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 University 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). Alternatives (candidates) a 2 a 1 a 3 a 4 4 2 6 3 5 1 Agents (voters) Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation
Problem Agents have preferences over alternatives. Alternatives (candidates) 1 : a 1 ≻ a 3 ≻ a 5 ≻ a 2 ≻ a 4 2 : a 1 ≻ a 2 ≻ a 4 ≻ a 3 ≻ a 5 a 2 a 1 3 : a 1 ≻ a 3 ≻ a 2 ≻ a 4 ≻ a 5 a 3 a 4 4 : a 3 ≻ a 1 ≻ a 4 ≻ a 2 ≻ a 5 5 : a 1 ≻ a 3 ≻ a 4 ≻ a 5 ≻ a 2 6 : a 3 ≻ a 5 ≻ a 1 ≻ a 2 ≻ a 4 4 2 6 3 5 1 Agents (voters) Piotr Skowron, Piotr Faliszewski, Arkadii Slinko Approximability of proportional representation
Problem We want to select K candidates (in the example K = 2). Alternatives (candidates) 1 : a 1 ≻ a 3 ≻ a 5 ≻ a 2 ≻ a 4 2 : a 1 ≻ a 2 ≻ a 4 ≻ a 3 ≻ a 5 a 2 a 1 3 : a 1 ≻ a 3 ≻ a 2 ≻ a 4 ≻ a 5 a 3 a 4 4 : a 3 ≻ a 1 ≻ a 4 ≻ a 2 ≻ a 5 5 : a 1 ≻ a 3 ≻ a 4 ≻ a 5 ≻ a 2 6 : a 3 ≻ a 5 ≻ a 1 ≻ a 2 ≻ a 4 4 2 6 3 5 1 Agents (voters) 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. Alternatives (candidates) 1 : a 1 ≻ a 3 ≻ a 5 ≻ a 2 ≻ a 4 2 : a 1 ≻ a 2 ≻ a 4 ≻ a 3 ≻ a 5 a 2 a 1 3 : a 1 ≻ a 3 ≻ a 2 ≻ a 4 ≻ a 5 a 3 a 4 4 : a 3 ≻ a 1 ≻ a 4 ≻ a 2 ≻ a 5 5 : a 1 ≻ a 3 ≻ a 4 ≻ a 5 ≻ a 2 6 : a 3 ≻ a 5 ≻ a 1 ≻ a 2 ≻ a 4 4 2 6 3 5 1 Agents (voters) 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). Alternatives (candidates) 1 : a 1 ≻ a 3 a 3 ≻ a 5 ≻ a 2 ≻ a 4 a 3 2 : a 1 a 1 ≻ a 2 ≻ a 4 ≻ a 3 ≻ a 5 a 1 a 2 a 1 a 3 a 4 3 : a 1 a 1 ≻ a 3 ≻ a 2 ≻ a 4 ≻ a 5 a 1 4 : a 3 ≻ a 1 a 1 ≻ a 4 ≻ a 2 ≻ a 5 a 1 5 : a 1 ≻ a 3 a 3 ≻ a 4 ≻ a 5 ≻ a 2 a 3 a 3 ≻ a 5 ≻ a 1 ≻ a 2 ≻ a 4 6 : a 3 a 3 4 2 6 3 5 1 Agents (voters) 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 : � α 1 , α 2 , . . . α m � α i means that the satisfaction of the agent v from the candidate that he/she puts in his/her i -th position is α i . 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 : a 1 ≻ a 3 a 3 a 3 ≻ a 5 ≻ a 2 ≻ a 4 2 : a 1 a 1 a 1 ≻ a 2 ≻ a 4 ≻ a 3 ≻ a 5 3 : a 1 a 1 a 1 ≻ a 3 ≻ a 2 ≻ a 4 ≻ a 5 4 : a 3 ≻ a 1 a 1 ≻ a 4 ≻ a 2 ≻ a 5 a 1 5 : a 1 ≻ a 3 a 3 a 3 ≻ a 4 ≻ a 5 ≻ a 2 6 : a 3 a 3 a 3 ≻ a 5 ≻ a 1 ≻ a 2 ≻ a 4 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 : a 1 ≻ a 3 a 3 a 3 ≻ a 5 ≻ a 2 ≻ a 4 2 : a 1 a 1 a 1 ≻ a 2 ≻ a 4 ≻ a 3 ≻ a 5 3 : a 1 a 1 a 1 ≻ a 3 ≻ a 2 ≻ a 4 ≻ a 5 4 : a 3 ≻ a 1 a 1 a 1 ≻ a 4 ≻ a 2 ≻ a 5 5 : a 1 ≻ a 3 a 3 a 3 ≻ a 4 ≻ a 5 ≻ a 2 6 : a 3 a 3 a 3 ≻ a 5 ≻ a 1 ≻ a 2 ≻ a 4 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 = { a 1 , a 2 , . . . , a m } . 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 = { a 1 , a 2 , . . . , a m } . 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 : a 1 a 1 a 1 ≻ a 3 ≻ a 5 ≻ a 2 ≻ a 4 2 : a 1 a 1 a 1 ≻ a 2 ≻ a 4 ≻ a 3 ≻ a 5 3 : a 1 a 1 a 1 ≻ a 3 ≻ a 2 ≻ a 4 ≻ a 5 4 : a 3 a 3 a 3 ≻ a 1 ≻ a 4 ≻ a 2 ≻ a 5 5 : a 1 a 1 a 1 ≻ a 3 ≻ a 4 ≻ a 5 ≻ a 2 6 : a 3 a 3 a 3 ≻ a 5 ≻ a 1 ≻ a 2 ≻ a 4 In the Chamberlin-Courant’s system the winners are a 1 i a 3 (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 : a 1 a 1 a 1 ≻ a 3 ≻ a 5 ≻ a 2 ≻ a 4 2 : a 1 a 1 a 1 ≻ a 2 ≻ a 4 ≻ a 3 ≻ a 5 3 : a 1 a 1 a 1 ≻ a 3 ≻ a 2 ≻ a 4 ≻ a 5 4 : a 3 a 3 a 3 ≻ a 1 ≻ a 4 ≻ a 2 ≻ a 5 5 : a 1 ≻ a 3 a 3 a 3 ≻ a 4 ≻ a 5 ≻ a 2 6 : a 3 a 3 a 3 ≻ a 5 ≻ a 1 ≻ a 2 ≻ a 4 In the Monroe’s system the winners are also a 1 i a 3 , 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
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