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

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

1st June 2013

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

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

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

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

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

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

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 Alternatively, we can think of dissatisfaction with the Borda rule of the form: 0, 1, . . . m − 1

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

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. For the dissatisfaction Borda score (0, 1, 2, 3, 4): agents 2, 3 i 6 have dissatisfaction 0, nd agents 1, 4 and 5 have dissatisfaction 1.

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

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

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. Analogously for dissatisfaction.

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

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

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

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

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

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

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. How about the approximation algorithms? Lu and Boutilier (2011) have shown the (1 − 1/e)-approximation algorithm for the Chamberlin-Courants system.

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

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. How about the approximation algorithms? Lu and Boutilier (2011) have shown the (1 − 1/e)-approximation algorithm for the Chamberlin-Courants system. Our question Can we get a better approximation for the Borda scoring function?

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

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. How about the approximation algorithms? Lu and Boutilier (2011) have shown the (1 − 1/e)-approximation algorithm for the Chamberlin-Courants system. Our question Can we get a better approximation for the Borda scoring function? Yes, we can!

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

Approximating satisfaction or dissatisfaction?

1 2-approximation algorithm for satisfaction.

1 :a1 ≻ a2 ≻ a3 ≻ a4 ≻ a5 a5 a5 ≻ a6 ≻ a7 ≻ a8 ≻ a9 2 :a1 ≻ a2 ≻ a3 ≻ a4 ≻ a5 a5 a5 ≻ a6 ≻ a7 ≻ a8 ≻ a9 3 :a1 ≻ a2 ≻ a3 ≻ a4 ≻ a5 a5 a5 ≻ a6 ≻ a7 ≻ a8 ≻ a9 4 :a1 ≻ a2 ≻ a3 ≻ a4 ≻ a5 a5 a5 ≻ a6 ≻ a7 ≻ a8 ≻ a9 5 :a1 ≻ a2 ≻ a3 ≻ a4 ≻ a5 a5 a5 ≻ a6 ≻ a7 ≻ a8 ≻ a9 6 :a1 ≻ a2 ≻ a3 ≻ a4 ≻ a5 a5 a5 ≻ a6 ≻ a7 ≻ a8 ≻ a9

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

Approximating satisfaction or dissatisfaction?

2-approximation algorithm for dissatisfaction. 1 :a1 a1 a1 ≻ a2 ≻ a3 ≻ a4 ≻ a5 ≻ a6 ≻ a7 ≻ a8 ≻ a9 2 :a1 a1 a1 ≻ a2 ≻ a3 ≻ a4 ≻ a5 ≻ a6 ≻ a7 ≻ a8 ≻ a9 3 :a1 a1 a1 ≻ a2 ≻ a3 ≻ a4 ≻ a5 ≻ a6 ≻ a7 ≻ a8 ≻ a9 4 :a1 a1 a1 ≻ a2 ≻ a3 ≻ a4 ≻ a5 ≻ a6 ≻ a7 ≻ a8 ≻ a9 5 :a1 a1 a1 ≻ a2 ≻ a3 ≻ a4 ≻ a5 ≻ a6 ≻ a7 ≻ a8 ≻ a9 6 :a1 a1 a1 ≻ a2 ≻ a3 ≻ a4 ≻ a5 ≻ a6 ≻ a7 ≻ a8 ≻ a9

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

Approximating satisfaction or dissatisfaction?

E.g., for parliamentary elections in Poland: m = 6000 alternatives, K = 460 winners. Assume that in the optimal solution every voter is on average represented by its 2nd top candidate. 2 2 2-approximation of dissatisfaction every voter is on average represented by its 3rd top candidate.

1 2 1 2 1 2-approximation of satisfaction every voter is on average

represented by its 3000th top candidate.

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

Results overview

Monroe/CC Rules

Misrepresentation (dissatisfaction)

(0, 1, 2, 3, …, m-1 )

Representation (satisfaction)

(m-1, m-2, …, 0 )

Utilitarian

(minimize the sum

• f dissatisfactions)

Egalitarian

(minimize the worst dissatisfaction)

Utilitarian Egalitarian Misrepresentation (dissatisfaction)

(0, 1, 2, 3, …, m-1 )

Representation (satisfaction)

(m-1, m-2, …, 0 )

Utilitarian

(minimize the sum

• f dissatisfactions)

Egalitarian

(minimize the worst dissatisfaction)

Egalitarian

• n

(

(m

• n

(

(m

Hard to approximate if P ≠ NP Utilitarian Good approximation algorithms!

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

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

The approximation algorithm for the Monroe’s system

Consider the satisfaction of the agents assigned in the (i + 1)-th turn.

1: 2: 3: 4: 5: 6: 7: 8:

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

The approximation algorithm for the Monroe’s system

After the i-th turn i n

K agents have already assigned

representatives.

1: 2: 3: 4: 5: 6: 7: 8:

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

The approximation algorithm for the Monroe’s system

After the i-th turn we also used i alternatives.

1: 2: 3: 4: 5: 6: 7: 8:

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

The approximation algorithm for the Monroe’s system

Among i + ⌈m−i

K−i ⌉ top positions there is at least ⌈m−i K−i ⌉ occurrences

• f the unused alternatives.

1: 2: 3: 4: 5: 6: 7: 8:

✲ ✛ ✲ ✛

i ⌈m−i

K−i ⌉

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

The approximation algorithm for the Monroe’s system

Pigeonhole principle: some unused alternative is at at least n

K

positions better than i + ⌈m−i

K−i ⌉ among yet-unassigned agents:

• n − i n

K

• · m − i

K − i = n K · (m − i) .

1: 2: 3: 4: 5: 6: 7: 8:

a1 a1 a1 a2 a3 a4 a5 a6 a2 a1 a1 a1 a7 a5 a4 a3 a1 a7 a6 a4 a3 a2 a8 a8 a8 a8 a8 a8

✲ ✛ ✲ ✛

i ⌈m−i

K−i ⌉

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

The approximation algorithm for the Monroe’s system

The satisfaction of the agents assigned in (i + 1)-th turn is at least i + ⌈m−i

K−i ⌉.

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

The approximation algorithm for the Monroe’s system

The satisfaction of the agents assigned in (i + 1)-th turn is at least i + ⌈m−i

K−i ⌉.

Simple summation over i gives the approximation guarantee. Monroe’s system For finding the winners in the Monroe’s system there is a (1 −

K−1 2(m−1) − HK K )-approximation algorithm.

This is a very good approximation if the number of the alternatives is greater than the number of winners (m >> K). For parliamentary elections in Poland it is ≈ 0.95-approximation.

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

Randomized approximation for the Monroe’s system

If K

m is large then it is likely that a large part of the randomly

sampled alternatives will be the alternatives from the optimal solution. And the optimal matching between the selected alternatives and the agents is can be done in polynomial time. Monroe’s system For finding the winners in the Monroe’s system there is a randmized (1

2(1 + K m − K 2 m2−m + K 3 m3−m2 ) − ǫ)-approximation

algorithm. Monroe’s system The combination of the deterministic and randomized algorithms gives the approximation guarantee equal (0.715 − ǫ).

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

Chamberlin-Courant’s rule

1: 2: 3: 4: 5: 6: 7: 8:

a1 a2 a5 a4 a8 a3 a9 a6 a7 a8 a2 a3 a4 a7 a1 a9 a5 a6 a5 a8 a4 a7 a9 a3 a7 a2 a1 a3 a6 a2 a1 a9 a8 a5 a7 a4 a1 a8 a6 a9 a2 a4 a3 a7 a5 a5 a6 a2 a7 a3 a9 a8 a4 a1 a4 a9 a7 a6 a1 a8 a5 a3 a2 a2 a5 a6 a7 a3 a1 a9 a8 a4

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

Chamberlin-Courant’s rule

We consider only top x = mw(K)

K

positions.

1: 2: 3: 4: 5: 6: 7: 8:

a1 a2 a5 a9 a5 a6 a7 a2 a1 a5 a7 a4 a3 a7 a5 a8 a4 a1 a5 a3 a2 a9 a8 a4

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

Chamberlin-Courant’s rule

And greedily select most frequent alternatives.

1: 2: 3: 4: 5: 6: 7: 8:

a1 a2 a5 a9 a5 a6 a7 a2 a1 a5 a7 a4 a3 a7 a5 a8 a4 a1 a5 a3 a2 a9 a8 a4

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

Chamberlin-Courant’s rule

Selected alternatives and their the represented agents are removed from consideration.

1: 2: 3: 4: 5: 6: 7: 8:

a7 a2 a1 a3 a7 a5 a8 a4 a1 a9 a8 a4

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

Chamberlin-Courant’s rule

We repeat the procedure until K alternatives are selected.

1: 2: 3: 4: 5: 6: 7: 8:

a7 a2 a1 a3 a7 a5 a8 a4 a1 a9 a8 a4

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

Chamberlin-Courant’s rule

We repeat the procedure until K alternatives are selected.

1: 2: 3: 4: 5: 6: 7: 8:

a3 a7 a5 a9 a8 a4

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

Chamberlin-Courant’s rule

Chamberlin-Courant’s rule There is a (1 − 2w(K)

K

)-approximation algorithm for the utilitarian Chamberlin-Courant’s rule (approximating satisfaction). For large K we get very good quality of the approximation (0.99 0.99 0.99 for parliamentary elections in Poland).

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

Chamberlin-Courant’s rule

Chamberlin-Courant’s rule There is a (1 − 2w(K)

K

)-approximation algorithm for the utilitarian Chamberlin-Courant’s rule (approximating satisfaction). For large K we get very good quality of the approximation (0.99 0.99 0.99 for parliamentary elections in Poland). But for small K we can find the optimal solution! Chamberlin-Courant’s rule There is a PTAS for utilitarian Chamberlin-Courant’s rule.

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

Truncated ballots

Nice observation – truncated ballots Our all algorithms require only the knowledge of a small number of the top positions of the agents.

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

Relation to other problems

Parliamentary elections. Resource allocation. Facility location problem. Cooperative game theory — coalition formation with externalities. Recommendation systems. Group actvity selection.

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

Conclusions

Negative results No constant approximation for egalitarian case nor for the approximating dissatisfaction – both for Monroe’s and Chamberlin-Courant’s rule. Chamberlin-Courant’s rule PTAS for utilitarian Chamberlin-Courant’s rule. Monroe’s system (1 −

K−1 2(m−1) − HK K )-approximation algorithm.

Monroe’s system The combination of the deterministic and randomized algorithms gives the approximation guarantee equal (0.715 − ǫ).

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

Questions?

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

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