across paths.

72. The main idea:
- Split off the tree into paths.
- Elicit along each path.
- Merge the information across paths.
If the number of leaves is t, then the tree can be
partitioned into t disjoint paths.

73. The main idea:
- Split off the tree into paths.
- Elicit along each path.
- Merge the information across paths.
This requires O(|Pi|) queries for the path Pi. Recall that the
paths constitute a partition of m nodes - so the overall
time here is O(m).

74. The main idea:
- Split off the tree into paths.
- Elicit along each path.
- Merge the information across paths.
Merging t lists of m elements in total can be done in time
O(m log t), using divide and conquer.

75. The main idea:
- Split off the tree into paths.
- Elicit along each path.
- Merge the information across paths.
Overall queries: O(m log t)

76. Does your profile have “d outliers” to single-peakedness?

77. Does your profile have “d outliers” to single-peakedness?
O(d log d) + O(m — d) + O(m)

78. Does your profile have “d outliers” to single-peakedness?
O(d log d) + O(m — d) + O(m)
the usual merge sort

79. Does your profile have “d outliers” to single-peakedness?
O(d log d) + O(m — d) + O(m)
the usual merge sort
the single-peaked algorithm

80. Does your profile have “d outliers” to single-peakedness?
O(d log d) + O(m — d) + O(m)
the usual merge sort
the single-peaked algorithm
(putting the outputs together)

81. A quick Lower Bound
for preferences single-peaked on trees.

83. Any path that doesn’t involve the root goes along one of the spines.

89. The number of orderings that are single peaked over
the subdivided star of depth d on t leaves: (t!)d

90. The number of orderings that are single peaked over
the subdivided star of depth d on t leaves: (t!)d
A standard adversary argument gives the following
lower bound: log((t!)d) = dt log t = m log t

92. 1. Vincent Conitzer. Eliciting single-peaked preferences
using comparison queries.
J. Artif. Intell. Res., 35:161–191, 2009.
2. Gabrielle Demange. Single-peaked orders on a tree.
Math. Soc. Sci, 3(4):389–396, 1982.
ReferenceS

93. Thank you!