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