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Title: The Complexity of Elections: A New Domain for Heuristic Computation


1
The Complexity of Elections A New Domain for
Heuristic Computation
  • Piotr Faliszewski
  • AGH University of Science and Technology, Kraków,
    Poland
  • faliszew_at_agh.edu.pl

2
Why Are Elections Important?
  • Politics
  • Electing leaders
  • Deciding policies, laws
  • Business settings
  • Stock holders make decisions in companies
  • University Senates, hiring decisions
  • Competitions (e.g., racing, Eurovision)
  • Multiagent systems
  • Meta-search engines
  • Planning

Elections aggregate the opinions of the voters
3
Gibbard-Satterthwaite Theorem
Computational issuesin elections
Cheating in elections
Winner determination
Manipulation
control
Running theelection
bribery
Possiblewinner
Campaignmanagement
4
Complexity Barrier Approach
Model Represent each cheating strategy as a
computational decision problem.
Complexity barrier approach If manipulating
elections is hard, then we can ignore the fact
that it is in principle possible.
Approach initiated by Bartholdi, Tovey, and Trick
in the late 80s and the early 90s
5
Complexity Barrier Results
  • Effects of complexity barrier research
  • Dozens of computational problems identified
  • Multiple standard election systems analyze
  • Quite thorough understanding of worst case
    complexity of elections
  • Complications
  • We would like some of the problems to be
    efficiently computable
  • Determining winners
  • Organizing a campaign
  • Worst-case analysis seems problematic

Room for heuristic computation!
6
Agenda
  • Why study elections?
  • Formal model
  • What are elections?
  • How to model votes?
  • What voting rules to use?
  • Case study Plurality bribery
  • Simplest case
  • Deterministic solution
  • Case study Campaign management
  • Campaign management as bribery
  • Why is it good to study?
  • What has been done? Where to take data from?
  • Conclusions and comments

7
How to Study Elections?
  • We need to answer several questions
  • What are elections?
  • How can we model voters preferences?
  • What election rules are we interested in?

A mathematical object, a pair E (C,V). C set
of candidates, V set of voters
A single top guy? A ranking of candidates?
Numerical values, utility a voter derives from
having a particular candidate elected?
An election rule is a function that given an
election returns the set of winners.
8
Election Model Example
C , , , ,
  • Election E (C,V)
  • C candidate set
  • V voter set
  • Who is the winner?
  • Plurality
  • Veto
  • Borda
  • k-approval
  • Approval
  • Copeland
  • Llull
  • Dodgson
  • Kemeny
  • Young

V gt gt gt gt , gt gt gt gt , gt gt gt gt
, gt gt gt gt , gt gt gt gt
families of scoring protocols
9
Election Model Example
C , , , ,
  • Election E (C,V)
  • C candidate set
  • V voter set

a ( 4 , 3 , 2 , 1 , 0 )
  • Who is the winner?
  • Borda count

V gt gt gt gt , gt gt gt gt , gt gt gt gt
, gt gt gt gt , gt gt gt gt
score( ) 13
score( ) 12
score( ) 9
score( ) 8
score( ) 8
10
Election Model Example
C , , , ,
  • Election E (C,V)
  • C candidate set
  • V voter set
  • Who is the winner?
  • Copeland

V gt gt gt gt , gt gt gt gt , gt gt gt gt
, gt gt gt gt , gt gt gt gt





11
Agenda
  • Why study elections?
  • Formal model
  • What are elections?
  • How to model votes?
  • What voting rules to use?
  • Case study Plurality bribery
  • Simplest case
  • Deterministic solution
  • Case study Campaign management
  • Campaign management as bribery
  • Why is it good to study?
  • Conclusions and comments

12
Computational issuesin elections
Cheating in elections
Winner determination
Manipulation
control
Running theelection
bribery
Possiblewinner
Campaignmanagement
13
Bribery
  • Notation ? an election system

Name ?-bribery. Given An election E (C,V),
a candidate p in C, and an integer k. Question
Can we make p a winner via modifying votes of at
most k voters?
Many flavors of the bribery problem.
  • ?-bribery
  • ?-bribery
  • ?-weighted-bribery
  • ?-weighted-bribery

14
Bribery in Plurality
  • Setting
  • We are given
  • An election E (C,V)
  • A preferred candidate p
  • A budget k
  • Question
  • Can we make p a winner by bribing voters of cost
    at most k?
  • Observation
  • It is sufficient to only look at each voters
    most preferred candidate

unpriced priced
unweighted
weighted
15
Bribery in Plurality
  • Example
  • What is the cheapest way to make Alice a winner?
  • Setting
  • We are given
  • An election E (C,V)
  • A preferred candidate p
  • A budget k
  • Question
  • Can we make p a winner by bribing voters of cost
    at most k?

16
Bribery in Plurality
  • Example
  • What is the cheapest way to make Alice a winner?
  • Setting
  • We are given
  • An election E (C,V)
  • A preferred candidate p
  • A budget k
  • Question
  • Can we make p a winner by bribing voters of cost
    at most k?

17
Bribery in Plurality
  • Example
  • What is the cheapest way to make Alice a winner?
  • Setting
  • We are given
  • An election E (C,V)
  • A preferred candidate p
  • A budget k
  • Question
  • Can we make p a winner by bribing voters of cost
    at most k?

18
Bribery in Plurality
  • Example
  • What is the cheapest way to make Alice a winner?
  • Setting
  • We are given
  • An election E (C,V)
  • A preferred candidate p
  • A budget k
  • Question
  • Can we make p a winner by bribing voters of cost
    at most k?

19
Bribery in Plurality
  • Setting
  • We are given
  • An election E (C,V)
  • A preferred candidate p
  • A budget k
  • Question
  • Can we make p a winner by bribing voters of cost
    at most k?

unpriced priced
unweighted
weighted
20
Computational Complexity
  • NP
  • Class of problems whose solutions are of
    polynomial size, and are verifiable in polynomial
    time
  • Some problems in NP
  • SAT is a given Boolean formula satisfiable?
  • Given a sequence of integers (s1, , sn), is
    there a subsequence that sums up to a given value
    K?
  • Does a graph have a clique of size n?
  • NP-com An NP problem is NP-complete if all NP
    problems reduce to it
  • Reduction between problems
  • A, B two problems
  • A reduces to B if there is a polynomial-time
    computable function f such thatx in A ? f(x)
    in B

f
A
B
f
21
plurality-weighted-bribery ? NP-com
Proof Reduction from the subset-sum problem
Subset-Sum
plurality-weighted-bribery
k 17
Input s1, s2, , sn
?34
12
5
12
1
10
6
10
6
Question Is there a subsequence that sums up to
exactly half.
5
1
22
plurality-weighted-bribery ? NP-com
Proof Reduction from the subset-sum problem
Subset-Sum
plurality-weighted-bribery
k 17
Input s1, s2, , sn
?34
5
12
1
10
6
1
12
6
Question Is there a subsequence that sums up to
exactly half.
10
5
23
Bribery in Plurality
  • Setting
  • We are given
  • An election E (C,V)
  • A preferred candidate p
  • A budget k
  • Question
  • Can we make p a winner by bribing voters of cost
    at most k?

unpriced priced
unweighted
weighted
24
Weighted Bribery in Plurality
  • plurality-weighted-bribery
  • Now voters have weights
  • What algorithms can we use?
  • Heuristics
  • Heaviest voter first
  • Winners heaviest voter first

25
Bribery in Plurality
  • Setting
  • We are given
  • An election E (C,V)
  • A preferred candidate p
  • A budget k
  • Question
  • Can we make p a winner by bribing voters of cost
    at most k?

unpriced priced
unweighted
weighted
26
Agenda
  • Why study elections?
  • Formal model
  • What are elections?
  • How to model votes?
  • What voting rules to use?
  • Case study Plurality bribery
  • Simplest case
  • Deterministic solution
  • Case study Campaign management
  • Campaign management as bribery
  • Why is it good to study?
  • What has been done? Where to take data from?
  • Conclusions and comments

27
Computational issuesin elections
Cheating in elections
Winner determination
Manipulation
control
Running theelection
bribery
Possiblewinner
Campaignmanagement
28
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

29
Bribery Models
  • Standard bribery
  • Payment gt full control over a vote
  • Nonuniform bribery
  • Payment depends on the amount of change

Problem How to represent the prices?
30
Swap Bribery
  • Price function p for each voter.

gt
gt
gt
p( , ) 5
31
Swap Bribery
  • Price function p for each voter.

gt
gt
gt
p( , ) 5
p( , ) 2
32
Swap Bribery
  • Price function p 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?

gt
gt
gt
p( , ) 2
33
Questions About Swap Bribery
  • Price of reaching a given vote?
  • Swap bribery and other voting problems?
  • Complexity of swap bribery?

gt
gt
gt
gt
gt
gt
Swap bribery
Voting problem
ltm
34
Relations Between Voting Problems
35
Results on Swap Bribery
  • Swap Bribery is easy for
  • Plurality
  • Veto
  • Swap bribery is NP-hard for
  • Borda
  • Copeland
  • Bucklin
  • Maximin
  • Ranked pairs
  • Plurality with runoff
  • STV
  • Swap bribery can model winner problems
  • Kemeny
  • Dodgson

36
Why Study Swap Bribery?
Swap bribery is hard for almost all voting systems
Swap bribery is a generalization of most
important problems
37
Agenda
  • Why study elections?
  • Formal model
  • What are elections?
  • How to model votes?
  • What voting rules to use?
  • Case study Plurality bribery
  • Simplest case
  • Deterministic solution
  • Case study Campaign management
  • Campaign management as bribery
  • Why is it good to study?
  • What has been done? Where to take data from?
  • Conclusions and comments

38
Dealing with Complexity
  • Approximation algorithms
  • Some success for manipulation
  • Limited types of swap bribery
  • Parameterized complexity attacks
  • Mostly smart brute-force algorithms
  • Heuristic attacks
  • So far, only on manipulation instances
  • Limited type of algorithms (ad-hoc approaches)
  • Manipulation ? solved (Toby Walsh)
  • others ? untouched

39
Where to Take Data From?
Generate data
  • Real Elections
  • difficult to obtain
  • additional sampling

Impartial culture (votes chosen Independently at
random)
Polya-Eggenberger Urn Model An urn with all m!
votes. Pick a vote at random, return it a
additional copies of the vote
And many others E.g., uniform single-peaked
votes
40
Agenda
  • Why study elections?
  • Formal model
  • What are elections?
  • How to model votes?
  • What voting rules to use?
  • Case study Plurality bribery
  • Simplest case
  • Deterministic solution
  • Case study Campaign management
  • Campaign management as bribery
  • Why is it good to study?
  • Conclusions and comments

41
Conclusions and Comments
  • Conclusions
  • Elections are a rich source of computational
    issues
  • Abundance of worst-case complexity results for
    many settings
  • Very few experimental results
  • Very few heuristic solutions
  • Swap bribery can be a natural problem to attack
    with heuristic methods ? most general problem
    studied.

42
Conclusions and Comments
  • Where to get more information about the field?
  • Book chapter A Richer Understanding of the
    Complexity of Election Systems
    (Faliszewski, Hemaspaandra, Hemaspaandra, Rothe)
  • Upcoming AI Magazine paper
  • AIs War on Manipulation Are We Winning?
    (Faliszewski, Procaccia)
  • Upcoming CACM paper
  • Using Complexity to Protect Elections
    (Faliszewski, Hemaspaandra, Hemaspaandra)
  • Google search
  • By problem complexity voting manipulation,
    bribery, swap bribery, possible winner, control
  • By people Conitzer, Procaccia, Walsh,
    Hemaspaandra, Elkind, Slinko, Faliszewski

43
Conclusions and Comments
  • Where is this kind of work published?
  • Conferences
  • AAAI (AAAI Conference on Artificial
    Intelligence)
  • AAMAS (Autonomous Agents and Multiagent Systems)
  • EC (ACM Conference on Electronic Commerce)
  • WINE (Workshop on Internet and Network Economics)
  • SAGT (Symposium on Algorithmic Game Theory)
  • ADT (Algorithmic Decision Theory)
  • Journals
  • Artificial Intelligence
  • Journal of AI Research
  • Social Choice and Welfare
  • Autonomous Agents and Multiagent Systems
  • Mathematical Social Sciences

44
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