The Complexity of Llull’s Thirteenth-Century Election Sys - PowerPoint Presentation

Slide1

The Complexity of Llull’s Thirteenth-Century Election System

Piotr FaliszewskiUniversity of Rochester

Jörg RotheInstitute für InformatikHeinrich-Heine-Univ. Düsseldorf

Lane A. HemaspaandraUniversity of Rochester

Edith HemaspaandraRochester Institute of Technology

The Hebrew University of Jerusalem, June 17, 2009

Henning Schnoor

Rochester Institute of TechnologySlide2

Outline

IntroductionComputational study of electionsBribery and controlLlull/Copeland Elections

Model of electionsRepresentation of votes Llull/Copeland ruleResultsBribery and microbriberyControl of electionsManipulation

Hi, I am Ramon Llull. I have come up with the election that these guys now study!Slide3

Introduction

Computational study of electionsApplications in AIMultiagent systemsMulticriterion decision making

Meta search-enginesPlanningApplications in political scienceComputational barrier to cheating in electionsManipulationBribery

Control

Computational agents can systematically analyze an election to find optimal behaviorSlide4

Introduction

Many ways to affect the result of electionEvildoer wants to make someone a winner or prevent someone from winningEvildoer knows everybody else’s votes

ManipulationCoalition of agents changes their voteto obtain their desired effect

BriberyExternal agent, the briber, chooses a group of voters and tells them what votes to castThe briber is limited by budget

ControlOrganizers of the election modify theirstructure to obtain the desired result

In my times it was enough that we all promised we would not cheat...Slide5

Introduction

Manipulation versus bribery and controlControl  attempted by the chair

Bribery  attempted by some outside agentManipulation  attempted by the voters themselvesSelf-interested voters

Bribed voters might choose to not follow what the briber said (unless the voting is public, or maybe not even then)

Okay… I can take the money, but I will vote as I like anyway!Slide6

Outline

IntroductionComputational study of electionsBribery and control

Llull/Copeland ElectionsModel of electionsRepresentation of votes Llull/Copeland ruleResultsBribery and microbriberyControl of electionsManipulation

Let me tell you a bit about my system....Slide7

Voting and Elections

Candidates and voters C = {c1, ..., c

n}V = {v1, ..., vm}

Each voter vi is represented via his or her preferences over C.Assumption: We know all the preferencesStrengthens negative results

Can be justified as wellVoting rule aggregates these preferences and outputs the set of winners.

Hi v

7

, I hope you are not one of those awful

people who support c

3

!

Hi, my name is v

7

.

How will they aggregate those votes?!Slide8

C = { , , }

Representing Preferences

Rational votersPreferences are strict total ordersNo cycles in single voter’s preference list

Example

> >Slide9

C = { , , }

Representing Preferences

Rational votersPreferences are strict total ordersNo cycles in single voter’s preference list

Not all voters are rational though!People often have cyclical preferences!Irrational voters are represented via preference tables.

Example

> >Slide10

C = { , , }

Representing Preferences

Rational votersPreferences are strict total ordersNo cycles in single voter’s preference list

Irrational preferences

Example

> >Slide11

C = { , , }

Representing Preferences

Rational votersPreferences are strict total ordersNo cycles in single voter’s preference list

Irrational preferences

Example

> >Slide12

C = { , , }

Representing Preferences

Rational votersPreferences are strict total ordersNo cycles in single voter’s preference list

Irrational preferences

Example

> >Slide13

C = { , , }

Representing Preferences

Rational votersPreferences are strict total ordersNo cycles in single voter’s preference list

Irrational preferences

Example

> >Slide14

C = { , , }

Representing Preferences

Rational votersPreferences are strict total ordersNo cycles in single voter’s preference list

Irrational preferences

> > >

Example

> >Slide15

Llull/Copeland Rule

The general ruleFor every pair of candidates, ci and cj,

perform a head-to-head plurality contest.The winner of the contest gets 1 pointThe loser gets zero points.At the end of the day, candidates with most points are the winners

Difference between various flavors of the Llull/Copeland rule?What happens if the head-to-head contest ends with a tie?Llull: Both get 1 point

Copeland0: Both get 0 pointsCopeland0.5

: Both get half a pointSlide16

Outline

IntroductionComputational study of electionsBribery and control

Llull/Copeland ElectionsModel of electionsRepresentation of votes Llull/Copeland ruleResultsBribery and microbribery

Control of electionsManipulation

Mr. Llull. Let us see just how resistant your system isSlide17

Bribery

E-bribery (E – an election system)

Given: A set of candidates C, a set of voters V specified via their preference lists, p in C, and budget kQuestion: Can we make p win via bribing at most k voters?E-$briberyAs above, but voters have prices and k is the spending limit.

E-weighted-bribery, E-weighted-$briberyAs the two above, but the voters have weights.

Hmm... I seem to have trouble with finding the right guys to bribe...Slide18

Bribery

E-bribery (E – an election system)

Given: A set of candidates C, a set of voters V specified via their preference lists, p in C, and budget kQuestion: Can we make p win via bribing at most k voters?E-$briberyAs above, but voters have prices and k is the spending limit.

E-weighted-bribery, E-weighted-$briberyAs the two above, but the voters have weights.

Result

Llull/Copeland rule is resistant (NP-hard; warning: a worst-case formalism) to all forms of bribery, both for irrational and rational voters

Mr. Agent. My system is resistant to bribery!Slide19

Microb

ribery

MicrobriberyWe pay for each small change we makeIf we want to make two flips on the preference table of the same voter then we pay 2 instead of 1Comes in the same flavors as briberyLimitationsCould be studied for

rational voters...... But we limit ourselves to the irrational

case.

We do not really need to change each vote completely...

Yeah... It’s easier to work with the preference Matrix™ ...

Preference table, I mean…Slide20

Microb

ribery

MicrobriberyWe pay for each small change we makeIf we want to make two flips on the preference table of the same voter then we pay 2 instead of 1Comes in the same flavors as briberyLimitationsCould be studied for

rational voters...... But we limit ourselves to the irrational

case.Results

Both Llull and Copeland0 are vulnerable to microbribery

Uh oh... How did they do that?!?!?Slide21

Microbribery in Copeland Elections

SettingC = {p=c0, c1,..., c

n}V = {v1, ..., vm}Voters vi are irrational

For each two candidates ci, cj:p

ij – number of flips that switch the head-on-head contest between them

ApproachIf possible, find a bribery that gives p at least B

points, ...

... and

everyone else at most B

points

Try all reasonable B’s

Validate B via min-cost flow problemSlide22

Proof Technique: Flow Networks

Notation:s(ci) – c

i score before briberyB – the point boundK – large number

capacity/cost

c

1

c

n

c

2

p

s

tSlide23

Proof Technique: Flow Networks

Notation:s(ci) – c

i score before briberyB – the point boundK – large number

capacity/cost

c

1

c

n

c

2

p

s

t

source

sink

mesh

source – models pre-

bribery scores

mesh – models bribery cost

sink – models bribery successSlide24

Proof Technique: Flow Networks

Notation:s(ci) – c

i score before briberyB – the point boundK – large number

capacity/cost

c

1

c

n

c

2

p

s

t

s(c

1

)/0

s(c

2

)/0

s(c

n

)/0

s(p)/0

B/0

B/K

B/K

B/K

source

sink

mesh

source – models pre-

bribery scores

mesh – models bribery cost

sink – models bribery successSlide25

Proof Technique: Flow Networks

Notation:s(ci) – c

i score before briberyB – the point boundK – large number

capacity/cost

c

1

c

n

c

2

p

s

t

s(c

1

)/0

s(c

2

)/0

s(c

n

)/0

s(p)/0

B/0

B/K

B/K

B/K

source

sink

mesh

1/p

10

1/p

21

1/p

20

1/p

2n

source – models pre-

bribery scores

mesh – models bribery cost

sink – models bribery successSlide26

Proof Technique: Flow Networks

Notation:s(ci) – c

i score before briberyB – the point boundK – large number

capacity/cost

c

1

c

n

c

2

p

s

t

s(c

1

)/0

s(c

2

)/0

s(c

n

)/0

s(p)/0

B/0

B/K

B/K

B/K

source

sink

mesh

1/p

10

1/p

21

1/p

20

1/p

2n

source – models pre-

bribery scores

mesh – models bribery cost

sink – models bribery successSlide27

Proof Technique: Flow Networks

Notation:s(ci) – c

i score before briberyB – the point boundK – large number

capacity/cost

c

1

c

n

c

2

p

s

t

s(c

1

)/0

s(c

2

)/0

s(c

n

)/0

s(p)/0

B/0

B/K

B/K

B/K

source

sink

mesh

1/p

10

1/p

21

1/p

20

1/p

2n

source – models pre-

bribery scores

mesh – models bribery cost

sink – models bribery successSlide28

Proof Technique: Flow Networks

Notation:s(ci) – c

i score before briberyB – the point boundK – large number

capacity/cost

c

1

c

n

c

2

p

s

t

s(c

1

)/0

s(c

2

)/0

s(c

n

)/0

s(p)/0

B/0

B/K

B/K

B/K

source

sink

mesh

1/p

10

1/p

21

1/p

20

1/p

2n

source – models pre-

bribery scores

mesh – models bribery cost

sink – models bribery successSlide29

Proof Technique: Flow Networks

Notation:s(ci) – c

i score before briberyB – the point boundK – large number

capacity/cost

c

1

c

n

c

2

p

s

t

s(c

1

)/0

s(c

2

)/0

s(c

n

)/0

s(p)/0

B/0

B/K

B/K

B/K

source

sink

mesh

1/p

10

1/p

21

1/p

20

1/p

2n

source – models pre-

bribery scores

mesh – models bribery cost

sink – models bribery successSlide30

Proof Technique: Flow Networks

Notation:s(ci) – c

i score before briberyB – the point boundK – large number

capacity/cost

c

1

c

n

c

2

p

s

t

s(c

1

)/0

s(c

2

)/0

s(c

n

)/0

s(p)/0

B/0

B/K

B/K

B/K

source

sink

mesh

1/p

10

1/p

21

1/p

20

1/p

2n

source – models pre-

bribery scores

mesh – models bribery cost

sink – models bribery success

Cost = K(n

(n+1

)/2

- p-score) + cost-of-bribery Slide31

Microbribery: Application

Round-robin tournamentEveryone plays with everyone elseBribery in round-robin tournaments

For every game there we knowExpected (default) resultThe price for changing itWe want a minimal price for our guy having most pointsRound-robin tournament exampleFIFA World Cup, group stage3 points for win

1 point for tie0 points for lossMicrobribery?Slide32

Microbribery: Application

Round-robin tournamentEveryone plays with everyone elseBribery in round-robin tournaments

For every game there we knowExpected (default) resultThe price for changing itWe want a minimal price for our guy having most pointsRound-robin tournament exampleFIFA World Cup, group stage3 points for win

1 point for tie0 points for lossMicrobribery?Applies directly!!

Given the table of expected results and prices …… simply run the Microbribery algorithmFor FIFA:

Simply use 1/3 as the tie value.Slide33

Outline

IntroductionComputational study of electionsBribery and control

Llull/Copeland ElectionsModel of electionsRepresentation of votes Llull/Copeland ruleResultsBribery and microbribery

Control of electionsManipulation

How will your system deal with my attempts to control, Mr. Llull...?Slide34

Control

Control of electionsThe chair of the election attempts to influence the result via modifying the structure of the electionConstructive control (CC) or destructive control (DC)

Candidate controlAdding candidates (AC)Deleting candidates (DC)Partition of candidates (PC/RPC)with or without the runoff

Voter controlAdding voters (AV)Deleting voters (DV)Partition of voters (PV)

My system is resistant to all types of constructive control!!

Okay, almost all. Slide35

Flavors of control

AC

– adding candidatesDC – deleting candidates(R)PC – (runoff) partition of candidatesAV

– adding votersDV – deleting votersPV – partition of voters

Results: Control

Llull

/ Copeland

0

Plurality

Control

CC

DC

CC

DC

AC

R

V

R

R

DC

R

V

R

R

(R)PC-TP

R

V

R

R

(R)PC-TE

R

V

R

R

PV-TP

R

R

V

V

PV-TE

R

R

R

R

AV

R

R

V

V

DV

R

R

V

V

CC

– constructive control

TP

– ties promote

DC

– destructive control

TE

– ties eliminate

R

– NP-complete

V

– P membershipSlide36

Outline

IntroductionComputational study of electionsBribery and control

Llull/Copeland ElectionsModel of electionsRepresentation of votes Llull/Copeland ruleResultsBribery and microbribery

Control of electionsManipulation

My coalition of agents must be able to somehow manipulate the system!!Slide37

Manipulation

Manipulation problemGiven: Set of candidates Set of honest voters (with their preferences)

Set of manipulatorsPreferred candidate pQuestion: Can the manipulators set their votes so as to guarantee p’s victory?Can’t avoid manipulation!

Gibbard-SatterthwaiteDugan-Schwarz

There Llull! By Gibbard-Satterthwaite result I know I can sometimes manipulate your system. Finally math gets useful!Slide38

Results: Manipulation

Unweighted manipulationNP-complete for all rational tie-handling values except

0, 0.5, 1Even for two manipulators!Proof ideasAlmost a microbribery instance

The two manipulators can only switch head-to-head results from tie to a victoryReduce from X3C,1-in-3-SAT

Can’t use microbribery to solve the problem because the two voters might not be able to implement the solution we would get

Haha Mr. Agent! Gibbard-Satterthwaite say that you may be able to influence my system, but you won’t know when or how!Slide39

Results: Manipulation

Unweighted manipulationNP-complete for all rational tie-handling values except

0, 0.5, 1Even for two manipulators!Aren’t we missing exactly the interesting values?What is magic about the values we miss?

1  no point in causing ties0.5  switching the result of a head-to-head contest from tie to win affect both candidates in a symmetric way

0  makes “transferring points” difficult

So… maybe there is hope for me…Slide40

Results: Manipulation

Weighted manipulationCompletely resolved for three candidatesIssues with for and more candidates

Unique vs. non-unique winnerTie-value 1 is hard to account for.Strange things happen for four candidates.Weighted, three candidates

 = 0

0<<1

 = 1

winner

NPC

NPC

P

unique

winner

NPC

P

P

Work in progressSlide41

Summary

Copeland/Llull electionsBroad resistance to briberyConstructive/DestructiveRational/Irrational

Broad resistance to controlConstructive candidate controlConstructive/Destructive voter controlRational/IrrationalVulnerabilityMicrobribery (irrational case)

Arrgh! Llull, my agents are practically helpless against your system!Slide42

Thank you!

I will happily answer your questions!