Detecting Possible Manipulators in Elections
Palash Dey
Coauthors : Neeldhara Misra and Y. Narahari
Department of Computer Science & Automation
Indian Institute of Science Bangalore
May 8, 2015
In Proceedings of the 14th International Conference on Autonomous Systems and
Multiagent Systems (AAMAS-15), pp. 1441 − 1450, Istanbul, Turkey, 2015
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Manipulation Detection in Voting
May 8, 2015
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Motivation
Voting
Applications of Voting
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
2 / 13
Motivation
Voting
Applications of Voting
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
2 / 13
Motivation
Voting
Applications of Voting
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
2 / 13
Motivation
Voting
Applications of Voting
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
2 / 13
Motivation
Manipulation in Election
Motivation
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Manipulation Detection in Voting
May 8, 2015
3 / 13
Motivation
Manipulation in Election
Motivation
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
3 / 13
Motivation
Manipulation in Election
Motivation
Manipulation: misreporting votes may lead to better
outcome
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
3 / 13
Motivation
Manipulation in Election
Motivation
Manipulation: misreporting votes may lead to better
outcome
‘Reasonable’ voting rules are manipulable [Gibbard,
1973; Satterthwaite, 1975]
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Manipulation Detection in Voting
May 8, 2015
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Motivation
Manipulation in Election
Motivation
Have non-manipulable rules in restricted domains
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
4 / 13
Motivation
Manipulation in Election
Motivation
Have non-manipulable rules in restricted domains
Computational intractability
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
4 / 13
Motivation
Manipulation in Election
Motivation
Have non-manipulable rules in restricted domains
Not sure about domain
Computational intractability
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
4 / 13
Motivation
Manipulation in Election
Motivation
Have non-manipulable rules in restricted domains
Not sure about domain
Computational intractability
Easy on average
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
4 / 13
Motivation
Manipulation in Election
Motivation
Have non-manipulable rules in restricted domains
Not sure about domain
Computational intractability
Easy on average
No satisfactory solution for
manipulation prevention
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
4 / 13
Motivation
Manipulation Detection
Motivation
I
Manipulation detection in real life:
Figure : London Olympic 2012
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Figure : Fifa World Cup 1982
Manipulation Detection in Voting
May 8, 2015
5 / 13
Motivation
Manipulation Detection
Motivation
I
Manipulation detection in real life:
Figure : London Olympic 2012
Figure : Fifa World Cup 1982
Formal study of manipulation detection
Main Contribution
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Manipulation Detection in Voting
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Preliminaries
Voting Setting
Setting
I A set C of m candidates
I
A set V of n votes
I
Vote - a complete order over C
I
Voting rule - r : L(C )n −→ C
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Manipulation Detection in Voting
May 8, 2015
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Preliminaries
Voting Setting
Setting
I A set C of m candidates
I
A set V of n votes
I
Vote - a complete order over C
I
Voting rule - r : L(C )n −→ C
Example
I
I
C = {x, y, z}
Votes
X
X
X
Vote 1: x > y > z
Vote 2: z > y > x
Vote 3: x > z > y
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Plurality rule: winner is
candidate with most top
positions
Plurality winner: x
Manipulation Detection in Voting
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Preliminaries
Example: Scoring Rules
Scoring Rule
I
Score vector: (α1 , . . . , αm ) ∈ Rm
I
A vote x1 > x2 > · · · > xm ⇒ xi gets score αi
I
Winner: candidate with highest score
Important Special Cases
I
Plurality: (1, 0, · · · , 0)
I
Veto: (0, · · · , 0, −1)
I
Borda: (m − 1, m − 2, · · · , 0)
Figure : Borda rule
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Manipulation Detection in Voting
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Problem Definition
Problem Formulation
Fix any voting rule r
Example
r(· · · ,
|{z}
,···) = x
reported preference of voter i
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Manipulation Detection in Voting
May 8, 2015
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Problem Definition
Problem Formulation
Fix any voting rule r
Example
r(· · · ,
,···) = x
|{z}
reported preference of voter i
r(· · · , · · · 0
|
x
0 · · ·
{z
y
0 · · · , · · · ) =
}
y
possible actual preference of voter i
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
8 / 13
Problem Definition
Problem Formulation
Fix any voting rule r
Example
r(· · · ,
,···) = x
|{z}
reported preference of voter i
r(· · · , · · · 0
|
x
0 · · ·
{z
y
0 · · · , · · · ) =
}
y
possible actual preference of voter i
Coalition of possible manipulators
A subset of voters M ⊂ V is a CPM if there exists 0M such that:
r(M , V \M ) 0M r(0M , V \M )
We call r(0M , V \M ) the actual winner
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
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Problem Definition
Problem Formulation
Fix any voting rule r
Input: election
M ⊂ V given. Question:
is M a coalition of
possible manipulators?
Find: a coalition of
possible manipulators M with |M | = k .
Actual winner
given: CPMW
Actual winner
given: CPMSW
Actual winner
not given: CPM
Actual winner
not given: CPMS
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
9 / 13
Problem Definition
Problem Formulation
Fix any voting rule r
Input: election
M ⊂ V given. Question:
is M a coalition of
possible manipulators?
Find: a coalition of
possible manipulators M with |M | = k .
Actual winner
given: CPMW
Actual winner
given: CPMSW
Actual winner
not given: CPM
Actual winner
not given: CPMS
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
9 / 13
Problem Definition
Problem Formulation
Fix any voting rule r
Input: election
M ⊂ V given. Question:
is M a coalition of
possible manipulators?
Find: a coalition of
possible manipulators M with |M | = k .
Actual winner
given: CPMW
Actual winner
given: CPMSW
Actual winner
not given: CPM
Actual winner
not given: CPMS
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
9 / 13
Problem Definition
Problem Formulation
Fix any voting rule r
Input: election
M ⊂ V given. Question:
is M a coalition of
possible manipulators?
Find: a coalition of
possible manipulators M with |M | = k .
Actual winner
given: CPMW
Actual winner
given: CPMSW
Actual winner
not given: CPM
Actual winner
not given: CPMS
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
9 / 13
Problem Definition
Problem Formulation
Fix any voting rule r
Input: election
M ⊂ V given. Question:
is M a coalition of
possible manipulators?
Find: a coalition of
possible manipulators M with |M | = k .
Actual winner
given: CPMW
Actual winner
given: CPMSW
Actual winner
not given: CPM
Actual winner
not given: CPMS
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
9 / 13
Problem Definition
Problem Formulation
Fix any voting rule r
Input: election
M ⊂ V given. Question:
is M a coalition of
possible manipulators?
Find: a coalition of
possible manipulators M with |M | = k .
Actual winner
given: CPMW
Actual winner
given: CPMSW
Actual winner
not given: CPM
Actual winner
not given: CPMS
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
9 / 13
Problem Definition
Problem Formulation
Fix any voting rule r
Input: election
M ⊂ V given. Question:
is M a coalition of
possible manipulators?
Find: a coalition of
possible manipulators M with |M | = k .
Actual winner
given: CPMW
Actual winner
given: CPMSW
Actual winner
not given: CPM
Actual winner
not given: CPMS
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
9 / 13
Results
Results
Voting Rule
CPM, k = 1
CPM
CPMW, k = 1
CPMW
Scoring Rules
P
?
P
?
Borda
P
P
P
P
k -approval
P
P
P
P
Maximin
P
?
P
NPC
Bucklin
P
P
P
P
STV
NPC
NPC
NPC
NPC
Borda: manipulation is NPC [Davies et al., 2011].
Borda: Detecting manipulation is easy.
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
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Future Work
Future Work
I
Verifying the number of false positives that this model catches in
a real or synthetic data set
I
Does any practical model on voters’ true preference reduce false
positives?
Dept of CSA (IISc)
Manipulation Detection in Voting
May 8, 2015
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Thanks You
Questions?
:
palash @ csa . iisc . ernet . in
Acknowledgement:
I
All the images in this presentation are taken from Google images
I
MHRD, India, for providing scholarship
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Manipulation Detection in Voting
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References
References
Davies, J., Katsirelos, G., Narodytska, N., and Walsh, T. (2011).
Complexity of and algorithms for borda manipulation. In
Proceedings of the International Conference on Artificial
Intelligence (AAAI), pages 657–662.
Gibbard, A. (1973). Manipulation of voting schemes: a general result.
Econometrica: Journal of the Econometric Society, pages 587–601.
Satterthwaite, M. (1975). Strategy-proofness and arrow’s conditions:
Existence and correspondence theorems for voting procedures and
social welfare functions. Journal of Economic Theory (JET),
10(2):187–217.
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Manipulation Detection in Voting
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