Transcript hemaspaandralane.pptx

The Complexity of Llull’s
Thirteenth-Century Election
System
Piotr Faliszewski
Edith Hemaspaandra
Lane A. Hemaspaandra
Jörg Rothe
University of Rochester
University of Rochester
Henning Schnoor
Rochester Institute of
Technology
Institute für Informatik
Heinrich-Heine-Univ.
Düsseldorf
Rochester Institute of
Technology
The Hebrew University of Jerusalem, June 17, 2009
Outline
 Introduction
 Computational study of elections
 Bribery and control
 Llull/Copeland Elections
 Model of elections
 Representation of votes
 Llull/Copeland rule
 Results
 Bribery and microbribery
 Control of elections
 Manipulation
Hi, I am Ramon
Llull. I have come
up with the
election that these
guys now study!
Introduction
 Computational study of elections

Applications in AI





Multiagent systems
Multicriterion decision making
Meta search-engines
Planning
Computational agents
can systematically
analyze an election to
find optimal behavior
Applications in political science
 Computational barrier to cheating in elections



Manipulation
Bribery
Control
Introduction

Many ways to affect the result of election

Evildoer wants to make someone a winner or prevent someone
from winning
Evildoer knows everybody else’s votes

Manipulation



Bribery



Coalition of agents changes their vote
to obtain their desired effect
External agent, the briber, chooses a group
of voters and tells them what votes to cast
The briber is limited by budget
Control

Organizers of the election modify their
structure to obtain the desired result
In my times it was
enough that we all
promised we
would not cheat...
Introduction
 Manipulation versus
bribery and control



Control  attempted by
the chair
Bribery  attempted by
some outside agent
Manipulation  attempted
by the voters themselves
Okay… I can take
the money, but I
will vote as I like
anyway!
 Self-interested voters

Bribed voters might choose
to not follow what the
briber said (unless the
voting is public, or maybe
not even then)
Outline
 Introduction
 Computational study of elections
 Bribery and control
 Llull/Copeland Elections
 Model of elections
 Representation of votes
 Llull/Copeland rule
 Results
 Bribery and microbribery
 Control of elections
 Manipulation
Let me tell
you a bit
about my
system....
Voting and Elections
 Candidates and voters


C = {c1, ..., cn}
V = {v1, ..., vm}
 Each voter vi is represented via his or
her preferences over C.

Assumption: We know all the
preferences
Hi, my
name is v7.
Hi v7, I hope you
are not one of
those awful
people who
support c3!
 Strengthens negative results
 Can be justified as well
 Voting rule aggregates these
preferences and outputs the set of
winners.
How will they
aggregate
those votes?!
Representing Preferences
C={
,
,
 Rational voters
 Preferences are strict total
orders
 No cycles in single voter’s
preference list
 Example
>
>
}
Representing Preferences
C={
,
,
 Rational voters
 Preferences are strict total
orders
 No cycles in single voter’s
preference list
 Example
>
>
}
 Not all voters are rational
though!


People often have
cyclical preferences!
Irrational voters are
represented via
preference tables.
Representing Preferences
 Irrational preferences
C={
,
,
 Rational voters
 Preferences are strict total
orders
 No cycles in single voter’s
preference list
 Example
>
>
}
Representing Preferences
 Irrational preferences
C={
,
,
 Rational voters
 Preferences are strict total
orders
 No cycles in single voter’s
preference list
 Example
>
>
}
Representing Preferences
 Irrational preferences
C={
,
,
 Rational voters
 Preferences are strict total
orders
 No cycles in single voter’s
preference list
 Example
>
>
}
Representing Preferences
 Irrational preferences
C={
,
,
 Rational voters
 Preferences are strict total
orders
 No cycles in single voter’s
preference list
 Example
>
>
}
Representing Preferences
 Irrational preferences
C={
,
,
}
 Rational voters
 Preferences are strict total
orders
 No cycles in single voter’s
preference list
 Example
>
>
>
>
>
Llull/Copeland Rule
 The general rule




For every pair of candidates, ci and cj,
perform a head-to-head plurality contest.
The winner of the contest gets 1 point
The 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
Llull:
Copeland0:
Copeland0.5:
if the head-to-head contest ends with a tie?
Both get 1 point
Both get 0 points
Both get half a point
Outline
 Introduction
 Computational study of elections
 Bribery and control
 Llull/Copeland Elections
 Model of elections
 Representation of votes
 Llull/Copeland rule
 Results
 Bribery and microbribery
 Control of elections
 Manipulation
Mr. Llull. Let us
see just how
resistant your
system is
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 k
Question: Can we make p
win via bribing at most k
voters?

E-$bribery


As above, but voters have
prices and k is the
spending limit.
E-weighted-bribery, Eweighted-$bribery

As the two above, but the
voters have weights.
Hmm... I seem
to have trouble
with finding the
right guys to
bribe...
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 k
Question: Can we make p
win via bribing at most k
voters?

E-$bribery


As above, but voters have
prices and k is the
spending limit.
E-weighted-bribery, Eweighted-$bribery

As 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!
Microbribery

Microbribery




We pay for each small
change we make
If we want to make two
flips on the preference
table of the same voter
then we pay 2 instead of 1
Comes in the same flavors
as bribery
Limitations


Could 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…
Microbribery

Microbribery




We pay for each small
change we make
If we want to make two
flips on the preference
table of the same voter
then we pay 2 instead of 1
Comes in the same flavors
as bribery
Limitations


Could 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?!?!?
Microbribery in Copeland Elections

Setting




C = {p=c0, c1,..., cn}
V = {v1, ..., vm}
Voters vi are irrational
For each two candidates ci, cj:


pij – number of flips that
switch the head-on-head
contest between them
Approach


If 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
problem
Proof Technique: Flow Networks
Notation:
p
capacity/cost
s(ci) – ci score before bribery
B – the point bound
K – large number
c1
s
c2
cn
t
Proof Technique: Flow Networks
Notation:
p
capacity/cost
s(ci) – ci score before bribery
B – the point bound
K – large number
c1
s
c2
t
source – models prebribery scores
mesh
sink
– models bribery
cost
– models bribery
success
cn
source
mesh
sink
Proof Technique: Flow Networks
Notation:
p
capacity/cost
s(ci) – ci score before bribery
B – the point bound
s(p)/0
K – large number
s
source – models prebribery scores
mesh
sink
s(c1)/0
B/K
s(c2)/0
B/K
c2
s(cn)/0
– models bribery
cost
– models bribery
success
c1
B/0
t
B/K
cn
source
mesh
sink
Proof Technique: Flow Networks
Notation:
capacity/cost
p
s(ci) – ci score before bribery
B – the point bound
s(p)/0
K – large number
1/p20
c1
1/p10
s(c1)/0
s
source – models prebribery scores
mesh
sink
s(cn)/0
– models bribery
cost
– models bribery
success
s(c2)/0
B/0
B/K
1/p21
c2
1/p2n
B/K
t
B/K
cn
source
mesh
sink
Proof Technique: Flow Networks
Notation:
capacity/cost
p
s(ci) – ci score before bribery
B – the point bound
s(p)/0
K – large number
1/p20
c1
1/p10
s(c1)/0
s
source – models prebribery scores
mesh
sink
s(cn)/0
– models bribery
cost
– models bribery
success
s(c2)/0
B/0
B/K
1/p21
c2
1/p2n
B/K
t
B/K
cn
source
mesh
sink
Proof Technique: Flow Networks
Notation:
capacity/cost
p
s(ci) – ci score before bribery
B – the point bound
s(p)/0
K – large number
1/p20
c1
1/p10
s(c1)/0
s
source – models prebribery scores
mesh
sink
s(cn)/0
– models bribery
cost
– models bribery
success
s(c2)/0
B/0
B/K
1/p21
c2
1/p2n
B/K
t
B/K
cn
source
mesh
sink
Proof Technique: Flow Networks
Notation:
capacity/cost
p
s(ci) – ci score before bribery
B – the point bound
s(p)/0
K – large number
1/p20
c1
1/p10
s(c1)/0
s
source – models prebribery scores
mesh
sink
s(cn)/0
– models bribery
cost
– models bribery
success
s(c2)/0
B/0
B/K
1/p21
c2
1/p2n
B/K
t
B/K
cn
source
mesh
sink
Proof Technique: Flow Networks
Notation:
capacity/cost
p
s(ci) – ci score before bribery
B – the point bound
s(p)/0
K – large number
1/p20
c1
1/p10
s(c1)/0
s
source – models prebribery scores
mesh
sink
s(cn)/0
– models bribery
cost
– models bribery
success
s(c2)/0
B/0
B/K
1/p21
c2
1/p2n
B/K
t
B/K
cn
source
mesh
sink
Proof Technique: Flow Networks
Notation:
capacity/cost
p
s(ci) – ci score before bribery
B – the point bound
s(p)/0
K – large number
1/p20
c1
1/p10
s(c1)/0
s
source – models prebribery scores
mesh
sink
s(cn)/0
– models bribery
cost
– models bribery
success
s(c2)/0
B/0
B/K
1/p21
c2
1/p2n
t
B/K
B/K
cn
source
mesh
sink
Cost = K(n(n+1)/2 - p-score) + cost-of-bribery
Microbribery: Application
 Round-robin tournament

Everyone plays with
everyone else
 Bribery in round-robin
tournaments

For every game there we
know



Expected (default) result
The price for changing it
We want a minimal price
for our guy having most
points
 Round-robin tournament
example

FIFA World Cup, group
stage



3 points for win
1 point for tie
0 points for loss
 Microbribery?
Microbribery: Application
 Round-robin tournament

Everyone plays with
everyone else
 Round-robin tournament
example

 Bribery in round-robin
tournaments

For every game there we
know



Expected (default) result
The price for changing it
We want a minimal price
for our guy having most
points
FIFA World Cup, group
stage



3 points for win
1 point for tie
0 points for loss
 Microbribery?




Applies directly!!
Given the table of expected
results and prices …
… simply run the
Microbribery algorithm
For FIFA: Simply use 1/3
as the tie value.
Outline
 Introduction
 Computational study of elections
 Bribery and control
 Llull/Copeland Elections
 Model of elections
 Representation of votes
 Llull/Copeland rule
 Results
 Bribery and microbribery
 Control of elections
 Manipulation
How will your
system deal with
my attempts to
control, Mr.
Llull...?
Control
 Control of elections


The chair of the election
attempts to influence
the result via modifying
the structure of the
election
Constructive control
(CC) or destructive
control (DC)
My system is
resistant to all types
of constructive
control!!
Okay, almost all.
 Candidate control



Adding candidates (AC)
Deleting candidates
(DC)
Partition of candidates
(PC/RPC)
 with or without the
runoff
 Voter control



Adding voters (AV)
Deleting voters (DV)
Partition of voters (PV)
R – NP-complete
Results: Control
V – P membership
Llull /
Copeland0
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
 Flavors of control
 AC – adding candidates
 DC – deleting candidates
 (R)PC – (runoff) partition of
candidates
 AV – adding voters
 DV – deleting voters
 PV – partition of voters
CC – constructive control
DC – destructive control
TP – ties promote
TE – ties eliminate
Outline
 Introduction
 Computational study of elections
 Bribery and control
 Llull/Copeland Elections
 Model of elections
 Representation of votes
 Llull/Copeland rule
 Results
 Bribery and microbribery
 Control of elections
 Manipulation
My coalition of
agents must be
able to somehow
manipulate the
system!!
Manipulation
 Manipulation problem


Given:
 Set of candidates
 Set of honest voters
(with their preferences)
 Set of manipulators
 Preferred candidate p
Question:
 Can the manipulators
set their votes so as to
guarantee p’s victory?
 Can’t avoid manipulation!


Gibbard-Satterthwaite
Dugan-Schwarz
There Llull! By GibbardSatterthwaite result I
know I can sometimes
manipulate your system.
Finally math gets useful!
Results: Manipulation
 Unweighted manipulation

NP-complete for all rational
tie-handling values except
 Proof ideas

Almost a microbribery
instance

The two manipulators can
only switch head-to-head
results from tie to a victory

Reduce 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
0, 0.5, 1

Even for two manipulators!
Haha Mr. Agent! GibbardSatterthwaite say that you
may be able to influence
my system, but you won’t
know when or how!
Results: Manipulation
 Unweighted manipulation


NP-complete for all rational
tie-handling values except
 What is magic about the
values we miss?
0, 0.5, 1

Even for two manipulators!

 Aren’t we missing exactly
the interesting values?

So… maybe
there is hope
for me…
1  no point in causing
ties
0.5  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
Results: Manipulation
 Weighted manipulation


Work in
progress
Completely resolved for
three candidates
Issues with for and more
candidates

Unique vs. non-unique
winner

Tie-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
Summary
 Copeland/Llull elections

Broad resistance to bribery
 Constructive/Destructive
 Rational/Irrational

Broad resistance to control
 Constructive candidate control
 Constructive/Destructive voter control
 Rational/Irrational

Vulnerability
 Microbribery (irrational case)
Arrgh! Llull, my
agents are practically
helpless against your
system!
Thank you!
I will happily
answer your
questions!