Voting, Algorithms, and Complexity
Palash Dey; Supervisors: Y. Narahari and Arnab Bhattacharyya; Indian Institute of Science, Bangalore, India.
14th International Conference on Autonomous Agents and Multiagent Systems
Voting Applications
Frugal Bribery
Figure : Movie ratings
Figure : Collaborative filtering
Motivation
Bribery in the setting where the briber is frugal in nature which is captured by her
inability to bribe certain types of voters.
Vulnerable votes
Let v1, · · · , vn be a set of voters which makes some candidate x win. Let y be
another candidate. A voter vi is called y -vulnerable if vi prefers y to x.
Frugal-bribery
Given a set of votes and a candidate y , does there exist a way to change the
y -vulnerable votes that makes y win?
Frugal-$bribery
Given a set of votes, a candidate y , costs of y -vulnerable votes, and a budget B,
does there exist a way to change some of the y -vulnerable votes subject to budget
constraint B that makes y win?
Key Results
We show that two problems are intractable for many common voting rules.
Kernelization for Possible Winner
Figure : Search engine
Figure : Crowd sourcing
Voting Setting
• V - a set of n voters. C - a set of m candidates.
• Complete vote - a complete order over C . L(C ) -
set of complete orders over C .
• Partial vote - a partial order over C .
• Voting rule - r : L(C )n −→ C .
¯ = (α1, · · · , αm) ∈ Rm.
Scoring rule : Defined by α
Candidate ranked at i th position gets score αi . Winner
is the candidate with highest score.
¯ = (m − 1, m −
Borda rule : Scoring rule with α
2, · · · , 1, 0).
Maximin rule : Winner is the candidate with minimum margin of victory in its worst pairwise election.
Copeland rule : Winner is the candidate with maximum pairwise wins.
Bucklin rule : Winner is the candidate getting majority within minimum number of top positions.
Margin of victory - smallest number of vote changes required to change winner.
Thesis Overview
Thesis: Voting, Algorithms, and Complexity
Election Manipulation
Partial Information
Manipulation detection [AAMAS 2015]
Kernelization for Possible
Winner [AAMAS 2015]
Frugal bribery
Manipulation under
partial information
Prediction Algorithms
Winner prediction
[AAMAS 2015]
Possible Winner
Given a voting rule r , a set of partial votes P, and a candidate c, does there exist
an completion of the partial votes in P to linear votes that makes the candidate c
win the election.
Coalitional Manipulation
Given a voting rule r , a set of complete votes P corresponding to the votes of the
non-manipulators, a non-empty set of manipulators M, and a candidate c, does
there exist a way to cast the manipulators’ votes such that c wins the election.
Question
What are the kernelization complexity of above problems?
Key Results
The Possible Winner problem does not admit a polynomial kernel where as
the Coalitional Manipulation problem admits a polynomial kernel.
Manipulation under Partial Information
Motivation
Manipulators usually have only partial knowledge about the truthful votes.
Weak Manipulation
Given a voting rule r , a set of partial votes P corresponding to the votes of the
non-manipulators, a non-empty set of manipulators M, and a candidate c, does
there exist a way to cast the manipulators’ votes such that c wins the election in
least one extension of the partial votes in P.
Strong Manipulation
Given a voting rule r , a set of partial votes P corresponding to the votes of the nonmanipulators, a non-empty set of manipulators M, and a candidate c, does there
exist a way to cast the manipulators’ votes such that c always wins the election in
every extension of the partial votes in P.
Key Results
Manipulation becomes intractable for most of the voting rules in this setting.
Winner Prediction
Motivation
Margin of victory
estimation [IJCAI 2015]
Manipulation Detection
Motivation
Coalition of possible manipulators:
Given an r -election, 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
Problem Formulation
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
Key Results
We study computational complexity of above problems for various voting rules. For the
Borda voting rule, the results are very promising.
Borda: manipulation is NPC.
Borda: Detecting manipulation is easy.
Predicting the winner of an election is a basic problem in social choice theory. We
study the sample complexity of this problem.
(ε, δ)–Winner determination
Given a voting rule and a set of n votes over a set of m candidates such that the
margin of victory is at least εn, determine the winner of the election with probability
at least 1 − δ.
Results
We obtain tight upper and lower bounds on the sample complexity for winner
prediction. Our results show that the winner can be predicted quite accurately by
sampling only a few votes.
Margin of Victory Estimation
Motivation
In polling, surveys, and audits, the sample size depends on the margin of victory of
the election. Hence, given an election, we need to estimate its margin of victory.
(c, ε, δ)–Margin of Victory
Given a r -election E, determine margin of victory (MoV), the margin of victory of E
with respect to r , within an additive error of at most c · MoV + εn with probability
at least 1 − δ.
Results
We obtain tight upper and lower bounds on the sample complexity for estimating
margin of victory. Our results show that the margin of victory of an election can
be estimated quite accurately by sampling only a few votes.