Fully Proportional Representation as Resource Allocation: - - PowerPoint PPT Presentation

▶

Nov 20, 2023 179 likes •832 views

Fully Proportional Representation as Resource Allocation: Approximability Results Piotr Skowron 1 , Piotr Faliszewski 2 , Arkadii Slinko 3 1 p.skowron@mimuw.edu.pl 2 faliszew@agh.edu.pl 3 a.slinko@auckland.ac.nz 1 Uniwersytet Warszawski 2 AGH 3

SLIDE 1

Fully Proportional Representation as Resource Allocation: Approximability Results

Piotr Skowron1, Piotr Faliszewski2, Arkadii Slinko3

1p.skowron@mimuw.edu.pl 2faliszew@agh.edu.pl 3a.slinko@auckland.ac.nz

1Uniwersytet Warszawski 2AGH 3University of Auckland

4 maja 2013

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 2

Problem

We want to find the representatives for the set of agents (we want to find the representatives for the society).

Agents (voters) Alternatives (candidates)

1 2 3 4 5 a1 a2 a3 a4 6

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 3

Problem

Agents have preferences over alternatives.

Agents (voters) Alternatives (candidates)

1 2 3 4 5 a1 a2 a3 a4

1 : a1 ≻ a3 ≻ a5 ≻ a2 ≻ a4 2 : a1 ≻ a2 ≻ a4 ≻ a3 ≻ a5 3 : a1 ≻ a3 ≻ a2 ≻ a4 ≻ a5 4 : a3 ≻ a1 ≻ a4 ≻ a2 ≻ a5 5 : a1 ≻ a3 ≻ a4 ≻ a5 ≻ a2

6 6 : a3 ≻ a5 ≻ a1 ≻ a2 ≻ a4

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 4

Problem

We want to select K candidates (in the example K = 2).

Agents (voters) Alternatives (candidates)

1 2 3 4 5 a1 a2 a3 a4

1 : a1 ≻ a3 ≻ a5 ≻ a2 ≻ a4 2 : a1 ≻ a2 ≻ a4 ≻ a3 ≻ a5 3 : a1 ≻ a3 ≻ a2 ≻ a4 ≻ a5 4 : a3 ≻ a1 ≻ a4 ≻ a2 ≻ a5 5 : a1 ≻ a3 ≻ a4 ≻ a5 ≻ a2

6 6 : a3 ≻ a5 ≻ a1 ≻ a2 ≻ a4

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 5

Problem

We want to select K candidates (in the example K = 2); and to assign each agent to exactly one representative.

Agents (voters) Alternatives (candidates)

1 2 3 4 5 a1 a2 a3 a4

1 : a1 ≻ a3 ≻ a5 ≻ a2 ≻ a4 2 : a1 ≻ a2 ≻ a4 ≻ a3 ≻ a5 3 : a1 ≻ a3 ≻ a2 ≻ a4 ≻ a5 4 : a3 ≻ a1 ≻ a4 ≻ a2 ≻ a5 5 : a1 ≻ a3 ≻ a4 ≻ a5 ≻ a2

6 6 : a3 ≻ a5 ≻ a1 ≻ a2 ≻ a4

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 6

Problem

Agents have certain satisfaction from the representatives (want to be represented by the candidates they prefer).

Agents (voters) Alternatives (candidates)

1 2 3 4 5 a1 a2 a3 a4

1 : a1 ≻ a3 a3 a3 ≻ a5 ≻ a2 ≻ a4 2 : a1 a1 a1 ≻ a2 ≻ a4 ≻ a3 ≻ a5 3 : a1 a1 a1 ≻ a3 ≻ a2 ≻ a4 ≻ a5 4 : a3 ≻ a1 a1 a1 ≻ a4 ≻ a2 ≻ a5 5 : a1 ≻ a3 a3 a3 ≻ a4 ≻ a5 ≻ a2

6 6 : a3 a3 a3 ≻ a5 ≻ a1 ≻ a2 ≻ a4

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 7

How to measure the satisfaction of the single agent

We can use positional scoring function: α1, α2, . . . αm

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 8

How to measure the satisfaction of the single agent

We can use positional scoring function: α1, α2, . . . αm αi means that the satisfaction of the agent v from the candidate that he/she puts in his/her i-th position is αi.

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 9

How to measure the satisfaction of the single agent

We can use positional scoring function: α1, α2, . . . αm αi means that the satisfaction of the agent v from the candidate that he/she puts in his/her i-th position is αi. A popular positional scoring function is the Borda score: m − 1, m − 2, . . . 0

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 10

How to measure the satisfaction of the single agent – example

1 :a1 ≻ a3 a3 a3 ≻ a5 ≻ a2 ≻ a4 2 :a1 a1 a1 ≻ a2 ≻ a4 ≻ a3 ≻ a5 3 :a1 a1 a1 ≻ a3 ≻ a2 ≻ a4 ≻ a5 4 :a3 ≻ a1 a1 a1 ≻ a4 ≻ a2 ≻ a5 5 :a1 ≻ a3 a3 a3 ≻ a4 ≻ a5 ≻ a2 6 :a3 a3 a3 ≻ a5 ≻ a1 ≻ a2 ≻ a4

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 11

How to measure the satisfaction of the single agent – example

1 :a1 ≻ a3 a3 a3 ≻ a5 ≻ a2 ≻ a4 2 :a1 a1 a1 ≻ a2 ≻ a4 ≻ a3 ≻ a5 3 :a1 a1 a1 ≻ a3 ≻ a2 ≻ a4 ≻ a5 4 :a3 ≻ a1 a1 a1 ≻ a4 ≻ a2 ≻ a5 5 :a1 ≻ a3 a3 a3 ≻ a4 ≻ a5 ≻ a2 6 :a3 a3 a3 ≻ a5 ≻ a1 ≻ a2 ≻ a4 For the Borda score (4, 3, 2, 1, 0): agents 2, 3 and 6 have satisfaction 4, and agents 1, 4 and 5 have satisfaction 3.

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 12

How to aggregate agents’ satisfaction

Utilitarian approach — the satisfaction of the agents is the sum of the satisfaction of the individual agents. Egalitarian approach — the satisfaction of the agents is the satisfaction of the least satisfied agent.

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 13

How to aggregate agents’ satisfaction

Example: 1 :a1 ≻ a3 a3 a3 ≻ a5 ≻ a2 ≻ a4 2 :a1 a1 a1 ≻ a2 ≻ a4 ≻ a3 ≻ a5 3 :a1 a1 a1 ≻ a3 ≻ a2 ≻ a4 ≻ a5 4 :a3 ≻ a1 a1 a1 ≻ a4 ≻ a2 ≻ a5 5 :a1 ≻ a3 a3 a3 ≻ a4 ≻ a5 ≻ a2 6 :a3 a3 a3 ≻ a5 ≻ a1 ≻ a2 ≻ a4

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 14

How to aggregate agents’ satisfaction

Example: 1 :a1 ≻ a3 a3 a3 ≻ a5 ≻ a2 ≻ a4 2 :a1 a1 a1 ≻ a2 ≻ a4 ≻ a3 ≻ a5 3 :a1 a1 a1 ≻ a3 ≻ a2 ≻ a4 ≻ a5 4 :a3 ≻ a1 a1 a1 ≻ a4 ≻ a2 ≻ a5 5 :a1 ≻ a3 a3 a3 ≻ a4 ≻ a5 ≻ a2 6 :a3 a3 a3 ≻ a5 ≻ a1 ≻ a2 ≻ a4 Utilitarian approach: 3 + 4 + 4 + 3 + 3 + 4 = 21.

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 15

How to aggregate agents’ satisfaction

Example: 1 :a1 ≻ a3 a3 a3 ≻ a5 ≻ a2 ≻ a4 2 :a1 a1 a1 ≻ a2 ≻ a4 ≻ a3 ≻ a5 3 :a1 a1 a1 ≻ a3 ≻ a2 ≻ a4 ≻ a5 4 :a3 ≻ a1 a1 a1 ≻ a4 ≻ a2 ≻ a5 5 :a1 ≻ a3 a3 a3 ≻ a4 ≻ a5 ≻ a2 6 :a3 a3 a3 ≻ a5 ≻ a1 ≻ a2 ≻ a4 Utilitarian approach: 3 + 4 + 4 + 3 + 3 + 4 = 21. Egalitarian approach: 3.

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 16

Chamberlin-Courant’s and Monroe’s systems

Chamberlin-Courant’s rule In the Chamberlin-Courant’s system we have: The set of the agents N = {1, 2, . . . n}. The set of the alternatives A = {a1, a2, . . . , am}. The preference profile — the orderings of all agent. We look for such a subset of alternatives W (winners) and such an assignment of the agents to the alternatives from W that: W = K. The satisfaction of the agents is maximized.

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 17

Chamberlin-Courant’s and Monroe’s systems

Chamberlin-Courant’s rule In the Chamberlin-Courant’s system we have: The set of the agents N = {1, 2, . . . n}. The set of the alternatives A = {a1, a2, . . . , am}. The preference profile — the orderings of all agent. We look for such a subset of alternatives W (winners) and such an assignment of the agents to the alternatives from W that: W = K. The satisfaction of the agents is maximized. Monroe’s system In the Monroe’s system we additionally require that every alternative is assigned to exactly the same number of the agents (with the possible difference equal to 1 if K does not divide the number of the agents n).

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 18

Chamberlin-Courant’s and Monroe’s systems – example

1 :a1 a1 a1 ≻ a3 ≻ a5 ≻ a2 ≻ a4 2 :a1 a1 a1 ≻ a2 ≻ a4 ≻ a3 ≻ a5 3 :a1 a1 a1 ≻ a3 ≻ a2 ≻ a4 ≻ a5 4 :a3 a3 a3 ≻ a1 ≻ a4 ≻ a2 ≻ a5 5 :a1 a1 a1 ≻ a3 ≻ a4 ≻ a5 ≻ a2 6 :a3 a3 a3 ≻ a5 ≻ a1 ≻ a2 ≻ a4 In the Chamberlin-Courant’s system the winners are a1 i a3 (maximizing the satisfaction of the agents, equal to 4 · 6 = 24).

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 19

Chamberlin-Courant’s and Monroe’s systems – example

1 :a1 a1 a1 ≻ a3 ≻ a5 ≻ a2 ≻ a4 2 :a1 a1 a1 ≻ a2 ≻ a4 ≻ a3 ≻ a5 3 :a1 a1 a1 ≻ a3 ≻ a2 ≻ a4 ≻ a5 4 :a3 a3 a3 ≻ a1 ≻ a4 ≻ a2 ≻ a5 5 :a1 ≻ a3 a3 a3 ≻ a4 ≻ a5 ≻ a2 6 :a3 a3 a3 ≻ a5 ≻ a1 ≻ a2 ≻ a4 In the Monroe’s system the winners are also a1 i a3, but now every winner must be assigned to 3 agents; thus, we get the satisfaction equal to 23.

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 20

Winner determination in both systems is difficult

The problems of winner determination in both systems are: NP-hard. Hard in terms of parametrized complexity theory.

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 21

Winner determination in both systems is difficult

The problems of winner determination in both systems are: NP-hard. Hard in terms of parametrized complexity theory. Still we have hope There are the approximation algorithms for both problems.

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 22

Winner determination in both systems is difficult

The problems of winner determination in both systems are: NP-hard. Hard in terms of parametrized complexity theory. Still we have hope There are the approximation algorithms for both problems. Our results We have shown the (1 − 1/e)-approximation algorithm for the Monroe’s system for arbitrary positional scoring function.

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 23

Winner determination in both systems is difficult

The problems of winner determination in both systems are: NP-hard. Hard in terms of parametrized complexity theory. Still we have hope There are the approximation algorithms for both problems. Our results We have shown the (1 − 1/e)-approximation algorithm for the Monroe’s system for arbitrary positional scoring function. Our results We asked, how all these approximation algorithms work in practice.

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 24

Winner determination in both systems is difficult

The problems of winner determination in both systems are: NP-hard. Hard in terms of parametrized complexity theory. Still we have hope There are the approximation algorithms for both problems. Our results We have shown the (1 − 1/e)-approximation algorithm for the Monroe’s system for arbitrary positional scoring function. Our results We asked, how all these approximation algorithms work in practice. Our results We asked, whether we can modify the algorithms to get better quality in practice, but still keeping the approximation guarantees.

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 25

The approximation algorithm for the Monroe’s system

The simple greedy algorithm:

1 Select the alternative a and the set of n

K voters so that the

total utility of these n

K voters from a would be the best.

2 Remove these n

K voters from the further consideration.

3 Repeat steps 1 and 2, until we find K winners. Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 26

The approximation algorithm for the Monroe’s system

Step 0 = ⇒ W = {}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 27

The approximation algorithm for the Monroe’s system

Step 0 = ⇒ W = {} Step 1 = ⇒ W = {w1}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 28

The approximation algorithm for the Monroe’s system

Step 0 = ⇒ W = {} Step 1 = ⇒ W = {w1} Step 2 = ⇒ W = {w1, w2}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 29

The approximation algorithm for the Monroe’s system

Step 0 = ⇒ W = {} Step 1 = ⇒ W = {w1} Step 2 = ⇒ W = {w1, w2} . . .

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 30

The approximation algorithm for the Monroe’s system

Step 0 = ⇒ W = {} Step 1 = ⇒ W = {w1} Step 2 = ⇒ W = {w1, w2} . . . Step K = ⇒ W = {w1, w2, . . . , wk}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 31

The approximation algorithm for the Monroe’s system –

ur modification

d – the parameter of the algorithm (example d = 3) Step 0 = ⇒ {}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 32

The approximation algorithm for the Monroe’s system –

ur modification

d – the parameter of the algorithm (example d = 3) Step 0 = ⇒ {} Step 1a = ⇒ {a1}, {a2}, {a3}, {a4}, {a5} . . . {am}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 33

The approximation algorithm for the Monroe’s system –

ur modification

d – the parameter of the algorithm (example d = 3) Step 0 = ⇒ {} Step 1a = ⇒ {a1}, {a2}, {a3}, {a4}, {a5} . . . {am} Step 1b = ⇒ {a1}, {a2}, {a3}, {a4}, {a4} . . . {am}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 34

The approximation algorithm for the Monroe’s system –

ur modification

d – the parameter of the algorithm (example d = 3) Step 0 = ⇒ {} Step 1a = ⇒ {a1}, {a2}, {a3}, {a4}, {a5} . . . {am} Step 1b = ⇒ {a1}, {a2}, {a3}, {a4}, {a4} . . . {am} Step 1c = ⇒ {a1}, {a3}, {a4}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 35

The approximation algorithm for the Monroe’s system –

ur modification

d – the parameter of the algorithm (example d = 3) Step 0 = ⇒ {} Step 1a = ⇒ {a1}, {a2}, {a3}, {a4}, {a5} . . . {am} Step 1b = ⇒ {a1}, {a2}, {a3}, {a4}, {a4} . . . {am} Step 1c = ⇒ {a1}, {a3}, {a4} Step 2a = ⇒ {a1, a2}, {a1, a3}, {a1, a4}, {a1, a5} . . . , {a1, am} {a3, a1}, {a3, a2}, {a3, a4}, {a3, a5} . . . {a3, am} {a4, a1}, {a4, a2}, {a4, a3}, {a4, a5} . . . {a4, am}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 36

The approximation algorithm for the Monroe’s system –

ur modification

d – the parameter of the algorithm (example d = 3) Step 0 = ⇒ {} Step 1a = ⇒ {a1}, {a2}, {a3}, {a4}, {a5} . . . {am} Step 1b = ⇒ {a1}, {a2}, {a3}, {a4}, {a4} . . . {am} Step 1c = ⇒ {a1}, {a3}, {a4} Step 2a = ⇒ {a1, a2}, {a1, a3}, {a1, a4}, {a1, a5} . . . , {a1, am} {a3, a1}, {a3, a2}, {a3, a4}, {a3, a5} . . . {a3, am} {a4, a1}, {a4, a2}, {a4, a3}, {a4, a5} . . . {a4, am} Step 2b = ⇒ {a1, a2}, {a1, a3}, {a1, a4}, {a1, a5} . . . {a1, am} {a3, a1}, {a3, a2}, {a3, a4}, {a3, a5} . . . {a3, am} {a4, a1}, {a4, a2}, {a4, a3}, {a4, a5} . . . {a4, am}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 37

The approximation algorithm for the Monroe’s system –

ur modification

d – the parameter of the algorithm (example d = 3) Step 0 = ⇒ {} Step 1a = ⇒ {a1}, {a2}, {a3}, {a4}, {a5} . . . {am} Step 1b = ⇒ {a1}, {a2}, {a3}, {a4}, {a4} . . . {am} Step 1c = ⇒ {a1}, {a3}, {a4} Step 2a = ⇒ {a1, a2}, {a1, a3}, {a1, a4}, {a1, a5} . . . , {a1, am} {a3, a1}, {a3, a2}, {a3, a4}, {a3, a5} . . . {a3, am} {a4, a1}, {a4, a2}, {a4, a3}, {a4, a5} . . . {a4, am} Step 2b = ⇒ {a1, a2}, {a1, a3}, {a1, a4}, {a1, a5} . . . {a1, am} {a3, a1}, {a3, a2}, {a3, a4}, {a3, a5} . . . {a3, am} {a4, a1}, {a4, a2}, {a4, a3}, {a4, a5} . . . {a4, am} Step 2c = ⇒ {a1, a3}, {a3, a2}, {a4, a5}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 38

The approximation algorithm for the Monroe’s system –

ur modification

d – the parameter of the algorithm (example d = 3) Step 2b = ⇒ {a1, a3}, {a3, a2}, {a4, a5}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 39

The approximation algorithm for the Monroe’s system –

ur modification

d – the parameter of the algorithm (example d = 3) Step 2b = ⇒ {a1, a3}, {a3, a2}, {a4, a5} Step 3a = ⇒ {a1, a3, a2}, {a1, a3, a4}, {a1, a3, a5} . . . {a1, a3, am} {a3, a2, a1}, {a3, a2, a4}, {a3, a2, a5} . . . {a3, a2, am} {a4, a5, a1}, {a4, a5, a2}, {a4, a5, a3} . . . {a4, a5, am}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 40

The approximation algorithm for the Monroe’s system –

ur modification

d – the parameter of the algorithm (example d = 3) Step 2b = ⇒ {a1, a3}, {a3, a2}, {a4, a5} Step 3a = ⇒ {a1, a3, a2}, {a1, a3, a4}, {a1, a3, a5} . . . {a1, a3, am} {a3, a2, a1}, {a3, a2, a4}, {a3, a2, a5} . . . {a3, a2, am} {a4, a5, a1}, {a4, a5, a2}, {a4, a5, a3} . . . {a4, a5, am} Step 3b = ⇒ {a1, a3, a2}, {a1, a3, a4}, {a1, a3, a5} . . . {a1, a3, am} {a3, a2, a1}, {a3, a2, a4}, {a3, a2, a5} . . . {a3, a2, am} {a4, a5, a1}, {a4, a5, a2}, {a4, a5, a3} . . . {a4, a5, am}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 41

The approximation algorithm for the Monroe’s system –

ur modification

d – the parameter of the algorithm (example d = 3) Step 2b = ⇒ {a1, a3}, {a3, a2}, {a4, a5} Step 3a = ⇒ {a1, a3, a2}, {a1, a3, a4}, {a1, a3, a5} . . . {a1, a3, am} {a3, a2, a1}, {a3, a2, a4}, {a3, a2, a5} . . . {a3, a2, am} {a4, a5, a1}, {a4, a5, a2}, {a4, a5, a3} . . . {a4, a5, am} Step 3b = ⇒ {a1, a3, a2}, {a1, a3, a4}, {a1, a3, a5} . . . {a1, a3, am} {a3, a2, a1}, {a3, a2, a4}, {a3, a2, a5} . . . {a3, a2, am} {a4, a5, a1}, {a4, a5, a2}, {a4, a5, a3} . . . {a4, a5, am} Step 3c = ⇒ {a1, a3, a4}, {a1, a3, am}, {a4, a5, a2}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 42

The approximation algorithm for the Monroe’s system –

ur modification

d – the parameter of the algorithm (example d = 3) Step 2b = ⇒ {a1, a3}, {a3, a2}, {a4, a5} Step 3a = ⇒ {a1, a3, a2}, {a1, a3, a4}, {a1, a3, a5} . . . {a1, a3, am} {a3, a2, a1}, {a3, a2, a4}, {a3, a2, a5} . . . {a3, a2, am} {a4, a5, a1}, {a4, a5, a2}, {a4, a5, a3} . . . {a4, a5, am} Step 3b = ⇒ {a1, a3, a2}, {a1, a3, a4}, {a1, a3, a5} . . . {a1, a3, am} {a3, a2, a1}, {a3, a2, a4}, {a3, a2, a5} . . . {a3, a2, am} {a4, a5, a1}, {a4, a5, a2}, {a4, a5, a3} . . . {a4, a5, am} Step 3c = ⇒ {a1, a3, a4}, {a1, a3, am}, {a4, a5, a2} . . .

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 43

The approximation algorithm for the Monroe’s system –

ur modification

d – the parameter of the algorithm (example d = 3) Step 2b = ⇒ {a1, a3}, {a3, a2}, {a4, a5} Step 3a = ⇒ {a1, a3, a2}, {a1, a3, a4}, {a1, a3, a5} . . . {a1, a3, am} {a3, a2, a1}, {a3, a2, a4}, {a3, a2, a5} . . . {a3, a2, am} {a4, a5, a1}, {a4, a5, a2}, {a4, a5, a3} . . . {a4, a5, am} Step 3b = ⇒ {a1, a3, a2}, {a1, a3, a4}, {a1, a3, a5} . . . {a1, a3, am} {a3, a2, a1}, {a3, a2, a4}, {a3, a2, a5} . . . {a3, a2, am} {a4, a5, a1}, {a4, a5, a2}, {a4, a5, a3} . . . {a4, a5, am} Step 3c = ⇒ {a1, a3, a4}, {a1, a3, am}, {a4, a5, a2} . . . Step K = ⇒ {ai1,1, ai1,2, . . . , ai1,K}, {ai2,1, ai2,2, . . . , ai2,K}, {ai3,1, ai3,2, . . . , ai3,K}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 44

The approximation algorithm for the Monroe’s system –

ur modification

d – the parameter of the algorithm (example d = 3) Step 2b = ⇒ {a1, a3}, {a3, a2}, {a4, a5} Step 3a = ⇒ {a1, a3, a2}, {a1, a3, a4}, {a1, a3, a5} . . . {a1, a3, am} {a3, a2, a1}, {a3, a2, a4}, {a3, a2, a5} . . . {a3, a2, am} {a4, a5, a1}, {a4, a5, a2}, {a4, a5, a3} . . . {a4, a5, am} Step 3b = ⇒ {a1, a3, a2}, {a1, a3, a4}, {a1, a3, a5} . . . {a1, a3, am} {a3, a2, a1}, {a3, a2, a4}, {a3, a2, a5} . . . {a3, a2, am} {a4, a5, a1}, {a4, a5, a2}, {a4, a5, a3} . . . {a4, a5, am} Step 3c = ⇒ {a1, a3, a4}, {a1, a3, am}, {a4, a5, a2} . . . Step K = ⇒ {ai1,1, ai1,2, . . . , ai1,K}, {ai2,1, ai2,2, . . . , ai2,K}, {ai3,1, ai3,2, . . . , ai3,K}

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 45

Experiments

We tested the algorithms on different real datasets describing preferences:

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 46

Experiments

We tested the algorithms on different real datasets describing preferences:

1 Sushi preferneces. Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 47

Experiments

We tested the algorithms on different real datasets describing preferences:

1 Sushi preferneces. 2 Movie preferences. Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 48

Experiments

We tested the algorithms on different real datasets describing preferences:

1 Sushi preferneces. 2 Movie preferences. 3 Course preferences. Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 49

Experiments

We tested the algorithms on different real datasets describing preferences:

1 Sushi preferneces. 2 Movie preferences. 3 Course preferences. 4 Figure skating judges preferences. Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 50

Experiments

We tested the algorithms on different real datasets describing preferences:

1 Sushi preferneces. 2 Movie preferences. 3 Course preferences. 4 Figure skating judges preferences.

... and on syntetic data:

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 51

Experiments

We tested the algorithms on different real datasets describing preferences:

1 Sushi preferneces. 2 Movie preferences. 3 Course preferences. 4 Figure skating judges preferences.

... and on syntetic data:

1 Impartial Culture. Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 52

Experiments

We tested the algorithms on different real datasets describing preferences:

1 Sushi preferneces. 2 Movie preferences. 3 Course preferences. 4 Figure skating judges preferences.

... and on syntetic data:

1 Impartial Culture. 2 Polya-Eggenberger urn model. Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 53

Experiments

We tested the algorithms on different real datasets describing preferences:

1 Sushi preferneces. 2 Movie preferences. 3 Course preferences. 4 Figure skating judges preferences.

... and on syntetic data:

1 Impartial Culture. 2 Polya-Eggenberger urn model. 3 Generalized Mallow’s model. Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 54

Experiments – results

The quality of the algorithms is very high.

Monroe CC A B C GM R C GM P R S1 0.94 0.94 0.94 0.99 ≈ 1.0 0.99 0.99 1.0 ≈ 1.0 0.99 0.99 S2 0.95 0.95 0.95 0.99 1.0 ≈ 1.0 0.99 1.0 ≈ 1.0 0.98 0.99 Mv 0.96 0.96 0.96 ≈ 1.0 1.0 ≈ 1.0 0.98 1.0 ≈ 1.0 0.96 ≈ 1.0 Cr 0.98 0.98 0.98 0.99 1.0 ≈ 1.0 0.99 1.0 ≈ 1.0 1.0 ≈ 1.0 Sk 0.99 0.99 0.99 ≈ 1.0 1.0 ≈ 1.0 0.94 1.0 ≈ 1.0 0.85 0.99 IC 0.94 0.94 0.94 0.99 ≈ 1.0 0.99 0.99 1.0 ≈ 1.0 0.99 0.99 Ml 0.94 0.94 0.94 0.99 1.0 0.99 0.99 1.0 ≈ 1.0 0.99 0.99 Ur 0.95 0.95 0.95 0.99 ≈ 1.0 0.99 0.99 1.0 0.99 0.97 0.99

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 55

Experiments – results

The quality of the algorithms is very high.

Monroe CC A B C GM R C GM P R S1 0.94 0.99 ≈ 1.0 0.99 0.99 1.0 ≈ 1.0 0.99 0.99 0.99 0.99 S2 0.95 0.99 1.0 ≈ 1.0 0.99 1.0 ≈ 1.0 0.98 0.98 0.98 0.99 Mv 0.96 ≈ 1.0 1.0 ≈ 1.0 0.98 1.0 ≈ 1.0 0.96 0.96 0.96 ≈ 1.0 Cr 0.98 0.99 1.0 ≈ 1.0 0.99 1.0 ≈ 1.0 1.0 1.0 1.0 ≈ 1.0 Sk 0.99 ≈ 1.0 1.0 ≈ 1.0 0.94 1.0 ≈ 1.0 0.85 0.85 0.85 0.99 IC 0.94 0.99 ≈ 1.0 0.99 0.99 1.0 ≈ 1.0 0.99 0.99 0.99 0.99 Ml 0.94 0.99 1.0 0.99 0.99 1.0 ≈ 1.0 0.99 0.99 0.99 0.99 Ur 0.95 0.99 ≈ 1.0 0.99 0.99 1.0 0.99 0.97 0.97 0.97 0.99

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 56

Experiments – results

The quality of the algorithms is very high.

Monroe CC A B C GM R C GM P R S1 0.94 0.99 ≈ 1.0 ≈ 1.0 ≈ 1.0 0.99 0.99 1.0 ≈ 1.0 0.99 0.99 S2 0.95 0.99 1.0 1.0 1.0 ≈ 1.0 0.99 1.0 ≈ 1.0 0.98 0.99 Mv 0.96 ≈ 1.0 1.0 1.0 1.0 ≈ 1.0 0.98 1.0 ≈ 1.0 0.96 ≈ 1.0 Cr 0.98 0.99 1.0 1.0 1.0 ≈ 1.0 0.99 1.0 ≈ 1.0 1.0 ≈ 1.0 Sk 0.99 ≈ 1.0 1.0 1.0 1.0 ≈ 1.0 0.94 1.0 ≈ 1.0 0.85 0.99 IC 0.94 0.99 ≈ 1.0 ≈ 1.0 ≈ 1.0 0.99 0.99 1.0 ≈ 1.0 0.99 0.99 Ml 0.94 0.99 1.0 1.0 1.0 0.99 0.99 1.0 ≈ 1.0 0.99 0.99 Ur 0.95 0.99 ≈ 1.0 ≈ 1.0 ≈ 1.0 0.99 0.99 1.0 0.99 0.97 0.99

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 57

Experiments – results

The quality of the algorithms is very high.

Monroe CC A B C GM R C GM P R S1 0.94 0.99 ≈ 1.0 0.99 0.99 1.0 1.0 1.0 ≈ 1.0 0.99 0.99 S2 0.95 0.99 1.0 ≈ 1.0 0.99 1.0 1.0 1.0 ≈ 1.0 0.98 0.99 Mv 0.96 ≈ 1.0 1.0 ≈ 1.0 0.98 1.0 1.0 1.0 ≈ 1.0 0.96 ≈ 1.0 Cr 0.98 0.99 1.0 ≈ 1.0 0.99 1.0 1.0 1.0 ≈ 1.0 1.0 ≈ 1.0 Sk 0.99 ≈ 1.0 1.0 ≈ 1.0 0.94 1.0 1.0 1.0 ≈ 1.0 0.85 0.99 IC 0.94 0.99 ≈ 1.0 0.99 0.99 1.0 1.0 1.0 ≈ 1.0 0.99 0.99 Ml 0.94 0.99 1.0 0.99 0.99 1.0 1.0 1.0 ≈ 1.0 0.99 0.99 Ur 0.95 0.99 ≈ 1.0 0.99 0.99 1.0 1.0 1.0 0.99 0.97 0.99

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 58

Experiments – results

The relation between the quality of the algorithms and the number

f the alternatives m – Chamberlin-Courant.

0.95 0.96 0.97 0.98 0.99 1 quality of the algorithm (C/Csim) 50 100 150 200 250 300 number of alternatives m Algorithm C for MV Algorithm R for MV Algorithm C for UR Algorithm R for UR

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 59

Experiments – results

The relation between the quality of the algorithms and the number

f the alternatives m – Monroe.

0.7 0.75 0.8 0.85 0.9 0.95 1 quality of the algorithm (C/Csim) 50 100 150 200 250 300 number of alternatives m Algorithm C for UR Algorithm A for UR Algorithm R for UR Algorithm C for MV Algorithm A for MV Algorithm R for MV

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 60

Experiments – results

The relation between the quality of the algorithms and the ratio k

m

– Chamberlin-Courant.

0.95 0.96 0.97 0.98 0.99 1 quality of the algorithm (C/Csim) 10 20 30 40 50 ratio K/m [%] Algorithm C for MV Algorithm C for UR Algorithm R for MV Algorithm R for UR

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 61

Experiments – results

The relation between the quality of the algorithms and the ratio k

m

– Monroe.

0.7 0.75 0.8 0.85 0.9 0.95 1 quality of the algorithm (C/Csim) 10 20 30 40 50 ratio K/m [%] Algorithm C for UR Algorithm A for UR Algorithm R for UR Algorithm C for MV Algorithm A for MV Algorithm R for MV

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 62

Experiments – results

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 63

Experiments – results

Main conclusion Finding the proportional representation is simple in practice!

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation

SLIDE 64

Questions? Also, feel free to send any questions to: p.skowron@mimuw.edu.pl.

Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation