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