The Complexity of Elections: A New Domain for Heuristic Computation Piotr Faliszewski AGH University of Science and Technology, Kraków, Poland [email protected]

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

Computational issues in elections

Gibbard Satterthwaite Theorem

Winner determination Cheating in elections Running the election Manipulation bribery control Possible winner Campaign management

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.

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!

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

How to Study Elections?

• We need to answer several questions: – What are elections?

A mathematical object, a pair E = (C,V).

C – set of candidates, V – set of voters – How can we model voters’ preferences?

A single top guy? A ranking of candidates? Numerical values, “utility” a voter derives from having a particular candidate elected?

– What election rules are we interested in?

An election rule is a function that given an election returns the set of winners.

Election Model: Example

C = { , • Election E = (C,V) C – candidate set V – voter set • Who is the winner?

V = { > – Plurality – Veto – Borda – k-approval – Approval – Copeland – Llull – Dodgson – Kemeny – Young – … families of scoring protocols > > > > , > > > > > , > > > > > , } > , > , > , > , > }

Election Model: Example

C = { , • Election E = (C,V) C – candidate set V – voter set • Who is the winner?

– Borda count

α = ( 4 , 3 , 2 , 1 , 0 )

V = { > , > , > , } > , score( ) = 13 score( ) = 12 score( ) = 9 score( ) = 8 score( ) = 8 > > > > > > > > > > , > , > , > > > > }

Election Model: Example

C = { , • Election E = (C,V) C – candidate set V – voter set • Who is the winner?

– Copeland V = { > , > > > , > > , } > , > , > > > > , > > > > > , > > > }

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

Computational issues in elections Winner determination Cheating in elections Running the election Manipulation bribery control Possible winner Campaign management

Bribery

Notation:

Ɛ – an election system Name:

Ɛ -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. Ɛ Ɛ Ɛ Ɛ -bribery -$bribery -weighted-bribery -weighted-$bribery

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?

unweighted

• Observation: – It is sufficient to only look at each voter’s most preferred candidate

weighted unpriced priced

Bribery in Plurality

• Example: – What is the cheapest way to make Alice a winner?

7 • 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?

6 5 4 3 0 2 1 Mary Bill John Alice

Bribery in Plurality

• Example: – What is the cheapest way to make Alice a winner?

7 • 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?

6 5 4 3 0 2 1 Mary Bill John Alice

Bribery in Plurality

• Example: – What is the cheapest way to make Alice a winner?

7 • 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?

6 5 4 3 0 2 1 Mary Bill John Alice

Bribery in Plurality

• Example: – What is the cheapest way to make Alice a winner?

7 • 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?

6 5 4 3 0 2 1 Mary Bill John Alice

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?

unweighted unpriced priced weighted

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 (s 1 , … , s n ), is 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 that x in A  f(x) in B A f f B

plurality-weighted-$bribery



NP-com

Proof: Reduction from the subset-sum problem Subset-Sum Input: s 1 , s 2 , … , s n  =34 plurality-weighted-$bribery k = 17 1 5 6 10 12 Question: Is there a subsequence that sums up to exactly half.

12 10 6 5 1

plurality-weighted-$bribery



NP-com

Proof: Reduction from the subset-sum problem Subset-Sum Input: s 1 , s 2 , … , s n  =34 plurality-weighted-$bribery k = 17 1 5 6 10 12 Question: Is there a subsequence that sums up to exactly half.

12 5 1 6 10

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?

unweighted unpriced priced weighted

Weighted Bribery in Plurality

• plurality-weighted-bribery – Now voters have weights • What algorithms can we use?

– Heuristics: • Heaviest voter first • Winner’s heaviest voter first 2 0 6 4 12 10 8 Mary Bill John

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?

unweighted unpriced priced weighted

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

Computational issues in elections Winner determination Cheating in elections Running the election Manipulation bribery control Possible winner Campaign management

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

Bribery 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 Bribery

• Price function π for each voter .

> > >

π( , ) = 5

Swap Bribery

• Price function π for each voter .

> > >

Swap Bribery

• Price function π for each voter .

> > >

• Swap bribery problem

– Given: E = (C,V), price function for each voter

Questions About Swap Bribery

• Price of reaching a given vote?

> > > > > >

• Swap bribery and other voting problems?

Voting problem < m Swap bribery • Complexity of swap bribery?

Relations Between Voting Problems

Results on Swap Bribery

• Swap Bribery is easy for: – Plurality – Veto • Swap bribery is NP-hard for: – Borda • Swap bribery can model winner problems: – Kemeny – Dodgson – Copeland – Bucklin – Maximin – Ranked pairs – Plurality with runoff – STV – …

Why Study Swap Bribery?

Swap bribery is hard for almost all voting systems Swap bribery is a generalization of most important problems

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

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

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…

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

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.

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:

AI’s 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

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

Thank you!

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