1. 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

2. Multiwinner Elections Single-Winner Elections
movies on a plane
voters’ preferences
voting rule
fundamentally
different!
making a shortlist
Seek the best, the most widely supported candidates
parliaments
[EFSS17] E. Elkind, P. Faliszewski, P. Skowron, A. Slinko, Properties of Multiwinner Voting Rules, Social Choice and Welfare, 2017

3. What Are We After? Excellence Diversity Proportionality Applicatons
Shortlisting
Jobs
Awards
Competitions
Movie selection
One movie per passenger
Proportional to interest?
Parliamentary elections
National level
Company level or internal level
Excellence
Similar candidates: treated identically
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.

4. What Are We After? Excellence Diversity Proportionality Applicatons
Shortlisting
Jobs
Awards
Competitions
Movie selection
One movie per passenger
Proportional to interest?
Parliamentary elections
National level
Company level or internal level
Excellence
Similar candidates: treated identically
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.

5. What Are We After? Excellence Diversity Proportionality Applicatons
Shortlisting
Jobs
Awards
Competitions
Movie selection
One movie per passenger
Proportional to interest?
Parliamentary elections
National level
Company level or internal level
Excellence
Similar candidates: treated identically
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.

6. Nature of the Committees
(Proportionality)

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

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

9. Traditional Approaches to Electing a Parliament
district 2
district 3
Single-member districts:
voters and candidates are divided into districts
each electoral district selects a single candidate
This is not proportional!
district 1
Disproportionality of the single-member constituency can be very large:
[BLLZ17] Y. Bachrach, O. Lev, Y. Lewenberg, Y. Zick, Misrepresentation in District Voting, IJCAI, 2016
committee of size k = 3

10. Traditional Approaches to Electing a Parliament
Some countries use open-list systems
In ``panachage’’ voters can vote for candidates from different parties
[LS16] J.-F. Laslier, K. Van der Straeten, Strategic voting in multi-winners elections with approval balloting: a theory for large electorates, Social Choice and Welfare, 2016
Party list systems:
voters vote for political parties rather than for individuals
Voters do not really have influence who’ll get to the parliament.
I should care more about my party than about the electorate

11. Proportionality in Approval-Based Model
ℓ-cohesive group of votets:
A group of at least ℓ⋅𝑛/𝑘 voters who all approve the same ℓ candidates
V1:
V2:
V3:
V4:
V5:
V6:
𝑘=2 and ℓ=1
V7:
V8:

12. Proportionality in Approval-Based Model
ℓ-cohesive group of votets:
A group of at least ℓ⋅𝑛/𝑘 voters who all approve the same ℓ candidates
V1:
V2:
V3:
V4:
V5:
V6:
𝑘=4 and ℓ=2
V7:
V8:

13. Proportionality in Approval-Based Model
ℓ-cohesive group of votets:
A group of at least ℓ⋅𝑛/𝑘 voters who all approve the same ℓ candidates
V1:
V2:
V3:
V4:
V5:
Justified Representation:
In each 1-cohesive group there should be a voter who approves a committee member
V6:
OK!
V7:
V8:
𝒌=𝟐

14. Proportionality in Approval-Based Model
ℓ-cohesive group of votets:
A group of at least ℓ⋅𝑛/𝑘 voters who all approve the same ℓ candidates
V1:
V2:
V3:
V4:
V5:
Justified Representation:
In each 1-cohesive group there should be a voter who approves a committee member
V6:
OK!
V7:
V8:
𝒌=𝟐

15. Proportionality in Approval-Based Model
ℓ-cohesive group of votets:
A group of at least ℓ⋅𝑛/𝑘 voters who all approve the same ℓ candidates
V1:
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
[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.
V2:
V3:
V4:
V5:
Justified Representation:
In each 1-cohesive group there should be a voter who approves a committee member
V6:
BAD!
V7:
V8:
𝒌=𝟐

16. Chamberlin-Courant Approval-CC “cover” as many voters as possible
Select k candidates which together maximize the total satisfaction
Approval-CC
“cover” as many voters as possible
V1:
V5:
V2:
V3:
V4: { }
[CC83] JR. Chamberlin, PN. Courant, Representative deliberations and representative decisions: Proportional representation and the Borda rule, American Political Science Review, 1983
[PRZ08] A. Procaccia, J. Rosenschein, A. Zohar, On the Complexity of Achieving Proportional Representation, Social Choice and Welfare 2008

17. Chamberlin-Courant ({ , }) = 3 ({ , }) = 2 ({ , }) = 4 Approval-CC
Select k candidates which together maximize the total satisfaction
Approval-CC
“cover” as many voters as possible
V1:
V5:
V2:
V3:
V4: { }
({ , }) = 3
({ , }) = 2
({ , }) = 4
[CC83] JR. Chamberlin, PN. Courant, Representative deliberations and representative decisions: Proportional representation and the Borda rule, American Political Science Review, 1983
[PRZ08] A. Procaccia, J. Rosenschein, A. Zohar, On the Complexity of Achieving Proportional Representation, Social Choice and Welfare 2008

18. Chamberlin-Courant ({ , }) = 3 ({ , }) = 2 ({ , }) = 4 Approval-CC
Select k candidates which together maximize the total satisfaction
Approval-CC
“cover” as many voters as possible
V1:
V5:
V2:
V3:
V4: { }
({ , }) = 3
({ , }) = 2
({ , }) = 4
[CC83] JR. Chamberlin, PN. Courant, Representative deliberations and representative decisions: Proportional representation and the Borda rule, American Political Science Review, 1983
[PRZ08] A. Procaccia, J. Rosenschein, A. Zohar, On the Complexity of Achieving Proportional Representation, Social Choice and Welfare 2008

19. Chamberlin-Courant ({ , }) = 3 ({ , }) = 2 ({ , }) = 4 Approval-CC
Select k candidates which together maximize the total satisfaction
Approval-CC
“cover” as many voters as possible
V1:
V5:
V2:
V3:
V4: { }
({ , }) = 3
({ , }) = 2
({ , }) = 4
[CC83] JR. Chamberlin, PN. Courant, Representative deliberations and representative decisions: Proportional representation and the Borda rule, American Political Science Review, 1983
[PRZ08] A. Procaccia, J. Rosenschein, A. Zohar, On the Complexity of Achieving Proportional Representation, Social Choice and Welfare 2008

20. Recall the definition of PAV
Assume a voter 𝑖 approves 𝑡 members of a committee 𝐶. Such voter gives score of …+ 1 𝑡 to 𝐶.
V1:
V2:
E.g., consider committee
V3:
V4:
Score from voter:
v1:
V5:
V6:
V7:
V8:

21. Recall the definition of PAV
Assume a voter 𝑖 approves 𝑡 members of a committee 𝐶. Such voter gives score of …+ 1 𝑡 to 𝐶.
V1:
V2:
E.g., consider committee
V3:
V4:
Score from voter:
v1:
v2:
V5:
V6:
V7:
V8:

22. Recall the definition of PAV
Assume a voter 𝑖 approves 𝑡 members of a committee 𝐶. Such voter gives score of …+ 1 𝑡 to 𝐶.
V1:
V2:
E.g., consider committee
V3:
V4:
Score from voter:
v1:
v2:
V5:
v3:
V6:
V7:
V8:

23. Recall the definition of PAV
Assume a voter 𝑖 approves 𝑡 members of a committee 𝐶. Such voter gives score of …+ 1 𝑡 to 𝐶.
V1:
V2:
E.g., consider committee
V3:
V4:
Score from voter:
v1:
A committee with the highest total score wins the election.
v2:
V5:
v3:
v4:
V6:
v5:
v6: 0
V7:
v7: 0
v8: 1
V8:
Total score = 8 5 6

24. Proportionality in Approval-Based Model
Rules satisfying EJR:
PAV is the only OWA-based rule which satisfies EJR [ABC+17]
Local search algorithm for PAV satisfies EJR [AH17, SLES17]
There exist some other rules satisfying EJR [SEL17]
[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.
[AH17] H. Aziz, S. Huang, A Polynomial-time Algorithm to Achieve Extended Justified Representation, Arxiv, 2017
[SLES17] P. Skowron, M. Lackner, E. Elkind, L. Sánchez-Fernández, Optimal Average Satisfaction and Extended Justified Representation in Polynomial Time, Arxiv, 2017.
[SEL17] L. Sánchez-Fernández, E. Elkind, M. Lackner, Committees providing EJR can be computed efficiently, Arxiv, 2017.
ℓ-cohesive group of votets:
A group of at least ℓ⋅𝑛/𝑘 voters who all approve the same ℓ candidates
V1:
V2:
V3:
V4:
V5:
Extended Justified Representation:
In each ℓ-cohesive group there should be a voter who approves at least ℓ committee members.
For 𝑘=4 one of the voters v1, v2, v3, v4 needs to approve at least 2 committee members.
V6:
V7:
V8:

25. Proportionality in Approval-Based Model
ℓ-cohesive group of votets:
A group of at least ℓ⋅𝑛/𝑘 voters who all approve the same ℓ candidates
V1:
Rules satisfying PJR:
PAV (the only OWA-based rule) [SEL+17]
Approval-based Monroe and Greedy Monroe [SEL+17]
Max-Phragmén [BFJL17]
Sequential-Phragmén [BFJL17]
[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.
[BFJL17] M. Brill, R. Freeman, S. Janson, M. Lackner, Phragmén's Voting Methods and Justified Representation, AAAI 2017.
V2:
V3:
V4:
Proportional JR:
For each ℓ-cohesive group 𝐺 there is a set of ℓ committee members 𝑆 such that each candidate from 𝑆 is approved by some voter from 𝐺.
V5:
OK!
V6:
V7:
V8:
𝒌=𝟒

26. 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.
{c1, c2, c3, c4}
{c5, c6}
{c1, c2, c3}
{c1, c2}
{c1, c6}

27. 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.
1 6
1 6
1 6
1 6
1 6
1 6
{c1, c2, c3, c4}
{c5, c6}
{c1, c2, c3}
{c1, c2}
{c1, c6}

28. 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.
1 5
1 5
1 5
1 5
1 5
1 6
1 6
1 6
1 6
1 6
1 6
{c1, c2, c3, c4}
{c5, c6}
{c1, c2, c3}
{c1, c2}
{c1, c6}

29. 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.
5 12
7 12
1 5
1 5
1 5
1 5
1 6
1 6
1 6
1 6
1 6
1 6
{c1, c2, c3, c4}
{c5, c6}
{c1, c2, c3}
{c1, c2}
{c1, c6}

30. 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:
[Jan16] S. Janson, Phragmén's and Thiele's election methods, Arxiv, 2016
Some properties of Phragmén's rule:
[BFJL17] M. Brill, R. Freeman, S. Janson, M. Lackner, Phragmén's Voting Methods and Justified Representation, AAAI 2017.
1 4
1 4
1 4
1 4
5 12
7 12
1 5
1 5
1 5
1 5
1 6
1 6
1 6
1 6
1 6
1 6
{c1, c2, c3, c4}
{c5, c6}
{c1, c2, c3}
{c1, c2}
{c1, c6}

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

32. Can We Use Multiwinner Rules for Party-Lists?
60 votes:
Party A
30 votes:
Party B
10 votes:
Party C
Sequential PAV:
How adding different candidates improves total PAV score:
candidate from Party A: 𝟔𝟎⋅𝟏 = 60
candidate from Party B: 30⋅1 = 30
candidate from Party C: 10⋅1 = 10

33. Can We Use Multiwinner Rules for Party-Lists?
60 votes:
Party A
30 votes:
Party B
10 votes:
Party C
Sequential PAV:
How adding different candidates improves total PAV score:
candidate from Party A: 𝟔𝟎⋅½ = 30
candidate from Party B: 3𝟎⋅𝟏 = 30
candidate from Party C: 10⋅1 = 10

34. Can We Use Multiwinner Rules for Party-Lists?
60 votes:
Party A
30 votes:
Party B
10 votes:
Party C
Sequential PAV:
How adding different candidates improves total PAV score:
candidate from Party A: 60⋅⅓ = 20
candidate from Party B: 3𝟎⋅𝟏 = 𝟑𝟎
candidate from Party C: 10⋅1 = 10

35. Can We Use Multiwinner Rules for Party-Lists?
60 votes:
Party A
30 votes:
Party B
10 votes:
Party C
Sequential PAV:
How adding different candidates improves total PAV score:
candidate from Party A: 𝟔𝟎⋅⅓ = 20
candidate from Party B: 30⋅ ½ = 15
candidate from Party C: 10⋅1 = 10

36. Can We Use Multiwinner Rules for Party-Lists?
60 votes:
Party A
30 votes:
Party B
10 votes:
Party C
Sequential PAV:
How adding different candidates improves total PAV score:
candidate from Party A: 𝟔𝟎⋅ ¼ = 15
candidate from Party B: 3𝟎⋅ ½ = 15
candidate from Party C: 10⋅1 = 10

37. Can We Use Multiwinner Rules for Party-Lists?
60 votes:
Party A
30 votes:
Party B
10 votes:
Party C
Sequential PAV:
How adding different candidates improves total PAV score:
candidate from Party A: 60⋅ ⅕ = 12.5
candidate from Party B: 3𝟎⋅ ½ = 15
candidate from Party C: 10⋅1 = 10

38. Can We Use Multiwinner Rules for Party-Lists?
60 votes:
Party A
30 votes:
Party B
10 votes:
Party C
Sequential PAV:
How adding different candidates improves total PAV score:
candidate from Party A: 𝟔𝟎⋅ ⅕ = 12.5
candidate from Party B: 30⋅ ⅓ = 10
candidate from Party C: 10⋅1 = 10

39. Can We Use Multiwinner Rules for Party-Lists?
60 votes:
Party A
30 votes:
Party B
10 votes:
Party C
Sequential PAV:
How adding different candidates improves total PAV score:
candidate from Party A: 𝟔𝟎⋅ ⅙ = 10
candidate from Party B: 3𝟎⋅ ⅓ = 10
candidate from Party C: 𝟏𝟎⋅𝟏 = 10

40. Can We Use Multiwinner Rules for Party-Lists?
60 votes:
Party A
30 votes:
Party B
10 votes:
Party C
Sequential PAV:
How adding different candidates improves total PAV score:
candidate from Party A: 60⋅ ⅟₇ = 8.5
candidate from Party B: 3𝟎⋅ ⅓ = 10
candidate from Party C: 𝟏𝟎⋅𝟏 = 10

41. Can We Use Multiwinner Rules for Party-Lists?
60 votes:
Party A
30 votes:
Party B
Sequential PAV when applied to party-list profiles gives proportional apportionment
10 votes:
Party C
Sequential PAV:
How adding different candidates improves total PAV score:
candidate from Party A: 60⋅ ⅟₇ = 8.5
candidate from Party B: 30⋅ ¼ = 7.5
candidate from Party C: 𝟏𝟎⋅𝟏 = 10

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

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

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

45. Relation between Multiwinner Rules and Apportionment Methods
PAV
Sequential PAV
D’Hondt method
Max-Phragmén
Variance minimizing Phragmén
Sainte-Laguë method
Monroe
Hamilton method
Greedy Monroe
[BLS17] M. Brill, J.-F. Laslier, P. Skowron, Multiwinner Approval Rules as Apportionment Methods, AAAI 2017.

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

47. 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
# votes 1 6 7 39 48
# votes 2
# votes 3
# votes 4
# votes 5
# votes 6
# votes 7

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
# votes 1 6 7 39 48
# votes 2 3
3.5
19.5 24
# votes 3 2
2.33 13 16
# votes 4
1.5
1.75
9.75 12
# votes 5
1.2
1.4
7.8
9.6
# votes 6 1
1.17
6.5
8.0
# votes 7
0.86
5.57
6.86

49. 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
# votes 1 6 7 39 48
# votes 2 3
3.5
19.5 24
# votes 3 2
2.33 13 16
# votes 4
1.5
1.75
9.75 12
# votes 5
1.2
1.4
7.8
9.6
# votes 6 1
1.17
6.5
8.0
# votes 7
0.86
5.57
6.86

50. 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
# votes 1 6 7 39 48
# votes 2 3
3.5
19.5 24
# votes 3 2
2.33 13 16
# votes 4
1.5
1.75
9.75 12
# votes 5
1.2
1.4
7.8
9.6
# votes 6 1
1.17
6.5
8.0
# votes 7
0.86
5.57
6.86

51. 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
# votes 1 6 7 39 48
# votes 2 3
3.5
19.5 24
# votes 3 2
2.33 13 16
# votes 4
1.5
1.75
9.75 12
# votes 5
1.2
1.4
7.8
9.6
# votes 6 1
1.17
6.5
8.0
# votes 7
0.86
5.57
6.86

52. 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
# votes 1 6 7 39 48
# votes 2 3
3.5
19.5 24
# votes 3 2
2.33 13 16
# votes 4
1.5
1.75
9.75 12
# votes 5
1.2
1.4
7.8
9.6
# votes 6 1
1.17
6.5
8.0
# votes 7
0.86
5.57
6.86

53. 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
# votes 1 6 7 39 48
# votes 2 3
3.5
19.5 24
# votes 3 2
2.33 13 16
# votes 4
1.5
1.75
9.75 12
# votes 5
1.2
1.4
7.8
9.6
# votes 6 1
1.17
6.5
8.0
# votes 7
0.86
5.57
6.86

54. 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
# votes 1 6 7 39 48
# votes 2 3
3.5
19.5 24
# votes 3 2
2.33 13 16
# votes 4
1.5
1.75
9.75 12
# votes 5
1.2
1.4
7.8
9.6
# votes 6 1
1.17
6.5
8.0
# votes 7
0.86
5.57
6.86

55. 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
# votes 1 6 7 39 48
# votes 2 3
3.5
19.5 24
# votes 3 2
2.33 13 16
# votes 4
1.5
1.75
9.75 12
# votes 5
1.2
1.4
7.8
9.6
# votes 6 1
1.17
6.5
8.0
# votes 7
0.86
5.57
6.86

56. 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
# votes 1 6 7 39 48
# votes 2 3
3.5
19.5 24
# votes 3 2
2.33 13 16
# votes 4
1.5
1.75
9.75 12
# votes 5
1.2
1.4
7.8
9.6
# votes 6 1
1.17
6.5
8.0
# votes 7
0.86
5.57
6.86

57. 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
# votes 1 6 7 39 48
# votes 2 3
3.5
19.5 24
# votes 3 2
2.33 13 16
# votes 4
1.5
1.75
9.75 12
# votes 5
1.2
1.4
7.8
9.6
# votes 6 1
1.17
6.5
8.0
# votes 7
0.86
5.57
6.86

58. 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
# votes 1 6 7 39 48
# votes 2 3
3.5
19.5 24
# votes 3 2
2.33 13 16
# votes 4
1.5
1.75
9.75 12
# votes 5
1.2
1.4
7.8
9.6
# votes 6 1
1.17
6.5
8.0
# votes 7
0.86
5.57
6.86
Party 1 gets 0 seats
Party 2 gets 0 seats
Party 3 gets 4 seats
Party 4 gets 6 seats

59. Relation between Multiwinner Rules and Apportionment Methods
PAV
Sequential PAV
D’Hondt method
Max-Phragmén
Variance minimizing Phragmén
Sainte-Laguë method
Monroe
Hamilton method
Greedy Monroe
[BLS17] M. Brill, J.-F. Laslier, P. Skowron, Multiwinner Approval Rules as Apportionment Methods, AAAI 2017.

60. 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
# votes 1 6 7 39 48
# votes 3 2
2.33 13 16
# votes 5
1.2
1.4
7.8
9.6
# votes 7
0.86 1
5.57
6.86
# votes 9
0.67
0.78
4.33
5.33
Party 1 gets 1 seat
Party 2 gets 1 seat
Party 3 gets 4 seats
Party 4 gets 4 seats

61. Relation between Multiwinner Rules and Apportionment Methods
PAV
Sequential PAV
D’Hondt method
Max-Phragmén
Variance minimizing Phragmén
Sainte-Laguë method
Monroe
Hamilton method
Greedy Monroe
[BLS17] M. Brill, J.-F. Laslier, P. Skowron, Multiwinner Approval Rules as Apportionment Methods, AAAI 2017.

62. Hamilton method of apportionment: Example
Party 1: 6 votes Party 2: 7 votes Party 3: 39 votes Party 4: 48 votes
First rounding down:
Party 1 gets ⌊6/10⌋ = 0 seats
Party 2 gets ⌊7/10⌋ = 0 seats
Party 3 gets ⌊39/10⌋ = 3 seats
Party 4 gets ⌊48/10⌋ = 4 seats
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.

63. 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.
[LS17] M. Lackner, P. Skowron, Consistent Approval-Based Multi-Winner Rules, Arxiv 2017.

64. 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.
[LS17] M. Lackner, P. Skowron, Consistent Approval-Based Multi-Winner Rules, Arxiv 2017.

65. 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.
[LS17] M. Lackner, P. Skowron, Consistent Approval-Based Multi-Winner Rules, Arxiv 2017.

66. Interlude: my view on new directions of research

67. PAV for 2-Euclidean Preferences
Computational Properties of PAV:
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, [SFL16] P. Skowron, P. Faliszewski, J. Lang, Finding a collective set of items: From proportional multirepresentation to group recommendation. Artif. Intell., 2016
Restricted domains: [Pet16] D. Peters, Single-Peakedness and Total Unimodularity: Efficiently Solve Voting Problems Without Even Trying, Arxiv, 2016.
Admits good approximation: [SFL16] P. Skowron, P. Faliszewski, J. Lang, Finding a collective set of items: From proportional multirepresentation to group recommendation. Artif. Intell., [BSS17] J. Byrka, P. Skowron, K. Sornat, Proportional Approval Voting, Harmonic k-median, and Negative Association, Arxiv, 2017.
FPT approximation schemes: [Sko16] P. Skowron, FPT Approximation Schemes for Maximizing Submodular Functions, WINE, 2016.
PAV for 2-Euclidean Preferences
uniform on a circle
uniform on a square
Gaussian
4 Gaussians

68. Ordinal Preferences and Proportionality

69. Too much focus on the top preferences.
SNTV
SNTV:
Select 𝑘 candidates with the highest plurality scores.
>
>
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V1:
>
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V2:
>
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V3:
Too much focus on the top preferences.
V4:
>
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>
For 𝑘=2:
V5:
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V6:
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V7:
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V8:
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>
>

70. SNTV for 2-Euclidean Preferences
uniform on a circle
uniform on a square
Gaussian
4 Gaussians
[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

71. Dummett’s Proportionality and STV
Assume there is a group of at least ℓ⋅𝑛/𝑘 voters who all have the same ℓ candidates on top. These ℓ candidates should be members of the winning committee.
>
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V1:
V2:
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A variant of Dummett’s proportionality [W94] (with quota ⌊𝑛/𝑘⌋ + 1 instead of 𝑛/𝑘) is satisfied by a variant of STV.
[W94] D. Woodall, Properties of preferential election rules, Voting Matters, 1994.
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V3:
V4:
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V5:
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V6:
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For 𝑘=4 the two candidates below must belong to the winning committee.
V7:
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V8:
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>

72. Recall the definition of STV
If there exists a candidate ranked first by at least 𝑛/𝑘 voters, take this candidate to the committee, remove this candidate from the election, and remove some of her 𝑛/𝑘 supporters (voters who rank her first).
Otherwise, remove the candidate with the lowest plurality score.
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V1:
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V2:
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V3:
V4:
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V5:
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V6:
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V7:
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V8:
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73. Recall the definition of STV
If there exists a candidate ranked first by at least 𝒏/𝒌 voters, take this candidate to the committee, remove this candidate from the election, and remove some of her 𝒏/𝒌 supporters (voters who rank her first).
Otherwise, remove the candidate with the lowest plurality score.
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V1:
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V2:
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V3:
V4:
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V5:
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V6:
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For 𝑘=2:
V7:
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V8:
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>

74. Recall the definition of STV
If there exists a candidate ranked first by at least 𝑛/𝑘 voters, take this candidate to the committee, remove this candidate from the election, and remove some of her 𝑛/𝑘 supporters (voters who rank her first).
Otherwise, remove the candidate with the lowest plurality score.
>
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V2:
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V3:
For 𝑘=2:
V7:
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V8:
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>
>

75. Recall the definition of STV
If there exists a candidate ranked first by at least 𝑛/𝑘 voters, take this candidate to the committee, remove this candidate from the election, and remove some of her 𝑛/𝑘 supporters (voters who rank her first).
Otherwise, remove the candidate with the lowest plurality score. x
>
V2:
>
> x
>
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>
V3:
For 𝑘=2: x
V7:
>
>
> x
V8:
>
>
>

76. Recall the definition of STV
If there exists a candidate ranked first by at least 𝑛/𝑘 voters, take this candidate to the committee, remove this candidate from the election, and remove some of her 𝑛/𝑘 supporters (voters who rank her first).
Otherwise, remove the candidate with the lowest plurality score. x
V2:
>
> x
>
>
V3:
For 𝑘=2: x
V7:
>
> x
V8:
>
>

77. Recall the definition of STV
If there exists a candidate ranked first by at least 𝑛/𝑘 voters, take this candidate to the committee, remove this candidate from the election, and remove some of her 𝑛/𝑘 supporters (voters who rank her first).
Otherwise, remove the candidate with the lowest plurality score. x
V2:
> x
>
V3:
For 𝑘=2: x
V7:
> x
V8:
>

78. Recall the definition of STV
If there exists a candidate ranked first by at least 𝑛/𝑘 voters, take this candidate to the committee, remove this candidate from the election, and remove some of her 𝑛/𝑘 supporters (voters who rank her first).
Otherwise, remove the candidate with the lowest plurality score.
V2:
V3:
For 𝑘=2:
V7:
V8:

79. STV for 2-Euclidean Preferences
uniform on a circle
uniform on a square
Gaussian
4 Gaussians
STV is often considered to be very well-suited for tasks that require proportional representation:
[TR00] N. Tideman, D. Richardson. Better voting methods through technology: The refinement-manageability trade-off in the Single Transferable Vote, Public Choice, 2000.
[EFSS17] 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

80. This is a very nonmonotonic rule!
Drawbacks of STV
This is a very nonmonotonic rule!
Consider 𝑘=1 and the following tie-breaking:
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The following profile:
10 votes:
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9 votes:
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1 vote:
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10 votes:
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81. This is a very nonmonotonic rule!
Drawbacks of STV
This is a very nonmonotonic rule!
Consider 𝑘=1 and the following tie-breaking:
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The following profile:
10 votes:
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9 votes:
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1 vote:
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10 votes:
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82. Solid Coalitions Property and the Monroe rule
If at least 𝑛/𝑘 voters rank some candidate on top, then this candidate should belong to the winning committee.
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V2:
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Solid coalitions property is satisfied by STV, SNTV, and by the greedy variant of the Monroe rule.
[EFSS17] E. Elkind, P. Faliszewski, P. Skowron, A. Slinko, Properties of Multiwinner Voting Rules, Social Choice and Welfare, 2017
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V4:
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For 𝑘=2 green lady must belong to the winning committee.
V7:
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V8:
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83. The Monroe rule Define the score for a committee: > > > >
Define the score for a committee:
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V2:
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Find the best assignment of voters to committee members so that:
Each committee member is assigned to roughly 𝒏/𝒌 voters.
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V3:
V4:
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V5:
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This assignment has score:
3+6⋅4+1=28
V6:
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V7:
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V8:
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84. The Monroe rule Define the score for a committee: > > > >
Define the score for a committee:
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V1:
V2:
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Find the best assignment of voters to committee members so that:
Each committee member is assigned to roughly 𝒏/𝒌 voters.
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V3:
V4:
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V5:
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This assignment has score:
3+6⋅4+1=28
V6:
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V7:
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V8:
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85. The Monroe rule Define the score for a committee: > > > >
Define the score for a committee:
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V1:
V2:
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Find the best assignment of voters to committee members so that:
Each committee member is assigned to roughly 𝒏/𝒌 voters.
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V3:
V4:
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V5:
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This would be a better assignment with score of 30.
V6:
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V7:
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But this assignment is unbalanced and so it is not valid!
V8:
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86. A committee with the best optimal valid assignment is winning.
The Monroe rule
Define the score for a committee:
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V1:
V2:
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Find the best assignment of voters to committee members so that:
Each committee member is assigned to roughly 𝒏/𝒌 voters.
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V3:
V4:
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V5:
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V6:
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A committee with the best optimal valid assignment is winning.
V7:
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V8:
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87. The Greedy Monroe rule For 𝑘 = 2: Repeat 𝑘 times:
Repeat 𝑘 times:
Find a group 𝐺 of 𝑛/𝑘 voters and a candidate 𝑐 such that the score of voters from 𝐺 from 𝑐 is maximal.
Remove candidate 𝑐 and the voters from 𝐺 from the election.
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V1:
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V2:
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V3:
V4:
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For 𝑘 = 2:
V5:
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V6:
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V7:
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V8:
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88. The Greedy Monroe rule For 𝑘 = 2: Repeat 𝑘 times:
Repeat 𝑘 times:
Find a group 𝐺 of 𝑛/𝑘 voters and a candidate 𝑐 such that the score of voters from 𝐺 from 𝑐 is maximal.
Remove candidate 𝑐 and the voters from 𝐺 from the election.
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V2:
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Greedy Monroe satisfies the Solid Coalitions property but the original Monroe’s rule does not.
Interesting example of a criterion where the ``approximate’’ variant of a rule is better than the original variant.
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V3:
V4:
For 𝑘 = 2:
V5:
V6:
V7:
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V8:
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89. How Good is the Greedy Monroe Rule?
Consider the situation right after the i-th iteration
i possibly unavailable positions
By pigeonhole principle, there is an unassigned candidate that n/k voters rank within the green area
(m-i)/(k-i) positions
v1:
vj:
vn:
in/k voters with assignment

90. How Good is the Greedy Monroe Rule?
Consider the situation right after the i-th iteration
i possibly unavailable positions
By pigeonhole principle, there is an unassigned candidate that n/k voters rank within the green area
(m-i)/(k-i) positions
v1:
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) )

91. How Good is the Greedy Monroe Rule?
Consider the situation right after the i-th iteration
i possibly unavailable positions
By pigeonhole principle, there is an unassigned candidate that n/k voters rank within the green area
(m-i)/(k-i) positions
v1:
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) – Hk/k fraction of maximum possible satisfaction!

92. 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 

93. Monroe for 2-Euclidean Preferences
Computational Properties of the Monroe rule:
NP-hard to compute: [PRZ08] 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: [BSU13] N. Betzler, A. Slinko, J. Uhlmann, On the Computation of Fully Proportional Representation. J. Artif. Intell. Res, 2013.
Admits relatively good approx. (i.e., Greedy Monroe): [SFS15] P. Skowron, P. Faliszewski, A. Slinko, Achieving fully proportional representation: Approximability results. Artif. Intell, 2015.
Monroe for 2-Euclidean Preferences
uniform on a circle
uniform on a square
Gaussian
4 Gaussians
Monroe:
Greedy 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

94. Further Topics

95. 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:
>
>
>
>
>
Committee of size 𝑘=1

96. 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:
>
>
>
>
>
Committee of size 𝑘=2

97. 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:
>
>
>
>
>
Committee monotonicity to some extend
contradicts proportionality:
Committee of size 𝑘=3
Example:
40 voters approve 𝑐 1 , 𝑐 2 , …, 𝑐 100 .
30 voters approve 𝑐 101 , 𝑐 102 , …, 𝑐 200 .
30 voters approve 𝑐 201 , 𝑐 202 , …, 𝑐 300 .
If we use a simple multiwinner approval voting, we will get ranking:
𝑐 1 ≻ 𝑐 2 ≻…≻ 𝑐 100 ≻ 𝑐 101 ≻…
left
right

98. 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:
How to approach the problem of finding a proportional ranking?
We can use insights from multiwinner elections to define formally the notion of proportionality of rankings,
There exists some committee monotonic proportional rules, which can be used in this case:
Sequential PAV
Sequential Phragmén’s Rule
Some greedy variants of proportional/diverse rules
>
>
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>
>
Recall Greedy CC:
Committee monotonicity to some extend
contradicts proportionality:
left
right

99. Interlude: my view on new directions of research

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

101. 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.

102. 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?

103. Thank You!