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Chamberlin-Courant on Restricted Domains Neeldhara Misra Recent Trends in Algorithms National Institute of Science Education and Research The standard Voting Setup The standard Voting Setup and typical computational problems. The standard

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Chamberlin-Courant on Restricted Domains

Neeldhara Misra

Recent Trends in Algorithms

National Institute of Science Education and Research

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The standard Voting Setup

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The standard Voting Setup

and typical computational problems.

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Single-peaked & Single-Crossing Preferences The standard Voting Setup

and typical computational problems.

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Single-peaked & Single-Crossing Preferences The standard Voting Setup

…better winner determination, greater resilience to manipulation, etc.

and typical computational problems.

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Almost Special Single-peaked & Single-Crossing Preferences The standard Voting Setup

…better winner determination, greater resilience to manipulation, etc.

and typical computational problems.

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Almost Special Single-peaked & Single-Crossing Preferences The standard Voting Setup

…better winner determination, greater resilience to manipulation, etc.

and typical computational problems. Getting realistic about domain restrictions.

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Almost Special Concluding Remarks Single-peaked & Single-Crossing Preferences The standard Voting Setup

…better winner determination, greater resilience to manipulation, etc.

and typical computational problems. Getting realistic about domain restrictions.

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Almost Special Concluding Remarks Single-peaked & Single-Crossing Preferences The standard Voting Setup

…better winner determination, greater resilience to manipulation, etc.

and typical computational problems. Getting realistic about domain restrictions. Red flags and research directions.

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The standard Voting Setup

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The standard Voting Setup

and typical computational problems.

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Candidates/Alternatives

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Voters express their preferences over alternatives (here, as rankings).

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Voters express their preferences over alternatives (could also be approval ballots).

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Social Choice Functions

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Social Welfare Functions

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&

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Multiwinner Voting Rules

&

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Typical problems

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Typical problems

What’s the “best” alternative?

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Typical problems

What’s the “best” alternative? What ranking most closely reflects the overall “societal” opinion?

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Typical problems

What’s the “best” alternative? What ranking most closely reflects the overall “societal” opinion? Do voters have incentives to lie about their preferences?

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Typical problems

What’s the “best” alternative? How does the removal or duplication of an alternative affect the outcome? What ranking most closely reflects the overall “societal” opinion? Do voters have incentives to lie about their preferences?

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Typical problems

What’s the “best” alternative? How does the removal or duplication of an alternative affect the outcome? What ranking most closely reflects the overall “societal” opinion? Winoer Determination Do voters have incentives to lie about their preferences?

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Typical problems

What’s the “best” alternative? How does the removal or duplication of an alternative affect the outcome? What ranking most closely reflects the overall “societal” opinion? Winoer Determination Do voters have incentives to lie about their preferences? Preference Aghregation

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Typical problems

What’s the “best” alternative? How does the removal or duplication of an alternative affect the outcome? What ranking most closely reflects the overall “societal” opinion? Winoer Determination Do voters have incentives to lie about their preferences? Manipulation Preference Aghregation

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Typical problems

What’s the “best” alternative? How does the removal or duplication of an alternative affect the outcome? What ranking most closely reflects the overall “societal” opinion? Winoer Determination Control Do voters have incentives to lie about their preferences? Manipulation Preference Aghregation

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Voting Rules

Some Examples

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Voting Rules

Plurality

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(Plurality)

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(Plurality)

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(Plurality)

The plurality winner can also be 
 among the least popular options.

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We say that a voter (or a group of voters) can manipulate if they can obtain a more desirable

  • utcome by misreporting their preferences.
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(Plurality)

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(Plurality)

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(Plurality)

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(Plurality)

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This scheme is intended 


  • nly for honest men.

Borda

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Voting Rules

STV

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(STV)

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Voting Rules

Condorcet

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An alternative that beats all the

  • thers in pairwise comparisons.
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(Condorcet)

An alternative that beats all the

  • thers in pairwise comparisons.
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An alternative that beats all the others in pairwise comparisons.

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An alternative that beats all the others in pairwise comparisons.

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An alternative that beats all the others in pairwise comparisons.

may not exist!

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Voting Rules

Dodgson

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Dodgson

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Dodgson

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Dodgson score of c Smallest #of swaps needed to 
 make c a Condorcet winner.

Dodgson

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Voting Rules

Kemeny Preference Aghregation

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Kemeny

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Kemeny score of a ranking Sum of pairwise agreements across all votes.

Kemeny

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Voting Rules

Chamberlin-Courant Multiwinoer

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Chamberlin-Courant

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&

Chamberlin-Courant

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&

Chamberlin-Courant

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&

Chamberlin-Courant

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CC-score score of a committee: maximum dissatisfaction across all votes.

Chamberlin-Courant

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CC-score score of a committee: maximum dissatisfaction across all votes. More precisely…

Chamberlin-Courant

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Voters Candidates

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Voters Candidates

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Voters Candidates

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Voters Candidates

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dissatisfaction of voter v = rank of best candidate from the committee in his vote

Voters Candidates

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Single-peaked & Single-Crossing Preferences

…better winner determination, greater resilience to manipulation, etc.

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Definition

The Theory of Committees and Elections. Black, D.,New York: Cambridge University Press, 1958

Single Peaked Preferences

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Left Right Center A B C D E F G

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Left Right Center A B C D E F G

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Left Right Center A B C D E F G E D C F G B A

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Left Right Center A B C D E F G E D C F G B A E D C F G B A

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Left Right Center A B C D E F G

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Left Right Center A B C D E F G

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Left Right Center A B C D E F G If an agent with single-peaked preferences prefers x to y,

  • ne of the following must be true:
  • x is the agent’s peak,
  • x and y are on opposite sides of the agent’s peak, or
  • x is closer to the peak than y.
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Left Right Center A B C D E F G The notion is popular for several reasons:

  • No Condorcet Cycles.
  • No incentive for an agent to misreport its preferences.
  • Identifiable in polynomial time.
  • Reasonable (?) model of actual elections.
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Single Peaked Preferences

Strategyproofness

The Theory of Committees and Elections. Black, D.,New York: Cambridge University Press, 1958

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A B C D E F G

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A B C D E F G

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A B C D E F G

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A B C D E F G

Claim: D beats all other candidates in pairwise elections.

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A B C D E F G

Claim: D beats all other candidates in pairwise elections.

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A B C D E F G

Claim: D beats all other candidates in pairwise elections.

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A B C D E F G

Claim: D beats all other candidates in pairwise elections.

(Peak to the left of D, F further than D)

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A B C D E F G

Claim: D beats all other candidates in pairwise elections.

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A B C D E F G

Claim: D beats all other candidates in pairwise elections.

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A B C D E F G

Claim: D beats all other candidates in pairwise elections.

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A B C D E F G

Claim: D beats all other candidates in pairwise elections.

(Peak to the right of D, B further than D)

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A B C D E F G

Claim: Choosing D also leaves nobody with any incentive to manipulate.

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A B C D E F G

Claim: Choosing D also leaves nobody with any incentive to manipulate.

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A B C D E F G

Claim: Choosing D also leaves nobody with any incentive to manipulate.

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A B C D E F G

Claim: Choosing D also leaves nobody with any incentive to manipulate.

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A B C D E F G

Claim: Choosing D also leaves nobody with any incentive to manipulate.

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A B C D E F G

Claim: Choosing D also leaves nobody with any incentive to manipulate.

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Chamberlin-Courant

Single Peaked Preferences

  • N. Betzler, A. Slinko, and J. Uhlmann. On the computation of fully proportional representation.

Journal of Artificial Intelligence Research, 47(1):475–519, 2013.

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A B C D E F G

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A B C D E F G

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A B C D E F G

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A B C D E F G

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A B C D E F G

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A B C D E F G

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A B C D E F G

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Determining the winner reduces to stabbing a set of intervals with k lines.

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Definition

Single Crossing Preferences

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A profile is single-crossing if it admits an ordering of the voters such that for every pair of candidates (a,b), either: a) all voters who prefer a over b appear before all voters who prefer b over a, or, b) all voters who prefer a over b appear after all voters who prefer b over a, or,

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The notion is popular for several reasons:

  • No Condorcet Cycles.
  • Identifiable in polynomial time.
  • Reasonable (?) model of actual elections.
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Chamberlin-Courant

Single Crossing Preferences

The Complexity of Fully Proportional Representation for Single-Crossing Electorates Skowron, SAGT, 2013

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dissatisfaction of voter v = rank of best candidate in the committee in his vote

Voters Candidates

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dissatisfaction of voter v = rank of best candidate in the committee in his vote

On single-crossing profiles, optimal CC solutions exhibit a “contiguous blocks property”.

Voters Candidates

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dissatisfaction of voter v = rank of best candidate in the committee in his vote

On single-crossing profiles, optimal CC solutions exhibit a “contiguous blocks property”.

Voters Candidates

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dissatisfaction of voter v = rank of best candidate in the committee in his vote

On single-crossing profiles, optimal CC solutions exhibit a “contiguous blocks property”.

Voters Candidates

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A[p,q,t] := best committee of size t from the first p candidates,
 accounting for the first q voters.

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A[p,q,t] := best committee of size t from the first p candidates,
 accounting for the first q voters.

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A[p,q,t] := best committee of size t from the first p candidates,
 accounting for the first q voters.

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A[p,q,t] := best committee of size t from the first p candidates,
 accounting for the first q voters.

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A[p,q,t] := best committee of size t from the first p candidates,
 accounting for the first q voters.

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A[p,q,t] := best committee of size t from the first p candidates,
 accounting for the first q voters.

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A[p,q,t] := best committee of size t from the first p candidates,
 accounting for the first q voters. (1) A[p,q-1,t] - when cq doesn’t belong to OPT.

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A[p,q,t] := best committee of size t from the first p candidates,
 accounting for the first q voters. (1) A[p,q-1,t] - when cq doesn’t belong to OPT.

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A[p,q,t] := best committee of size t from the first p candidates,
 accounting for the first q voters. (1) A[p,q-1,t] - when cq doesn’t belong to OPT. (2) A[p-x,q-1,t-1] - when cq does belong to OPT.

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A[p,q,t] := best committee of size t from the first p candidates,
 accounting for the first q voters. (1) A[p,q-1,t] - when cq doesn’t belong to OPT. (2) A[p-x,q-1,t-1] - when cq does belong to OPT.

(Guess all possible choices for x.)

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A[p,q,t] := best committee of size t from the first p candidates,
 accounting for the first q voters. (1) A[p,q-1,t] - when cq doesn’t belong to OPT. (2) A[p-x,q-1,t-1] - when cq does belong to OPT.

(Guess all possible choices for x.)

Min{ }

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Almost Special

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Almost Special

Getting realistic about domain restrictions.

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The single-peaked and single-crossing domains have been generalised to notions of single-peaked and single- crossing on trees. The generalised domains continue to exhibit many of the nice properties we saw today.

Generalizing the Single-Crossing Property on Lines and Trees to Intermediate Preferences on Median Graphs, Clearwater, Puppe, and Slinko, IJCAI 2015 Single-peaked orders on a tree, Gabrielle Demange,


  • Math. Soc. Sci, 3(4), 1982.
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The single-peaked and single-crossing domains have been generalised to notions of single-peaked-width and single- crossing-width. Here, it is common that algorithms that work in the single- peaked or single-crossing settings can be generalised to profiles of width w at an expense that is exponential in w.

Kemeny Elections with Bounded Single-peaked or Single-crossing Width, Cornaz, Galand, and Spanjaard, IJCAI 2013

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Profiles that are “close” to being single-peaked or single- crossing (closeness measured usually in terms of candidate

  • r voter deletion) have also been studied.

It’s typically NP-complete to determine the optimal distance, but FPT and approximation algorithms are known.

Are There Any Nicely Structured Preference Profiles Nearby? Bredereck, Chen, and Woeginger, AAAI 2013 On Detecting Nearly Structured Preference Profiles Elkind and Lackner, AAAI 2014 Computational aspects of nearly single-peaked electorates, Erdélyi, Lackner, and Pfandler, AAAI 2013

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Summary from: On Detecting Nearly Structured Preference Profiles Elkind and Lackner, AAAI 2014

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On profiles that are k candidates or k voters away from the single- peaked and single-crossing domains, CC admits efficient algorithms:

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On profiles that are k candidates or k voters away from the single- peaked and single-crossing domains, CC admits efficient algorithms: For profiles that are k candidates away from being single-peaked or single-crossing, we have algorithms whose running time is FPT in k.

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On profiles that are k candidates or k voters away from the single- peaked and single-crossing domains, CC admits efficient algorithms: For profiles that are k voters away from being single- peaked or single- crossing, we have algorithms that are XP in k.

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One could also generalize SP/SC notions to profiles with multiple peaks/crossings, instances that can partitioned into a small number of disjoint sub-instances which are themselves SP or SC, and so on.

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One could also generalize SP/SC notions to profiles with multiple peaks/crossings, instances that can partitioned into a small number of disjoint sub-instances which are themselves SP or SC, and so on. Checklist of questions to ask when broadening a domain:

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One could also generalize SP/SC notions to profiles with multiple peaks/crossings, instances that can partitioned into a small number of disjoint sub-instances which are themselves SP or SC, and so on. Checklist of questions to ask when broadening a domain: (1) Efficient recognition.

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One could also generalize SP/SC notions to profiles with multiple peaks/crossings, instances that can partitioned into a small number of disjoint sub-instances which are themselves SP or SC, and so on. Checklist of questions to ask when broadening a domain: (1) Efficient recognition. (2) Algorithmic utility.

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One could also generalize SP/SC notions to profiles with multiple peaks/crossings, instances that can partitioned into a small number of disjoint sub-instances which are themselves SP or SC, and so on. Checklist of questions to ask when broadening a domain: (1) Efficient recognition. (2) Algorithmic utility. (3) Preservation of nice axiomatic properties.

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CONCLUDING REMARKS

Red flags and research directions.

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The Dark Side: Domain restrictions also have some side- effects: problems like manipulation, bribery, and so forth also become easy!

Bypassing Combinatorial Protections: Polynomial-Time Algorithms for Single-Peaked Electorates, Brandt et al; AAAI 2010 The Shield that Never Was: Societies with Single-Peaked Preferences are More Open to Manipulation and Control, Faliszewski et al; TARK 2009

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Directions for future work

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Directions for future work

Parameterizing by “distance to tractability”.

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Multidimensional domain restrictions.

Directions for future work

Parameterizing by “distance to tractability”.

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Multidimensional domain restrictions. Generalize structure in dichotomous 
 preference domains to trichotomous and beyond.

Directions for future work

Parameterizing by “distance to tractability”.

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Multidimensional domain restrictions. Generalize structure in dichotomous 
 preference domains to trichotomous and beyond. Consider completely new domain restrictions.

Directions for future work

Parameterizing by “distance to tractability”.

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Multidimensional domain restrictions. Generalize structure in dichotomous 
 preference domains to trichotomous and beyond. Consider completely new domain restrictions. Investigate the impact of structured preferences in other settings: 
 matchings and fair division.

Directions for future work

Parameterizing by “distance to tractability”.

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Thank you!

The Handbook of Computational Social Choice, Brandt, Conitzer, Endriss, Lang and Procaccia; 2016 Structured preferences. Elkind, Lackner, and Peters — Trends in Computational Social Choice; (2017): 187-207.