Multiwinner Voting Theory Experiments and Data Piotr Faliszewski

Multiwinner Voting: Theory, Experiments and Data and Theory! Piotr Faliszewski AGH University of Science

Multiwinner Elections Single-Winner Elections movies on a plane fundamentally different! voters’ preferences voting rule

Single-Winner Scoring Rules A single-winner scoring function: f(i) = score for position i The

Committee Elections: Examples SNTV: Pick k candidates with top plurality scores (α 1) C={

Committee Elections: Examples STV: Elimination process based on plurality scores (eliminate lowest scores; add

Committee Elections: Examples k-Borda: Pick k candidates with top Borda scores (β) C={ ,

Generating the Histograms AGH Comp. Sci. Dept. (8 -core subcluster) Easy rules (P-time) (k-Borda,

Committee Scoring Rules Consider a preference order: a committee Position of the committee =

Committee Scoring Rules Examples SNTV: f(i 1, . . . , ik) = α

k=2 Committee Scoring Rules Examples SNTV: f(i 1, . . . , ik) =

k=2 Committee Scoring Rules : 15 : 10 : 11 : 9 : 12

k=3 Committee Scoring Rules Examples 1 1 1 0 0 SNTV: f(i 1, .

Committee Scoring Rules Examples Bloc: f(i 1, . . . , ik) = αk(i

Committee Scoring Rules Examples Proportional Approval Voting (as CSR): f(i 1, . . .

Nature of the Committees (Individual Excellence)

Excellence: Main Idea Given: A group of candidates and preferences of the judges/voters Task:

Excellence: Main Idea Committee Selection Given: A group of candidates and preferences of the

Committee Scoring Rules Problem with Bloc Committee Scoring Rules β- V A P OWA-based

Committee Scoring Rules β- α α le Bloc t -P AV C -C k

Committee Scoring Rules β- > > > > [FSST 16] P. Faliszewski, P. Skowron,

Committee Scoring Rules Theorem A committee scoring rule is noncrossing monotone if and only

Committee Scoring Rules Committee SNT monotone V noncrossing monotone Bloc ble [FSST 16] P.

Is SNTV really good for individual excellence? SNTV k-Borda

Nature of the Committees (Diveristy/Coverage)

Applications Requiring Diversity/Coverage Instead of finding the “best” candidates (recall Excellence) we aim at

Axioms for Diversity: Narrow Top A rule satisfies the narrow top criterion if whenever

Axioms for Diversity: Narrow Top [FSST 16] P. Faliszewski, P. Skowron, A. Slinko, N.

Axiom: Noncrossing monotonicity Noncrossing Monotonicity: If a member of the winning committee is moved

Axiom: Top-member monotonicity Top-Member Monotonicity: If the highest ranked member of the winning committee

Axiom: Top-member monotonicity Noncrossing Monotonicity: If a member of the winning committee is moved

Axioms: Narrow Top + Top Member Monotonicity [FSST 16] P. Faliszewski, P. Skowron, A.

Axioms: Narrow Top + Top Member Monotonicity α t -P AV top-member + C

Chamberlin—Courant is good for diversity

How to deal with hardness of Chamberlin —Courant?

Greedy CC 4 Input: E = (C, V) — an election k — size

Algorithm P Rank 1 2 3 v 1 v 2 x m Goal: pick

1 -Best: PTAS (Borda Utilities) Rank 1 2 3 v 1 v 2 x

1 -Best: PTAS (Borda Utilities) Rank 1 2 3 v 1 v 2 vn

State of the Art Syntax VS Semantics Committee Scoring Rules OWA-based SNT da k-Bor

Take-Home Message s n o i t a c i l p y ap

Thank You! https: //github. com/elektronaj/MW 2 D Multiwinner Voting: A New Challenge for Social

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Multiwinner Voting: Theory, Experiments and Data Piotr Faliszewski AGH University of Science and Technology Kraków, Poland faliszew@agh. edu. pl Based on joint works with Edith Elkind (University of Oxford, UK), Jerome Lang (Dauphine Paris, FR), Jean-Francois Laslier (Paris School of Economics, FR), Piotr Skowron (Technische Universtaet Berlin, DE), Arkadii Slinko (University of Auckland, NZ), Nimrod Talmon (Weizmann Institute, IL)

Multiwinner Voting: Theory, Experiments and Data and Theory! Piotr Faliszewski AGH University of Science and Technology Kraków, Poland faliszew@agh. edu. pl Based on joint works with Edith Elkind (University of Oxford, UK), Jerome Lang (Dauphine Paris, FR), Jean-Francois Laslier (Paris School of Economics, FR), Piotr Skowron (Technische Universtaet Berlin, DE), Arkadii Slinko (University of Auckland, NZ), Nimrod Talmon (Weizmann Institute, IL)

Multiwinner Elections Single-Winner Elections movies on a plane fundamentally different! voters’ preferences voting rule t n e m a i rl pa Choosing presidents, scheduling, sports/competitions Seek the best, the most widely supported candidates s making a shortlist

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

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

Committee Elections: Examples SNTV: Pick k candidates with top plurality scores (α 1) C={ , , , V = (v 1, … , v 6) 1 V 1: V 2: V 3: V 4: V 5: V 6: 0 0 , 0 0 }

Committee Elections: Examples STV: Elimination process based on plurality scores (eliminate lowest scores; add to committee after reaching over n/(k+1) points) C={ , , , V = (v 1, … , v 6) V 1: V 2: V 3: V 4: V 5: V 6: , }

Committee Elections: Examples k-Borda: Pick k candidates with top Borda scores (β) C={ , , , V = (v 1, … , v 6) 4 V 1: V 2: V 3: V 4: V 5: V 6: 3 2 , 1 0 }

SNTV STV k-Borda

Generating the Histograms AGH Comp. Sci. Dept. (8 -core subcluster) Easy rules (P-time) (k-Borda, SNTV, …) 10 000 elections x rules x distributions Store and never look at them again (1 -3 TB) Aggregate results 10 000 elections x rules x distributions Weizmann Institute (heavy-duty cluster) NP-hard rules (CPLEX ILP-based alg. )

We Want to Understand Committee Rules

Committee Scoring Rules Consider a preference order: a committee Position of the committee = (1, 3, 4 ) f(i 1, i 2, …, ik) = the score of the committee Assuming i 1 < i 2 < … < ik [EFSS 17] E. Elkind, P. Faliszewski, P. Skowron, A. Slinko, Properties of Multiwinner Voting Rules, Social Choice and Welfare, 2017 [SFS 16] P. Skowron, P. Faliszewski, A. Slinko, Axiomatic Characterization of Committee Scoring Rules, ar. Xiv 2016

Committee Scoring Rules Examples SNTV: f(i 1, . . . , ik) = α 1(i 1) + α 1(i 2) … + α 1(ik) k-Borda: f(i 1, . . . , ik) = β(i 1) + β(i 2) +. . + β(ik) Bloc: f(i 1, . . . , ik) = αk(i 1) + αk(i 2) +. . + αk(ik) Chamberlin—Courant (β-CC): f(i 1, . . . , ik) = β(i 1) Proportional Approval Voting (as CSR): f(i 1, . . . , ik) = αk(i 1) + 1/2αk(i 2) +. . . + 1/k αk(ik)

k=2 Committee Scoring Rules Examples SNTV: f(i 1, . . . , ik) = α 1(i 1) + α 1(i 2) … + α 1(ik) k-Borda: f(i 1, . . . , ik) = β(i 1) + β(i 2) +. . + β(ik) Bloc: f(i 1, . . . , ik) = αk(i 1) + αk(i 2) +. . + αk(ik) Chamberlin—Courant (β-CC): f(i 1, . . . , ik) = β(i 1) Proportional Approval Voting (as CSR): f(i 1, . . . , ik) = αk(i 1) + 1/2αk(i 2) +. . . + 1/k αk(ik) [Tul 67] G. Tullock, Towards a Mathematics of Politics, Univ. of Michigan Press, 1967 : 2 : 1 : 0 1 V 1: V 2: V 3: V 4: V 5: V 6: 0 0

k=2 Committee Scoring Rules : 15 : 10 : 11 : 9 : 12 Examples SNTV: f(i 1, . . . , ik) = α 1(i 1) k-Borda: f(i 1, . . . , ik) = β(i 1) + β(i 2) +. . + β(ik) Bloc: f(i 1, . . . , ik) = αk(i 1) + αk(i 2) +. . + αk(ik) Chamberlin—Courant (β-CC): f(i 1, . . . , ik) = β(i 1) Proportional Approval Voting (as CSR): f(i 1, . . . , ik) = αk(i 1) + 1/2αk(i 2) +. . . + 1/k αk(ik) [Deb 92] B. Debord, An Axiomatic Characterization of Borda’s k-Choice Function, SC&W 1992 4 V 1: V 2: V 3: V 4: V 5: V 6: 3 2 1 0

k=2 Committee Scoring Rules Examples SNTV: f(i 1, . . . , ik) = α 1(i 1) k-Borda: f(i 1, . . . , ik) = β(i 1) + β(i 2) +. . + β(ik) Bloc: f(i 1, . . . , ik) = αk(i 1) + αk(i 2) +. . + αk(ik) Chamberlin—Courant (β-CC): f(i 1, . . . , ik) = β(i 1) Proportional Approval Voting (as CSR): f(i 1, . . . , ik) = αk(i 1) + 1/2αk(i 2) +. . . + 1/k αk(ik) : 2 : 2 : 1 : 4 1 V 1: V 2: V 3: V 4: V 5: V 6: 1 0 0 0

SNTV STV Bloc k-Borda

SNTV STV CC Bloc k-Borda

k=3 Committee Scoring Rules Examples 1 1 1 0 0 SNTV: f(i 1, . . . , ik) = α 1(i 1) k-Borda: f(i 1, . . . , ik) = β(i 1) + β(i 2) +. . + β(ik) Bloc: f(i 1, . . . , ik) = αk(i 1) + αk(i 2) +. . + αk(ik) score({ , , }) = 1∙ 1 + 1/2∙ 1 + 1/3∙ 0 = 3/2 Chamberlin—Courant (β-CC): f(i 1, . . . , ik) = β(i 1) Proportional Approval Voting (as CSR): f(i 1, . . . , ik) = αk(i 1) + 1/2αk(i 2) +. . . + 1/k αk(ik) [Thi 95] N. Thiele, Om Flerfoldsvalg, Oversigt over det Kongelige Danske Videnskabernes Selskabs Forhandlinger, 1895 [Kil 10] M. Kilgour, Approval Balloting for Multi-Winner Elections, Hanbook of Approval Voting 2010

Committee Scoring Rules Examples SNTV: f(i 1, . . . , ik) = α 1(i 1) k-Borda: f(i 1, . . . , ik) = β(i 1) + β(i 2) +. . + β(ik) Bloc: f(i 1, . . . , ik) = αk(i 1) + αk(i 2) +. . + αk(ik) Chamberlin—Courant (β-CC): f(i 1, . . . , ik) = β(i 1) Proportional Approval Voting (as CSR): f(i 1, . . . , ik) = αk(i 1) + 1/2αk(i 2) +. . . + 1/k αk(ik) [EFSS 17] E. Elkind, P. Faliszewski, P. Skowron, A. Slinko, Properties of Multiwinner Voting Rules, Social Choice and Welfare, 2017 Consistency If W is a winning committee under two elections, E 1 and E 2, then W is a winning committee under E 1+E 2 (and only such committees win in E 1+E 2) Theorem All committee scoring rules satisfy consistency Candidate Monotonicity If a member of a winning committee W is shifted forward in some vote, this candidate will still belong to some winning committee (but maybe not W) Theorem All committee scoring rules satsify candidate monotonicity

Committee Scoring Rules Examples SNTV: f(i 1, . . . , ik) = α 1(i 1) k-Borda: f(i 1, . . . , ik) = β(i 1) + β(i 2) +. . + β(ik) > > > Bloc: f(i 1, . . . , ik) = αk(i 1) + αk(i 2) +. . + αk(ik) Chamberlin—Courant (β-CC): f(i 1, . . . , ik) = β(i 1) Proportional Approval Voting (as CSR): f(i 1, . . . , ik) = αk(i 1) + 1/2αk(i 2) +. . . + 1/k αk(ik) [EFSS 17] E. Elkind, P. Faliszewski, P. Skowron, A. Slinko, Properties of Multiwinner Voting Rules, Social Choice and Welfare, 2017 Candidate Monotonicity If a member of a winning committee W is shifted forward in some vote, this candidate will still belong to some winning committee (but maybe not W) Theorem All committee scoring rules satsify candidate monotonicity

Committee Scoring Rules Examples SNTV: f(i 1, . . . , ik) = α 1(i 1) + α 1(i 2) +. . . + α 1(ik) Committee Scoring Rules k-Borda: f(i 1, . . . , ik) = β(i 1) + β(i 2) +. . + β(ik) Proportional Approval Voting (as CSR): f(i 1, . . . , ik) = αk(i 1) + 1/2αk(i 2) +. . . + 1/k αk(ik) [FSST 16] P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Committtee Scoring Rules: Axiomatic Classification and Hierarchy, AAAI-2016 V da y akl we rable a sep Chamberlin—Courant (β-CC): f(i 1, . . . , ik) = β(i 1) SNT k-Bor Bloc: f(i 1, . . . , ik) = αk(i 1) + αk(i 2) +. . + αk(ik) Bloc

Committee Scoring Rules Examples Bloc: f(i 1, . . . , ik) = αk(i 1) + αk(i 2) +. . + αk(ik) Proportional Approval Voting (as CSR): f(i 1, . . . , ik) = αk(i 1) + 1/2αk(i 2) +. . . + 1/k αk(ik) [FSST 16] P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Committtee Scoring Rules: Axiomatic Classification and Hierarchy, AAAI-2016 SNT V y akl we rable a sep Chamberlin—Courant (β-CC): f(i 1, . . . , ik) = β(i 1) representation β-CC focused da k-Borda: f(i 1, . . . , ik) = β(i 1) + β(i 2) +. . + β(ik) k-Bor SNTV: f(i 1, . . . , ik) = α 1(i 1) Committee Scoring Rules α Bloc C -C k

Committee Scoring Rules Examples Proportional Approval Voting (as CSR): f(i 1, . . . , ik) = αk(i 1) + 1/2αk(i 2) +. . . + 1/k αk(ik) [FSST 16] P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Committtee Scoring Rules: Axiomatic Classification and Hierarchy, AAAI-2016 V y akl we rable a sep Chamberlin—Courant (β-CC): f(i 1, . . . , ik) = β(i 1) SNT α Bloc C -C k α t o co p k -P un k AV tin g Bloc: f(i 1, . . . , ik) = αk(i 1) + αk(i 2) +. . + αk(ik) representation β-CC focused da k-Borda: f(i 1, . . . , ik) = β(i 1) + β(i 2) +. . + β(ik) k-Bor SNTV: f(i 1, . . . , ik) = α 1(i 1) Committee Scoring Rules

Committee Scoring Rules Examples Proportional Approval Voting (as CSR): f(i 1, . . . , ik) = αk(i 1) + 1/2αk(i 2) +. . . + 1/k αk(ik) [FSST 16] P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Committtee Scoring Rules: Axiomatic Classification and Hierarchy, AAAI-2016 α representation β-CC focused SNT V y akl we rable a sep Chamberlin—Courant (β-CC): f(i 1, . . . , ik) = β(i 1) OWA-based α Bloc t -P AV C -C k α t o co p k -P un k AV tin g Bloc: f(i 1, . . . , ik) = αk(i 1) + αk(i 2) +. . + αk(ik) β- da k-Borda: f(i 1, . . . , ik) = β(i 1) + β(i 2) +. . + β(ik) V A P k-Bor SNTV: f(i 1, . . . , ik) = α 1(i 1) Committee Scoring Rules

Nature of the Committees (Individual Excellence)

Excellence: Main Idea Given: A group of candidates and preferences of the judges/voters Task: Select k best candidates (the finalists)

Excellence: Main Idea Committee Selection Given: A group of candidates and preferences of the judges/voters Task: Select k best candidates (the finalists) Final Selection Later: Select the overall winner

Committee Scoring Rules Problem with Bloc Committee Scoring Rules β- V A P OWA-based representation β-CC focused SNT V k-Bor da y akl we rable a sep [FSST 16] P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Committtee Scoring Rules: Axiomatic Classification and Hierarchy, AAAI-2016 α α Bloc t -P AV C -C k α t o co p k -P un k AV tin g Committee Monotonicity: If a candidate is selected for a committee of size k, then this candidate is also selected for committee of size k+1

Committee Scoring Rules β- α α le Bloc t -P AV C -C k α t o co p k -P un k AV tin g da rab [FSST 16] P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Committtee Scoring Rules: Axiomatic Classification and Hierarchy, AAAI-2016 SNT committee V monotone a sep SNTV: f(i 1, . . . , ik) = α 1(i 1) + α 1(i 2) +. . . + α 1(ik) k-Borda: f(i 1, . . . , ik) = β(i 1) + β(i 2) +. . + β(ik) OWA-based representation β-CC focused Theorem A committee scoring rule is committee monotone if and only if it is separable. Separable Rules V A P k-Bor Committee Monotonicity: If a candidate is selected for a committee of size k, then this candidate is also selected for committee of size k+1

Committee Scoring Rules β- > > > > [FSST 16] P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Committtee Scoring Rules: Axiomatic Classification and Hierarchy, AAAI-2016 le > rab > a sep > SNT committee V monotone da > α representation β-CC focused k-Bor > V A P OWA-based α Bloc t -P AV C -C k α t o co p k -P un k AV tin g Noncrossing Monotonicity: If a member of the winning committee is moved forward (without passing another committee member), the committee is still winning

Committee Scoring Rules Theorem A committee scoring rule is noncrossing monotone if and only if it is weakly separable. α α le Bloc t -P AV C -C k α t o co p k -P un k AV tin g rab [FSST 16] P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Committtee Scoring Rules: Axiomatic Classification and Hierarchy, AAAI-2016 SNT committee V monotone a sep SNTV: f(i 1, . . . , ik) = α 1(i 1) Bloc: f(i 1, . . . , ik) = αk(i 1) + αk(i 2) +. . + αk(ik) k-Borda: f(i 1, . . . , ik) = β(i 1) + β(i 2) +. . + β(ik) OWA-based representation β-CC focused da Weakly Separable Rules β- V A P k-Bor Noncrossing Monotonicity: If a member of the winning committee is moved forward (without passing another committee member), the committee is still winning

Committee Scoring Rules Committee SNT monotone V noncrossing monotone Bloc ble [FSST 16] P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Committtee Scoring Rules: Axiomatic Classification and Hierarchy, AAAI-2016 representation β-CC focused ara sep SNTV: f(i 1, . . . , ik) = α 1(i 1) Bloc: f(i 1, . . . , ik) = αk(i 1) + αk(i 2) +. . + αk(ik) k-Borda: f(i 1, . . . , ik) = β(i 1) + β(i 2) +. . + β(ik) α Weakly separable α t -P AV C -C k α t o co p k -P un k AV tin g Weakly Separable Rules β- V A P OWA-based da Theorem A committee scoring rule is noncrossing monotone if and only if it is weakly separable. Committee Scoring Rules k-Bor Noncrossing Monotonicity: If a member of the winning committee is moved forward (without passing another committee member), the committee is still winning

Is SNTV really good for individual excellence? SNTV k-Borda

Nature of the Committees (Diveristy/Coverage)

Applications Requiring Diversity/Coverage Instead of finding the “best” candidates (recall Excellence) we aim at covering all views of the electorate Some applications: Where to place facilities? Which products to produce? Which products to advertise?

Axioms for Diversity: Narrow Top A rule satisfies the narrow top criterion if whenever there is a set W of k candidates such that each voter ranks first a member of W, then W is a winning committee 4 3 2 1 0 V 1: > > V 2: > > V 3: > > V 4: > > β-CC and SNTV β-CC satisfies narrow top k-Borda (e. g. , ) does not k-Borda : 10 : 8 : 4 SNTV : 2 E. Elkind, P. Faliszewski, P. Skowron, A. Slinko: Properties of Multiwinner Voting Rules, SC&W, 2017 : 0

Axioms for Diversity: Narrow Top [FSST 16] P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Committtee Scoring Rules: Axiomatic Classification and Hierarchy, AAAI-2016 α representation β-CC focused t -P AV narrow top SNT consistent -CC V αk y akl we rable a sep SNTV: f(i 1, . . . , ik) = α 1(i 1) β-CC: f(i 1, . . . , ik) = β(i 1) OWA-based Bloc α t o co p k -P un k AV tin g Representation-Focused Rules β- V A P da Theorem If a committee scoring rule is representation-focused then it is narrow-top consistent. Committee Scoring Rules k-Bor Narrow Top A rule satisfies the narrow top criterion if whenever there is a set W of k candidates such that each voter ranks first a member of W, then W is a winning committee

Axiom: Noncrossing monotonicity Noncrossing Monotonicity: If a member of the winning committee is moved forward (without passing another committee member), the committee is still winning > > > > > β-CC fails noncrossing monotonicity

Axiom: Top-member monotonicity Top-Member Monotonicity: If the highest ranked member of the winning committee is moved forward, the committee still wins. > > > > > score( , ) =X

Axiom: Top-member monotonicity Top-Member Monotonicity: If the highest ranked member of the winning committee is moved forward, the committee still wins. > > > > > β-CC satisfies top-member monotonicity score( , ) = X +1 The shift gives the same number of points to every committee where the candidate is top member

Axiom: Top-member monotonicity Noncrossing Monotonicity: If a member of the winning committee is moved forward (without passing another committee member), the committee is still winning Committee Scoring Rules β- V A P OWA-based α representation β-CC focused noncrossing V monotone k-Bor da y akl we rable a sep [FSST 16] P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Committtee Scoring Rules: Axiomatic Classification and Hierarchy, AAAI-2016 Bloc α AV C -C k α t o co p k -P un k AV tin g SNT t -P

Axiom: Top-member monotonicity Top-Member Monotonicity: If the highest ranked member of the winning committee is moved forward, the committee still wins. Committee Scoring Rules OWA-based α V A t -P P β representation AV top-member focused β-CC monotone CC SNT noncrossing V αk monotone k-Bor da y akl we rable a sep [FSST 16] P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Committtee Scoring Rules: Axiomatic Classification and Hierarchy, AAAI-2016 Bloc α t o co p k -P un k AV tin g Noncrossing Monotonicity: If a member of the winning committee is moved forward (without passing another committee member), the committee is still winning

Axioms: Narrow Top + Top Member Monotonicity [FSST 16] P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Committtee Scoring Rules: Axiomatic Classification and Hierarchy, AAAI-2016 noncrossing Bloc monotone α t o co p k -P un k AV tin g α V A t -P P β representation AV top-member focused β-CC monotone SNTnarrow top -CC k V α consistent y akl we rable a sep Narrow Top A rule satisfies the narrow top criterion if whenever there is a set W of k candidates such that each voter ranks first a member of W, then W is a winning committee OWA-based da Top-Member Monotonicity: If the highest ranked member of the winning committee is moved forward, the committee still wins. Committee Scoring Rules k-Bor Noncrossing Monotonicity: If a member of the winning committee is moved forward (without passing another committee member), the committee is still winning

Axioms: Narrow Top + Top Member Monotonicity α t -P AV top-member + C C SNT k narrow topα V noncrossing Bloc monotone α t o co p k -P un k AV tin g Theorem A committee scoring rule is representation focused if and only if it is topmember monotone and consistent with the narrow-top principle. OWA-based representation β-CC focused y akl we rable a sep Narrow Top A rule satisfies the narrow top criterion if whenever there is a set W of k candidates such that each voter ranks first a member of W, then W is a winning committee β- V A P da Top-Member Monotonicity: If the highest ranked member of the winning committee is moved forward, the committee still wins. Committee Scoring Rules k-Bor Noncrossing Monotonicity: If a member of the winning committee is moved forward (without passing another committee member), the committee is still winning

Chamberlin—Courant is good for diversity

How to deal with hardness of Chamberlin —Courant?

Greedy CC 4 Input: E = (C, V) — an election k — size of the parliament V 1 : Algorithm: V 2 : for i = 1 to k do: pick a candidate c that increases the score of the assignment most return the computed assignment V 3 : V 4 : V 5 : V 6 : This is a 1 -1/e approximation algorithm 3 2 1 0

Greedy CC 4 V 1 : V 2 : V 3 : V 4 : V 5 : V 6 : 3 2 1 0

CC Greedy. CC

Algorithm P Rank 1 2 3 v 1 v 2 x m Goal: pick K winners among m candidates, to get the highest utility Initialize: Forget about the whole profile beyond rank x: x = mw(K) / K (w(K) is Lambert’s W function, O(log K)) Loop: Keep picking the candidate that appears in the „available” part of the profile most frequently. vn

1 -Best: PTAS (Borda Utilities) Rank 1 2 3 v 1 v 2 x m Goal: pick K winners among m candidates, to get the highest utility Initialize: Forget about the whole profile beyond rank x: x = mw(K) / K (w(K) is Lambert’s W function, O(log K)) Loop: Keep picking the candidate that appears in the „available” part of the profile most frequently. vn

1 -Best: PTAS (Borda Utilities) Rank 1 2 3 v 1 v 2 vn x m

CC Greedy. CC Algorithm P

Ranging. CC Rank 1 2 3 v 1 v 2 vn x m

CC Greedy. CC Algorithm P Ranging. CC

Summary

State of the Art Syntax VS Semantics Committee Scoring Rules OWA-based SNT da k-Bor t -P β representation AV β-CCfocused narrow top C SNTconsistent k-C α V y akl we rable a sep Bloc C -k C α co top k -P un k AV tin g α V AV α V A -P Bloc α t o co p k -P un k AV tin g representation β-CCfocused t -P da α k-Bor V A P β-

Take-Home Message s n o i t a c i l p y ap Man New sub-field Multiwinner Elections ics t a m e h t a Nice M

Thank You! https: //github. com/elektronaj/MW 2 D Multiwinner Voting: A New Challenge for Social Choice Theory, P. Faliszewski, P. Skowron, A. Slinko, N. Talmon, Trends in Computational Social Choice, 2017