Multiwinner Elections: Theory and Experiments Piotr Faliszewski AGH - PowerPoint PPT Presentation

Multiwinner Elections: Theory and Experiments Piotr Faliszewski AGH University Kraków, Poland Based on joint work with Edith Elkind (University of Oxford), Jerome Lang (Universite Paris Dauphine), Piotr Skowron (Uniwersytet Warszawski & Google Polska), Arkadii Slinko (University of Auckland), Lan Yu (Google Inc.), Robert Schaefer (AGH), and Nimrod Talmon (TU Berlin, Niemcy) Supported by NCN grant 2012/06/M/ST1/00358

Multiwinner Elections? I can only put 2 movies on the entertainment system… which ones to pick? All these people want a job in our company … have to make a shortlist!

How to choose a parliament? Single-winner districts 100 / 0 100 / 0 49 / 51

How to choose a parliament? Single-winner districts 100 / 0 100 / 0 49 / 51 25% support sufficient to form a majority government

How to choose a parliament? Single-winner districts Party lists I have to be nice to my party leader to get into 25% support sufficient the parliament to form a majority government

Agenda 1. Introduction 2. Multiwinner elections – Election model – Basic rules and how they work 3. Committee scoring rules – Analogues of single-winner scoring rules – Important subclasses of CSRs – Complexity results – Example of an axiomatic approach 4. Conclusions

Election Model • Election E = (C, V) C = { , , , , } – C – set of candidates V = (v 1 , … , v 6 ) – V – set of voters • Parameter k V 1 : – k – the committee size V 2 : • … and a voting rule … V 3 : V 4 : V 5 : V 6 :

Main Families of Multiwinner Rules Multiwinner voting rules Rules based on Rules based on preference orders approval ballots Proportional Rules based on representation the Condorcet k-winner rules principle extensions of single-winner rules

Election Model • Election E = (C, V) C = { , , , , } – C – set of candidates V = (v 1 , … , v 6 ) – V – set of voters 1 0 0 0 0 • Parameter k V 1 : – k – the committee size V 2 : • … and a voting rule … V 3 : V 4 : V 5 : SNTV V 6 :

Election Model • Election E = (C, V) C = { , , , , } – C – set of candidates V = (v 1 , … , v 6 ) – V – set of voters 1 1 0 0 0 • Parameter k V 1 : – k – the committee size V 2 : • … and a voting rule … V 3 : V 4 : V 5 : Bloc V 6 :

Election Model • Election E = (C, V) C = { , , , , } – C – set of candidates V = (v 1 , … , v 6 ) – V – set of voters 4 3 2 1 0 • Parameter k V 1 : – k – the committee size V 2 : • … and a voting rule … V 3 : V 4 : V 5 : k-Borda V 6 :

Proportional Representation Rule of Chamberlin — Courant Choosing a parliament is a resource allocation problems V 1 : V 2 : Candidates = Resources Voting rule assigns V 3 : candidates to the voters V 4 : V 5 : V 6 :

Proportional Representation Rule of Chamberlin — Courant Choosing a parliament is a resource allocation problems 4 3 2 1 0 V 1 : Chamberlin-Courant V 2 : Pick k candidates and assign them to the voters V 3 : to maximze the score that the voters give to their representatives V 4 : V 5 : V 6 :

How Do These Rules Work: k-Borda

How Do These Rules Work: Bloc

How Do These Rules Work: Chamberlin-Courant

Single-Winner Scoring Rules A single-winner scoring function: C = { , , , , } V = (v 1 , … , v 6 ) f(i) = score for position i The candidate with the highest 4 3 2 1 0 sum of scores is the winner V 1 : V 2 : Examples: Borda score V 3 : B(i) = m-i V 4 : t-Approval score V 5 : A t (i) = 1 if i ≤ t and 0 otherwise V 6 :

Committee Scoring Rules Consider a preference order: winning committee Position of the winning committee = (1, 3, 4 ) f(i 1 , i 2 , …, i k ) = the score of the committee Assuming i 1 < i 2 < … < i k

Committee Scoring Rules Committee scoring function: C = { , , , , } V = (v 1 , … , v 6 ) f(i 1 , i 2 , …, i k ) = score for pos. (i 1 , i 2 , …, i k ) The committee with the highest 1 0 0 0 0 sum of scores is the winner V 1 : V 2 : Examples: f SNTV (i 1 , i 2 , …, i k ) = A 1 (i 1 ) + … + A 1 (i k ) V 3 : V 4 : V 5 : V 6 :

Committee Scoring Rules Committee scoring function: C = { , , , , } V = (v 1 , … , v 6 ) f(i 1 , i 2 , …, i k ) = score for pos. (i 1 , i 2 , …, i k ) The committee with the highest 4 3 2 1 0 sum of scores is the winner V 1 : V 2 : Examples: f SNTV (i 1 , i 2 , …, i k ) = A 1 (i 1 ) + … + A 1 (i k ) V 3 : f k-Borda (i 1 , i 2 , …, i k ) = B(i 1 ) + … + B( i k ) V 4 : V 5 : V 6 :

Committee Scoring Rules Committee scoring function: C = { , , , , } V = (v 1 , … , v 6 ) f(i 1 , i 2 , …, i k ) = score for pos. (i 1 , i 2 , …, i k ) The committee with the highest 1 1 0 0 0 sum of scores is the winner V 1 : V 2 : Examples: f SNTV (i 1 , i 2 , …, i k ) = A 1 (i 1 ) + … + A 1 (i k ) V 3 : f k-Borda (i 1 , i 2 , …, i k ) = B(i 1 ) + … + B( i k ) V 4 : f Bloc (i 1 , i 2 , …, i k ) = A k (i 1 ) + … + A k (i k ) V 5 : V 6 :

Committee Scoring Rules Committee scoring function: C = { , , , , } V = (v 1 , … , v 6 ) f(i 1 , i 2 , …, i k ) = score for pos. (i 1 , i 2 , …, i k ) The committee with the highest 4 3 2 1 0 sum of scores is the winner V 1 : V 2 : Examples: f SNTV (i 1 , i 2 , …, i k ) = A 1 (i 1 ) + … + A 1 (i k ) V 3 : f k-Borda (i 1 , i 2 , …, i k ) = B(i 1 ) + … + B( i k ) V 4 : f Bloc (i 1 , i 2 , …, i k ) = A k (i 1 ) + … + A k (i k ) V 5 : f CC (i 1 , i 2 , …, i k ) = B(i 1 ) V 6 :

Committee Scoring Rules Committee scoring function: Basic classes of CSRs f(i 1 , i 2 , …, i k ) = score for pos. (i 1 , i 2 , …, i k ) Separable rules: The committee with the highest f(i 1 , i 2 , …, i k ) = g(i 1 ) + … + g( i k ) sum of scores is the winner Weakly separable rules: Examples: f(i 1 , i 2 , …, i k ) = h k (i 1 ) + … + h k (i k ) f SNTV (i 1 , i 2 , …, i k ) = A 1 (i 1 ) + … + A 1 (i k ) Representation focused rules: f k-Borda (i 1 , i 2 , …, i k ) = B(i 1 ) + … + B( i k ) f(i 1 , i 2 , …, i k ) = q(i 1 ) f Bloc (i 1 , i 2 , …, i k ) = A k (i 1 ) + … + A k (i k ) f CC (i 1 , i 2 , …, i k ) = B(i 1 )

Committee Scoring Rules Committee scoring function: Basic classes of CSRs f(i 1 , i 2 , …, i k ) = score for pos. (i 1 , i 2 , …, i k ) Separable rules: The committee with the highest f(i 1 , i 2 , …, i k ) = g(i 1 ) + … + g( i k ) sum of scores is the winner Weakly separable rules: Examples: f(i 1 , i 2 , …, i k ) = h k (i 1 ) + … + h k (i k ) f SNTV (i 1 , i 2 , …, i k ) = A 1 (i 1 ) + … + A 1 (i k ) Representation focused rules: f k-Borda (i 1 , i 2 , …, i k ) = B(i 1 ) + … + B( i k ) f(i 1 , i 2 , …, i k ) = q(i 1 ) f Bloc (i 1 , i 2 , …, i k ) = A k (i 1 ) + … + A k (i k ) f CC (i 1 , i 2 , …, i k ) = B(i 1 )

Committee Scoring Rules Committee scoring function: Basic classes of CSRs f(i 1 , i 2 , …, i k ) = score for pos. (i 1 , i 2 , …, i k ) Separable rules: The committee with the highest f(i 1 , i 2 , …, i k ) = g(i 1 ) + … + g( i k ) sum of scores is the winner Weakly separable rules: Examples: f(i 1 , i 2 , …, i k ) = h k (i 1 ) + … + h k (i k ) f SNTV (i 1 , i 2 , …, i k ) = A 1 (i 1 ) + … + A 1 (i k ) Representation focused rules: f k-Borda (i 1 , i 2 , …, i k ) = B(i 1 ) + … + B( i k ) f(i 1 , i 2 , …, i k ) = q(i 1 ) f Bloc (i 1 , i 2 , …, i k ) = A k (i 1 ) + … + A k (i k ) f CC (i 1 , i 2 , …, i k ) = B(i 1 )

Committee Scoring Rules Committee scoring function: Basic classes of CSRs f(i 1 , i 2 , …, i k ) = score for pos. (i 1 , i 2 , …, i k ) Separable rules: The committee with the highest f(i 1 , i 2 , …, i k ) = g(i 1 ) + … + g( i k ) sum of scores is the winner Weakly separable rules: Examples: f(i 1 , i 2 , …, i k ) = h k (i 1 ) + … + h k (i k ) f SNTV (i 1 , i 2 , …, i k ) = A 1 (i 1 ) + … + A 1 (i k ) Representation focused rules: f k-Borda (i 1 , i 2 , …, i k ) = B(i 1 ) + … + B( i k ) f(i 1 , i 2 , …, i k ) = q(i 1 ) f Bloc (i 1 , i 2 , …, i k ) = A k (i 1 ) + … + A k (i k ) f CC (i 1 , i 2 , …, i k ) = B(i 1 )

Committee Scoring Rules Committee scoring function: Basic classes of CSRs f(i 1 , i 2 , …, i k ) = score for pos. (i 1 , i 2 , …, i k ) Separable rules: The committee with the highest f(i 1 , i 2 , …, i k ) = g(i 1 ) + … + g( i k ) sum of scores is the winner Weakly separable rules: Examples: f(i 1 , i 2 , …, i k ) = h k (i 1 ) + … + h k (i k ) f SNTV (i 1 , i 2 , …, i k ) = A 1 (i 1 ) + … + A 1 (i k ) Representation focused rules: f k-Borda (i 1 , i 2 , …, i k ) = B(i 1 ) + … + B( i k ) f(i 1 , i 2 , …, i k ) = q(i 1 ) f Bloc (i 1 , i 2 , …, i k ) = A k (i 1 ) + … + A k (i k ) f CC (i 1 , i 2 , …, i k ) = B(i 1 )

OWA-Based Committee Scoring Rules An OWA operator is a sequence of k numbers W = (w 1 , … , w) Given a single-winner scoring rule g and OWA opertor W , we define CSR: f(i 1 , i 2 , …, i k ) = w 1 g(i 1 ) + w 2 g(i 2 ) + … + w k g(i k )