Campaign Management via
Bribery
Piotr Faliszewski
AGH University of Science
and Technology, Poland
Joint work with Edith Elkind and Arkadii Slinko
COMSOC and Voting
Computational social choice
- group decision making
◦ Manipulation
◦ Control
◦ Bribery
Bribery vs Campaign Management

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
Agenda

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 Model

Election E = (C,V)
◦ C – the set of candidates
◦ V – the set of voters
A candidate set
Election Model

Election E = (C,V)
◦ C – the set of candidates
◦ V – the set of voters
>
>
A vote (preference order)
>
Election Model

Election E = (C,V)
◦ C – the set of candidates
◦ V – the set of voters
3
2
1
=6
0
> other>elections
> systems
Many
=5
studied! E.g, Plurality, k-approval,
=4
maximin, Copeland
>
>
>
>
>
>
=3
Bribery 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 Bribery

Price function π for each voter.
>
>
>
π(
,
)=5
Swap Bribery

Price function π for each voter.
>
>
>
π( , )π(= 2 ,
)=5
Swap Bribery

Price 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 Bribery

Price of reaching a given vote?
>

> >
> >
Swap bribery and other voting problems?
Voting problem

>
<m
Complexity of swap bribery?
Swap bribery
Relations Between Voting Problems
The Complexity of Swap Bribery
Voting rule
Swap bribery
Plurality
P
Veto
P
k-approval
NP-com
Borda
NP-com
Maximin
NP-com
Copeland
NP-com
Limit the
number of
voters?
Limit the
number of
candidates?
Limit the
types of
swaps?
Shift Bribery

Allowed swaps:
◦ Have to involve our candidate

Realistic?
◦ As bribery: Yes
◦ Also: as a campaigning model!

Gain in complexity?
The Complexity of Swap Bribery
Voting rule
Swap bribery
Shift 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
The Complexity of Swap Bribery
Voting rule
Swap bribery
Shift bribery
Approx.ratio
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)
The Complexity of Swap Bribery
Voting rule
Swap bribery
Shift bribery
Approx.ratio
―
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 Algorithm
Why 2-approximation?
> >
αi+1 αi
>
The Algorithm
Why 2-approximation?
> >
>
αi+1 αi
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 Algorithm
Why 2-approximation?
> >
>
αi+1 αi
Operation of the algorithm
1. Guess a cost k
2. Get most points for
at cost k
3. Guess a cost k’ <= k
gains αi+1 – αi points
loses αi+1 – αi points
Getting 2x the points for
than the best bribery gives
is sufficient to win
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?
How much does
optimal solution
shift candidate p
in each vote?
v1
v2
v3
v4
v5
O – the optimal solution  gives p some T points
Operation of the algorithm
1.
2.
3.
4.
Guess a cost k
Get most points for p at cost k
Guess a cost k’ <= k
Get most points for p at cost k’
Why Does the Algorithm Work?
How much does
optimal solution
shift candidate p
in each vote?
v1
v2
v3
v4
v5
O – the optimal solution  gives p some T points
Why Does the Algorithm Work?
How much does
optimal solution
shift candidate p
in each vote?
v1
v2
v3
v4
v5
O – the optimal solution  gives p some T points
S – solution that gives most points at cost k
Why Does the Algorithm Work?
How much does
optimal solution
shift candidate p
in each vote?
v1
v2
v3
v4
v5
O – the optimal solution  gives p some T points
S – solution that gives most points at cost k
min(O,S) – min shift of the two in each vote
gives 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?
How much does
optimal solution
shift candidate p
in each vote?
v1
v2
v3
v4
v5
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!
The Algorithm (General Case)
2-approximation algorithm
for unary prices
Scaling argument + twists
2+ε-approximation scheme
for any prices
Bootstrapping-flavored argument
2-approximation algorithm
for any prices
The Algorithm
Why 2-approximation?
> >
>
αi+1 αi
Operation of the algorithm
1. Guess a cost k
2. Get most points for
at cost k
3. Guess a cost k’ <= k
gains αi+1 – αi points
loses αi+1 – αi points
4. Get most points for
at cost k’
Is this algorithm still a 2approximation? Unclear!
Conclusions

Swap 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