Campaign Management via Bribery Piotr Faliszewski AGH University of Science and Technology, Poland...
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Transcript of Campaign Management via Bribery Piotr Faliszewski AGH University of Science and Technology, Poland...
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