Proportionality under Multiwinner Elections Piotr Skowron TU Berlin

Proportionality under Multiwinner Elections Piotr Skowron TU Berlin Germany Based on tutorial of myself and: Piotr Faliszewski AGH University Kraków, Poland Nimrod Talmon Weizmann Instutute of Science, Israel

Single-Winner Elections Multiwinner Elections movies on a plane voters’ preferences fundamentally different! voting rule Seek the best, the most widely supported candidates t n e m a i rl pa [EFSS 17] E. Elkind, P. Faliszewski, P. Skowron, A. Slinko, Properties of Multiwinner Voting Rules, Social Choice and Welfare, 2017 s making a shortlist

What Are We After? Applicatons • Shortlisting • • Similar candidates: treated identically Movie selection • • • Jobs Awards Competitions Excellence One movie per passenger Proportional to interest? Parliamentary elections • • National level Company level or internal level Diversity Similar candidates: at most one gets in Proportionality Similar candidates: outcome proportional to support Axioms • Committee Monotonicity • Dummet’s Proportionality • . . . Algorithms • Efficient • • • Poly-time FPT Approx. Heuristics Hardness • • • NP-hardness W[]-hard. Inapprox.

Nature of the Committees (Proportionality)

Traditional Approaches to Electing a Parliament Single-member districts: • voters and candidates are divided into districts • each electoral district selects a single candidate

Traditional Approaches to Electing a Parliament Single-member districts: • voters and candidates are divided into districts • each electoral district selects a single candidate district 2 district 3 district 1 committee of size k = 3

Traditional Approaches to Electing a Parliament Single-member districts: • voters and candidates are divided into districts • each electoral district This is not proportional! selects a single candidate district 1 district 2 district 3 Disproportionality of the single-member constituency can be very large: [BLLZ 17] Y. Bachrach, O. Lev, Y. Lewenberg, Y. Zick, Misrepresentation in District Voting, committee of size k = 3 IJCAI, 2016

Traditional Approaches to Electing a Parliament • Some countries use open-list systems In ``panachage’’ voters can vote for candidates Party list systems: Voters do not really have influence from different parties • voters vote for political parties who’ll get to the parliament. [LS 16] J. -F. Laslier, K. Van der Straeten, Strategic rather than for individuals voting in multi-winners elections with approval • balloting: a theory for large electorates, Social Choice and Welfare, 2016 Distribution of votes for parties I should care more about my party than about the electorate Distribution of k = 95 seats 10 45 20 20

Proportionality in Approval-Based Model V 1: V 2: V 3: V 4: V 5: V 6: V 7: V 8:

Proportionality in Approval-Based Model V 1: V 2: V 3: V 4: Justified Representation: In each 1 -cohesive group there should be a voter who approves a committee member V 5: V 6: OK! V 7: V 8:

Proportionality in Approval-Based Model V 1: Rules satisfying JR: • Greedy Approval Chamberlin-Courant • Approval Monroe • Proportional Approval Voting Rules not satisfying JR: • Approval Voting • Greedy Proportional Approval Voting • Satisfaction Approval Voting • Minimax Approval Voting Justified Representation: V 2: V 3: V 4: V 5: V 6: [ABC+17] H. Aziz, M. Brill, V. Conitzer, E. Elkind, R. Freeman, T. Walsh, Justified representation in approval-based committee voting, Social Choice and Welfare, 2017. V 7: In each 1 -cohesive group there should be a voter who approves a committee member V 8: BAD!

Chamberlin-Courant Select k candidates which together maximize the total satisfaction Approval-CC V 1: { } V 2: { V 3: { } } } V 4: { V 5: { “cover” as many voters as possible } [CC 83] JR. Chamberlin, PN. Courant, Representative deliberations and representative decisions: Proportional representation and the Borda rule, American Political Science Review, 1983 [PRZ 08] A. Procaccia, J. Rosenschein, A. Zohar, On the Complexity of Achieving Proportional Representation, Social Choice and Welfare 2008

Chamberlin-Courant Select k candidates which together maximize the total satisfaction Approval-CC V 1: { } V 2: { V 3: { } } } V 4: { V 5: { “cover” as many voters as possible ({ , }) = 3 ({ , }) = 2 ({ , }) = 4 } [CC 83] JR. Chamberlin, PN. Courant, Representative deliberations and representative decisions: Proportional representation and the Borda rule, American Political Science Review, 1983 [PRZ 08] A. Procaccia, J. Rosenschein, A. Zohar, On the Complexity of Achieving Proportional Representation, Social Choice and Welfare 2008

Recall the definition of PAV V 1: V 2: E. g. , consider committee V 3: V 4: V 5: V 6: V 7: V 8:

Recall the definition of PAV V 1: V 2: E. g. , consider committee V 4: V 3: A committee with the highest total score wins the election. V 5: V 6: V 7: V 8:

Proportionality in Approval-Based Model Rules satisfying EJR: • PAV is the only OWA-based rule which satisfies EJR [ABC+17] V 1: • Local search algorithm for PAV satisfies EJR [AH 17, SLES 17] • There exist some other rules satisfying EJR [SEL 17] V 2: [ABC+17] H. Aziz, M. Brill, V. Conitzer, E. Elkind, R. Freeman, T. V 3: Walsh, Justified representation in approval-based committee voting, Social Choice and Welfare, 2017. V 4: [AH 17] H. Aziz, S. Huang, A Polynomial-time Algorithm to Achieve Extended Justified Representation, Arxiv, 2017 V 5: [SLES 17] P. Skowron, M. Lackner, E. Elkind, L. Sánchez-Fernández, Optimal Average Satisfaction and Extended Justified Representation in Polynomial Time, Arxiv, 2017. V 6: [SEL 17] L. Sánchez-Fernández, E. Elkind, M. Lackner, Committees V 7: providing EJR can be computed efficiently, Arxiv, 2017. V 8:

Proportionality in Approval-Based Model V 1: Rules satisfying PJR: V 2: • PAV (the only OWA-based rule) [SEL+17] • Approval-based Monroe and Greedy Monroe [SEL+17] V 3: • Max-Phragmén [BFJL 17] • Sequential-Phragmén [BFJL 17] V 4: [SEL+17] L. Sánchez Fernández, E. Elkind, M. Lackner, N. Fernández García, J. Arias-Fisteus, P. Basanta-Val, P. Skowron, Proportional Justified Representation, AAAI 2016. V 5: [BFJL 17] M. Brill, R. Freeman, S. Janson, M. Lackner, Phragmén's V 6: Voting Methods and Justified Representation, AAAI 2017. V 7: V 8: OK!

Phragmén’s Rule: the Sequential Variant • Each candidate has one unit of ``load’’. • A committee member distributes her load to the voters approving her so that the load of the voter with the highest load is minimized. • In each step the candidate that minimizes the max load is selected. 0. 7 0. 6 c 1 0. 5 0. 4 c 2 0. 3 0. 2 c 6 0. 1 0 v 1 v 2 {c 1, c 2, c 3, c 4} v 3 v 4 {c 5, c 6} v 5 v 6 {c 1, c 2, c 3} {c 1, c 2} v 7 {c 1, c 6} c 3

Phragmén’s Rule: the Sequential Variant • Each candidate has one unit of ``load’’. • A committee member distributes her load to the voters approving her so that the load of the voter with the highest load is minimized. • In each step the candidate that minimizes the max load is selected. 0. 7 0. 6 c 1 0. 5 0. 4 c 2 0. 3 0. 2 0. 1 v 1 v 2 v 3 v 5 v 6 v 7 0 {c 1, c 2, c 3, c 4} v 4 {c 5, c 6} {c 1, c 2, c 3} {c 1, c 2} {c 1, c 6} c 6 c 3

Phragmén’s Rule: the Sequential Variant • Each candidate has one unit of ``load’’. • A committee member distributes her load to the voters approving her so that the load of the voter with the highest load is minimized. • In each step the candidate that minimizes the max load is selected. 0. 7 0. 6 c 1 0. 5 0. 4 0. 3 c 2 v 1 v 2 v 3 v 5 v 6 v 7 0. 2 0. 1 0 {c 1, c 2, c 3, c 4} v 4 {c 5, c 6} {c 1, c 2, c 3} {c 1, c 2} {c 1, c 6} c 6 c 3

Phragmén’s Rule: the Sequential Variant • Each candidate has one unit of ``load’’. • A committee member distributes her load to the voters approving her so that the load of the voter with the highest load is minimized. • In each step the candidate that minimizes the max load is selected. 0. 7 0. 6 c 1 0. 5 0. 4 0. 3 1/ 5 v 1 v 2 v 3 0. 2 0. 1 c 2 c 6 v 5 v 6 v 7 0 {c 1, c 2, c 3, c 4} v 4 {c 5, c 6} {c 1, c 2, c 3} {c 1, c 2} {c 1, c 6} c 3

Phragmén’s Rule: the Sequential Variant • Each candidate has one unit of ``load’’. • A committee member distributes her load to the voters approving her so that the load of the voter with the highest load is minimized. • In each step the candidate that minimizes the max load is selected. Original Phragmén’s works are in Swedish. Survery on Phragmén's and Thiele's election methods: 0. 7 [Jan 16] S. Janson, Phragmén's and Thiele's election methods, Arxiv, 2016 0. 5 Some properties of Phragmén's rule: [BFJL 17] M. Brill, R. Freeman, S. Janson, M. Lackner, Phragmén's Voting Methods and Justified Representation, AAAI 2017. 1/ 5 0. 4 0. 3 0. 2 0. 1 c 1 v 1 v 2 v 3 v 5 v 6 v 7 0 {c 1, c 2, c 3, c 4} v 4 {c 5, c 6} {c 1, c 2, c 3} {c 1, c 2} {c 1, c 6} c 2 c 6 c 3

Can We Use Multiwinner Rules for Party-Lists? 60 votes: Party A 30 votes: Party B 10 votes: Party C

Can We Use Multiwinner Rules for Party-Lists? 60 votes: Party A 30 votes: Party B Sequential PAV when applied to party 10 votes: Party C -list profiles gives proportional apportionment

Can We Use Multiwinner Rules for Party-Lists? How about Approval Chamberlin—Courant? 60 votes: Approval Party A Chamberlin—Courant when 30 votes: 10 votes: applied to party-list profiles can be very disproportional Party B Party C

Can We Use Multiwinner Rules for Party-Lists? How about Approval Chamberlin—Courant? 60 votes: Party A 30 votes: Party B 10 votes: Party C Diversity versus Proportionality 1 vote for each of:

Can We Use Multiwinner Rules for Party-Lists? How about Approval Voting? 60 votes: Excellence versus Proportionality Party A 30 votes: Party B 10 votes: Party C

Relation between Multiwinner Rules and Apportionment Methods PAV Sequential PAV D’Hondt method Max-Phragmén Variance minimizing Phragmén Monroe Sainte-Laguë method Hamilton method Greedy Monroe [BLS 17] M. Brill, J. -F. Laslier, P. Skowron, Multiwinner Approval Rules as Apportionment Methods, AAAI 2017.

D’Hondt method of apportionment: Example Party 1: 6 votes Party 2: 7 votes Party 3: 39 votes Party 4: 48 votes

D’Hondt method of apportionment: Example Party 1: 6 votes Party 2: 7 votes Party 3: 39 votes Party 4: 48 votes Party 1 6 Party 2 7 Party 3 Party 4 39 48

D’Hondt method of apportionment: Example Party 1: 6 votes Party 2: 7 votes Party 3: 39 votes Party 4: 48 votes Party 1 Party 2 Party 3 Party 4 6 7 39 48 3 3. 5 19. 5 24 2 2. 33 13 16 1. 5 1. 75 9. 75 12 1. 4 7. 8 9. 6 1 1. 17 6. 5 8. 0 0. 86 1 5. 57 6. 86

D’Hondt method of apportionment: Example Party 1: 6 votes Party 2: 7 votes Party 3: 39 votes Party 4: 48 votes Party 1 Party 2 Party 3 Party 4 39 48 6 Party 1 gets 0 seats 7 Party 2 gets 0 seats 3 Party 3 gets 4 seats 3. 5 19. 5 Party 4 gets 6 seats 2 2. 33 24 13 16 1. 5 1. 75 9. 75 12 1. 4 7. 8 9. 6 1 1. 17 6. 5 8. 0 0. 86 1 5. 57 6. 86

Relation between Multiwinner Rules and Apportionment Methods PAV Sequential PAV D’Hondt method Max-Phragmén Variance minimizing Phragmén Monroe Sainte-Laguë method Hamilton method Greedy Monroe [BLS 17] M. Brill, J. -F. Laslier, P. Skowron, Multiwinner Approval Rules as Apportionment Methods, AAAI 2017.

Sainte-Laguë method of apportionment: Example Party 1: 6 votes Party 2: 7 votes Party 3: 39 votes Party 4: 48 votes Party 1 Party 2 Party 3 Party 4 6 7 39 48 Party 1 gets 1 seat 2 Party 2 gets 1 seat 2. 33 13 Party 3 gets 4 seats 1. 2 1. 4 Party 4 gets 4 seats 7. 8 9. 6 16 0. 86 1 5. 57 6. 86 0. 67 0. 78 4. 33 5. 33

Relation between Multiwinner Rules and Apportionment Methods PAV Sequential PAV D’Hondt method Max-Phragmén Variance minimizing Phragmén Monroe Sainte-Laguë method Hamilton method Greedy Monroe [BLS 17] M. Brill, J. -F. Laslier, P. Skowron, Multiwinner Approval Rules as Apportionment Methods, AAAI 2017.

Hamilton method of apportionment: Example Party 1: 6 votes Party 2: 7 votes Party 3: 39 votes Party 4: 48 votes Party 1 gets 0 seat Party 2 gets 1 seat Party 3 gets 4 seats Party 4 gets 5 seats There are 3 remaining seats. Sort the parties by the remainders: Party 3 (with remainder 9) > Party 4 (with remainder 8) > Party 2 (with remainder 7) > Party 1 (with remainder 6) And take 3 parties with 3 largest remainders.

Relation between Multiwinner Rules and Apportionment Methods Theorem: Multi-Winner Approval Voting is the only ABC ranking rule that satisfies symmetry, consistency, continuity and disjoint equality. Theorem: Proportional Approval Voting is the only ABC ranking rule that satisfies symmetry, consistency, continuity and D’Hondt proportionality. Theorem: The Approval Chamberlin–Courant rule is (almost) the only ABC ranking rule that satisfies symmetry, consistency, weak efficiency, continuity and disjoint diversity. [LS 17] M. Lackner, P. Skowron, Consistent Approval-Based Multi-Winner Rules, Arxiv 2017.

Interlude: my view on new directions of research

Computational Properties of PAV: PAV for 2 -Euclidean Preferences • NP-hard to compute: [AGG+15] H. Aziz, S. Gaspers, J. Gudmundsson, S. Mackenzie, N. Mattei, T. Walsh, Computational Aspects of Multi-Winner Approval Voting, AAMAS, 2015. [SFL 16] P. Skowron, P. Faliszewski, J. Lang, Finding a collective set of items: uniform on From proportional multirepresentation to group recommendation. Artif. Intell. , Gaussian 4 Gaussians a square 2016 a circle • Restricted domains: [Pet 16] D. Peters, Single-Peakedness and Total Unimodularity: Efficiently Solve Voting Problems Without Even Trying, Arxiv, 2016. • Admits good approximation: [SFL 16] P. Skowron, P. Faliszewski, J. Lang, Finding a collective set of items: From proportional multirepresentation to group recommendation. Artif. Intell. , 2016 [BSS 17] J. Byrka, P. Skowron, K. Sornat, Proportional Approval Voting, Harmonic k-median, and Negative Association, Arxiv, 2017. • FPT approximation schemes: [Sko 16] P. Skowron, FPT Approximation Schemes for Maximizing Submodular

Ordinal Preferences and Proportionality

SNTV V 1: > > V 2: > > V 3: > > > > > V 4: > Too much focus on the top preferences. V 5: > V 6: > > V 7: > > V 8: > >

SNTV for 2 -Euclidean Preferences uniform on a circle Gaussian 4 Gaussians uniform on a square [EFL+17] E. Elkind, P. Faliszewski, JR. Laslier, P. Skowron, A. Slinko, and N. Talmon: What Do Multiwinner Voting Rules Do? An Experiment Over the Two-Dimensional Euclidean Domain, AAAI 2017

Dummett’s Proportionality and STV V 1: > > V 2: > > V 3: > > V 4: > > V 5: > > V 6: > > V 7: > > V 8: > >

Recall the definition of STV V 1: > > V 2: > > V 3: > > V 4: > > V 5: > > V 6: > > V 7: > > V 8: > >

Recall the definition of STV V 2: > > > V 3: > > > V 7: > > > V 8: > > >

Recall the definition of STV x> V 2: > > V 3: > x> V 7: > > x> V 8: > > x

Recall the definition of STV V 2: > V 3: > V 7: x> V 8: > x >x > > x>

Recall the definition of STV V 2: V 3: V 7: V 8: x >x > x> x>

Recall the definition of STV V 2: V 3: V 7: V 8:

STV for 2 -Euclidean Preferences uniform on a circle Gaussian 4 Gaussians uniform on a square STV is often considered to be very well-suited for tasks that require proportional representation: [TR 00] N. Tideman, D. Richardson. Better voting methods through technology: The refinement-manageability trade-off in the Single Transferable Vote, Public Choice, 2000. [EFSS 17] E. Elkind, P. Faliszewski, P. Skowron, A. Slinko, Properties of Multiwinner Voting Rules, Social Choice and Welfare, 2017 [EFL+17] E. Elkind, P. Faliszewski, JR. Laslier, P. Skowron, A. Slinko, and N. Talmon: What Do Multiwinner Voting Rules Do? An Experiment Over the Two-Dimensional Euclidean Domain, AAAI 2017

Drawbacks of STV This is a very nonmonotonic rule! > The following profile: 10 votes: > > > 9 votes: > > > 1 vote: > > > 10 votes: > >

Solid Coalitions Property and the Monroe rule V 1: > > V 2: > > > > V 6: > > V 7: > > V 8: > > Solid coalitions property is satisfied by STV, SNTV, V 3: > and by the greedy variant of the Monroe rule. [EFSS 17] E. Elkind, P. Faliszewski, P. Skowron, A. V 4: > Slinko, Properties of Multiwinner Voting Rules, Social Choice and Welfare, 2017 V 5:

The Monroe rule Define the score for a committee: V 1: > > V 2: > > V 3: > > V 4: > > V 5: > > V 6: > > V 7: > > V 8: > > 4 3 2 1 0

The Monroe rule Define the score for a committee: V 1: > > V 2: > > V 3: > > V 4: > > V 5: > > V 6: > > V 7: > > V 8: > > This would be a better assignment with score of 30. But this assignment is unbalanced and so it is not valid! 4 3 2 1 0

The Monroe rule Define the score for a committee: V 1: > > V 2: > > V 3: > > V 4: > > V 5: > > V 6: > > V 7: > > V 8: > > A committee with the best optimal valid assignment is winning. 4 3 2 1 0

The Greedy Monroe rule 4 3 2 1 0 V 1: > > V 2: > > V 3: > > V 4: > > V 5: > > V 6: > > V 7: > > V 8: > >

The Greedy Monroe rule 4 3 2 1 0 V 2: > > Greedy Monroe satisfies the Solid Coalitions property but V 3: > > the original Monroe’s rule does not. > > V 4: Interesting example of a criterion where the ``approximate’’ variant of a rule is better than the original variant. V 5: V 6: V 7: > > V 8: > >

How Good is the Greedy Monroe Rule? Consider the situation right after the i-th iteration By pigeonhole principle, there is an unassigned candidate that n/k voters rank within the green area i possibly unavailable (m-i)/(k-i) positions v 1: vj : vn: in/k voters with assignment

How Good is the Greedy Monroe Rule? Consider the situation right after the i-th iteration By pigeonhole principle, there is an unassigned candidate that n/k voters rank within the green area i possibly unavailable (m-i)/(k-i) positions v 1: vj : vn: in/k voters with assignment In the i-th iteration we pick a candidate who gives his/her voters total satisfaction at least: n/k ( m – i – (m-i)/(k-i) )

How Good is the Greedy Monroe Rule? Consider the situation right after the i-th iteration By pigeonhole principle, there is an unassigned candidate that n/k voters rank within the green area i possibly unavailable (m-i)/(k-i) positions v 1: vj : vn: in/k voters with assignment In the i-th iteration we pick a candidate who gives his/her voters total satisfaction at least: n/k ( m – i – (m-i)/(k-i) ) We achieve: 1 – (k-1)/2(m-1) – H fraction of m k /k aximum possib le satisfaction!

Okay, but is it really a good result? • Polish parliamentary elections: – k = 460, m = 6000 – We reach about 96% of maximal possible satisfaction – On the avarage, every voter is represented by someone this voter prefers to 96% of the candidates • Problems? – … the system assumes that each voter would rank 6000 candidates!! – There are workarounds

Monroe for 2 -Euclidean Preferences uniform on Computational a circle Gaussian Properties of 4 Gaussians the Monroe uniform on rule: a square • NP-hard to compute: [PRZ 08] A. Procaccia, J. Rosenschein, A. Zohar, On the complexity of achieving proportional representation, Social Choice and Welfare, 2008. • No FPT algorithms for many natural parameters: [BSU 13] N. Betzler, A. Slinko, J. Uhlmann, On the Computation of Fully Monroe: Proportional Representation. J. Artif. Intell. Res, 2013. • Admits relatively good approx. (i. e. , Greedy Monroe): [SFS 15] P. Skowron, P. Faliszewski, A. Slinko, Achieving fully proportional Greedy representation: Approximability results. Artif. Intell, 2015. Monroe: [EFL+17] E. Elkind, P. Faliszewski, JR. Laslier, P. Skowron, A. Slinko, and N. Talmon: What Do Multiwinner Voting Rules Do? An Experiment Over the Two-Dimensional Euclidean Domain, AAAI 2017

Further Topics

Proportional Rankings Committee monotonicity If a rule picks some set W of winners when electing parliament of size k, this rule should pick W’ such that W W’, when electing a parliament of size k+1. Committee monotonicity can be used to create a ranking: > > >

Proportional Rankings Committee monotonicity If a rule picks some set W of winners when electing parliament of size k, this rule should pick W’ such that W W’, when electing a parliament of size k+1. Committee monotonicity to some extend contradicts proportionality: Committee monotonicity can be used to create a ranking: > left right > >

Proportional Rankings Committee monotonicity can be used to Committee monotonicity create a ranking: How to approach the problem of finding a proportional If a rule picks some set W of winners ranking? when electing parliament of size k, this rule should pick W’ such that W W’, • We can use insights from multiwinner elections to define > > > when electing a parliament of size k+1. formally the notion of proportionality of rankings, • There exists some committee monotonic proportional Recall Greedy CC: Committee monotonicity to some extend rules, which can be used in this case: contradicts proportionality: • Sequential PAV • Sequential Phragmén’s Rule • Some greedy variants of proportional/diverse rules left right

Interlude: my view on new directions of research

Interlude: my view on new directions of research Back to fundamentals: we should better understand the properties.

Interlude: my view on new directions of research Back to fundamentals: we should better understand the properties. New models allowing to assess which properties are desired.

Interlude: my view on new directions of research Back to fundamentals: we should better understand the properties. New models allowing to assess which properties are desired. E. g. , which distribution is better?

Thank You!