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Campaign Campaign Management via Management via
BriberyBribery
Piotr FaliszewskiAGH University of
Scienceand Technology, Poland
Joint work with Edith Elkind and Arkadii Slinko
◦ Manipulation
◦ Control
◦ Bribery
COMSOC and VotingCOMSOC and Voting
Computational social choice- group decision making
BriberyBribery
Bribery
◦ Invest money to change votes
◦ Some votes are cheaper than others
◦ Want to spend as little as possible
Campaign management◦ Invest money to
change voters’ minds
◦ Some voters are easier to convince
◦ The campaign should be as cheap as possible
vs Campaign vs Campaign ManagementManagement
AgendaAgenda Introduction
◦ Standard model of elections◦ Election systems
Swap bribery◦ Cost model◦ Basic problems◦ Complexity of swap bribery
Shift bribery◦ Why useful?◦ Algorithms for shift bribery
Conlusions and open problems
Election ModelElection ModelElection E = (C,V)
◦ C – the set of candidates◦ V – the set of voters
A candidate set
Election ModelElection ModelElection E = (C,V)
◦ C – the set of candidates◦ V – the set of voters
A vote (preference order)
> > >
Election ModelElection ModelElection E = (C,V)
◦ C – the set of candidates◦ V – the set of voters
> > >
> > >
> > >
3 2 1 0
Borda count
= 6
= 5
= 4
= 3
Many other elections systems studied! E.g, Plurality, k-approval, maximin, Copeland
Many other elections systems studied! E.g, Plurality, k-approval, maximin, Copeland
Bribery ModelsBribery Models
Standard bribery◦ Payment ==> full control over a vote
Nonuniform bribery◦ Payment depends on the amount of change
Problem: How to represent the prices?
Swap BriberySwap BriberyPrice function π for each voter.
> > >
π( , ) = 5
Swap BriberySwap BriberyPrice function π for each voter.
> > >
π( , ) = 2π( , ) = 5
Swap BriberySwap BriberyPrice function π for each voter.
Swap bribery problem◦ Given: E = (C,V), price function for each
voter◦ Question: What is the cheapest sequence of
swaps that makes our guy a winner?
> > >
π( , ) = 2
Questions About Swap Questions About Swap BriberyBriberyPrice of reaching a given vote?
Swap bribery and other voting problems?
Complexity of swap bribery?
> > > > > >
Voting problem Swap bribery<m
Relations Between Voting Relations Between Voting ProblemsProblems
The Complexity of Swap BriberyThe Complexity of Swap Bribery
Voting rule Swap bribery
Plurality P
Veto P
k-approval NP-com
Borda NP-com
Maximin NP-com
Copeland NP-comLimit the
number of candidates
?
Limit the number of candidates
?
Limit the number
of voters?
Limit the number
of voters?
Limit the types of swaps?
Limit the types of swaps?
Shift BriberyShift BriberyAllowed swaps:
◦ Have to involve our candidate
Realistic?◦ As bribery: Yes◦ Also: as a campaigning model!
Gain in complexity?
Voting rule Swap bribery Shift bribery
The Complexity of Swap BriberyThe Complexity of Swap Bribery
Plurality P P
Veto P P
k-approval NP-com P
Borda NP-com NP-com
Maximin NP-com NP-com
Copeland NP-com NP-com
Voting rule Swap bribery Shift bribery Approx.ratio
The Complexity of Swap BriberyThe Complexity of Swap Bribery
Plurality P P ―
Veto P P ―
k-approval NP-com P ―
Borda NP-com NP-com 2
Maximin NP-com NP-com O(logm)
Copeland NP-com NP-com O(m)
Voting rule Swap bribery Shift bribery Approx.ratio
The Complexity of Swap BriberyThe Complexity of Swap Bribery
Plurality P P ―
Veto P P ―
k-approval NP-com P ―
Borda NP-com NP-com 2
Maximin NP-com NP-com O(logm)
Copeland NP-com NP-com O(m)
Single algorithm for all scoring protocols, even if weighted!
The AlgorithmThe Algorithm
Why 2-approximation?
> > >αiαi+1
The AlgorithmThe Algorithm
Why 2-approximation?
> > >αiαi+1
gains αi+1 – αi points
loses αi+1 – αi points
Getting 2x the points for than the best bribery gives is sufficient to win
The AlgorithmThe Algorithm
Why 2-approximation?
> > >αiαi+1
gains αi+1 – αi points
loses αi+1 – αi points
Getting 2x the points for than the best bribery gives is sufficient to win
Operation of the algorithm
1.Guess a cost k
2.Get most points for at cost k
3.Guess a cost k’ <= k
4.Get most points for at cost k’
This is a 2-approximation… but works in polynomial time only if prices are encoded in unary
Why Does the Algorithm Work?Why Does the Algorithm Work?
Operation of the algorithm
1.Guess a cost k2.Get most points for p at cost k3.Guess a cost k’ <= k4.Get most points for p at cost k’
How much does optimal solution shift candidate p in each vote?
O – the optimal solution gives p some T points
v1 v5v3 v4v2
Why Does the Algorithm Work?Why Does the Algorithm Work?
How much does optimal solution shift candidate p in each vote?
O – the optimal solution gives p some T points
v1 v5v3 v4v2
Why Does the Algorithm Work?Why Does the Algorithm Work?
How much does optimal solution shift candidate p in each vote?
O – the optimal solution gives p some T points
v1 v5v3 v4v2
S – solution that gives most points at cost k
Why Does the Algorithm Work?Why Does the Algorithm Work?
How much does optimal solution shift candidate p in each vote?
O – the optimal solution gives p some T points
v1 v5v3 v4v2
S – solution that gives most points at cost k
min(O,S) – min shift of the two in each votegives some D points to p
Now it is possible to complete min(O,S) in two independent ways:1.By continuing as S does (getting at least T-D points extra)2.By continuing as O does (getting T-D points extra)
Why Does the Algorithm Work?Why Does the Algorithm Work?
How much does optimal solution shift candidate p in each vote?
Now it is possible to complete min(O,S) in two independent ways:1.By continuing as S does (getting at least T-D points extra)2.By continuing as O does (getting T-D points extra)
After we perform shifts from min(O,S), there is a way to make p win by shifts that give him T-D points
Thus, any shift that gives him 2(T-D) points, makes him a winner.
It is easy to find these 2(T-D) points. We’re done!
v1 v5v3 v4v2
The Algorithm (General Case)The Algorithm (General Case)
2-approximation algorithm for unary
prices
2+ε-approximation scheme for any prices
2-approximation algorithm for any
prices
Scaling argument + twists
Bootstrapping-flavored argument
The AlgorithmThe Algorithm
Why 2-approximation?
> > >αiαi+1
gains αi+1 – αi points
loses αi+1 – αi points
Operation of the algorithm
1.Guess a cost k
2.Get most points for at cost k
3.Guess a cost k’ <= k
4.Get most points for at cost k’
Is this algorithm still a 2-approximation? Unclear!
ConclusionsConclusionsSwap bribery
◦ Interesting model◦ Many hardness results◦ Connection to possible winner
Special cases◦ Fixed #candidates, fixed #voters boring◦ Shift bribery
Realistic Lowers the complexity Interesting approximation issues