Sincere-Strategy Preference-Based Approval Voting Fully Resists - PowerPoint PPT Presentation

Introduction Approval Voting and its Versions Results Sincere-Strategy Preference-Based Approval Voting Fully Resists Constructive Control and Broadly Resists Destructive Control G´ abor Erd´ elyi Markus Nowak J¨ org Rothe Institut f¨ ur Informatik, Heinrich-Heine-Universit¨ at D¨ usseldorf, Germany COMSOC-08, Liverpool, September 2008 Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Outline Approval Voting and its Versions Introduction Results Outline Introduction 1 Approval Voting and its Versions 2 Approval Voting (AV) Sincere-Strategy Preference-Based Approval Voting (SP-AV) Results 3 Parts of this paper were presented at MFCS 2008. Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Outline Approval Voting and its Versions Introduction Results Introduction Artificial systems: Hybridization - Fully resists control (Hemaspaandra, Hemaspaandra, Rothe - IJCAI 2007) Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Outline Approval Voting and its Versions Introduction Results Introduction Artificial systems: Hybridization - Fully resists control (Hemaspaandra, Hemaspaandra, Rothe - IJCAI 2007) Natural systems: 3 × R 4 × I 7 × V Condorcet 4 × R 9 × I 9 × V Approval 14 × R 0 × I 8 × V Llull 15 × R 0 × I 7 × V Copeland 16 × R 0 × I 6 × V Plurality Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Outline Approval Voting and its Versions Introduction Results Introduction Artificial systems: Hybridization - Fully resists control (Hemaspaandra, Hemaspaandra, Rothe - IJCAI 2007) Natural systems: 3 × R 4 × I 7 × V Condorcet 4 × R 9 × I 9 × V Approval 14 × R 0 × I 8 × V Llull 15 × R 0 × I 7 × V Copeland 16 × R 0 × I 6 × V Plurality Can we do better? Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Voting Systems Set of candidates and voters: C = { c 1 , . . . , c m } V = { v 1 , . . . , v n } Voter preferences over C can be represented as preference lists (rankings) approval/disapproval vectors Voting rule aggregates the preferences and outputs the set of winners unique winner nonunique winner Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Approval Voting Introduced by Brams and Fishburn Each voter specifies his or her 0 − 1 approval vector: 1 represents approval 0 represents disapproval Ignores preference rankings The winners are the candidates with the highest score Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Example for Approval Voting Example Set of voters: V = { v 1 , . . . , v 10 } Set of candidates: c 1 = chicken c 2 = fish c 3 = pork c 4 = rump steak c 5 = tofu Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Example for Approval Voting Example Set of voters: The votes: V = { v 1 , . . . , v 10 } v 1 = v 2 = 00010 Set of candidates: v 3 = 00110 c 1 = chicken v 4 = 11110 c 2 = fish v 5 = v 6 = v 7 = v 8 = 11011 c 3 = pork v 9 = 01001 c 4 = rump steak v 10 = 00001 c 5 = tofu Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Example for Approval Voting Example The result of the voting: The votes: score ( c 1 ) = 5 v 1 = v 2 = 00010 v 3 = 00110 score ( c 2 ) = 6 v 4 = 11110 score ( c 3 ) = 2 v 5 = v 6 = v 7 = v 8 = 11011 v 9 = 01001 score ( c 4 ) = 8 v 10 = 00001 score ( c 5 ) = 6 Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Rules Proposed by Brams and Sanver Each voter has a preference ranking, a tie free linear ordering of all candidates: c 4 > c 1 > c 3 > c 5 > c 2 > c 6 Line between acceptable and inacceptable candidates: c 4 c 1 | c 3 c 5 c 2 c 6 The winners are the candidates with the highest score Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Notations The set of candidates S v that voter v approves of is an AV strategy The list of all strategies is an AV strategy profile Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Conventions Admissibility: v ’s most preferred candidate ∈ S v v ’s least preferred candidate / ∈ S v Sincerity: no gaps in the approval strategies Sincere strategy is always admissible for at least 2 candidates if ∅ � = S v � = C Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Example for SP-AV Example v 1 = v 2 = c 4 | c 1 c 3 c 5 c 2 v 3 = c 3 c 4 | c 2 c 5 c 1 v 4 = c 3 c 1 c 2 c 4 | c 5 v 5 = v 6 = c 5 c 4 c 1 c 2 | c 3 v 7 = v 8 = c 1 c 5 c 2 c 4 | c 3 v 9 = c 2 c 5 | c 1 c 3 c 4 v 10 = c 5 | c 1 c 4 c 3 c 2 Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Example for SP-AV Example v 1 = v 2 = c 4 | c 1 c 3 c 5 c 2 The result of the voting: v 3 = c 3 c 4 | c 2 c 5 c 1 score ( c 1 ) = 5 v 4 = c 3 c 1 c 2 c 4 | c 5 score ( c 2 ) = 6 v 5 = v 6 = c 5 c 4 c 1 c 2 | c 3 score ( c 3 ) = 2 v 7 = v 8 = c 1 c 5 c 2 c 4 | c 3 score ( c 4 ) = 8 score ( c 5 ) = 6 v 9 = c 2 c 5 | c 1 c 3 c 4 v 10 = c 5 | c 1 c 4 c 3 c 2 Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Violations Against the Conventions Violations against admissible AV strategies in control via: Deleting Candidates, Partition of Candidates, Partition of Voters. Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Example for Re-Enforcing the Conventions Example Nonvegetarian food: v 1 = v 2 = c 4 | c 1 c 3 c 2 v 3 = c 3 c 4 | c 2 c 1 v 4 = v 4 = c 3 c 1 c 2 | c 4 c 3 c 1 c 2 c 4 | ← − v 5 = v 6 = c 4 c 1 c 2 | c 3 v 7 = v 8 = c 1 c 2 c 4 | c 3 v 9 = c 2 | c 1 c 3 c 4 v 10 = v 10 = c 1 | c 4 c 3 c 2 | c 1 c 4 c 3 c 2 ← − Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Example for Re-Enforcing the Conventions Example Nonvegetarian food: v 1 = v 2 = c 4 | c 1 c 3 c 2 Result of ( C 1 , V ) : v 3 = c 3 c 4 | c 2 c 1 score ( c 1 ) = 7 v 4 = c 3 c 1 c 2 | c 4 score ( c 2 ) = 6 v 5 = v 6 = c 4 c 1 c 2 | c 3 score ( c 3 ) = 2 v 7 = v 8 = c 1 c 2 c 4 | c 3 score ( c 4 ) = 7 v 9 = c 2 | c 1 c 3 c 4 v 10 = c 1 | c 4 c 3 c 2 Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Example for Re-Enforcing the Conventions Example v 1 = v 2 = c 4 | c 1 c 5 v 3 = c 4 | c 5 c 1 v 4 = c 1 c 4 | c 5 v 5 = v 6 = v 5 = v 6 = c 5 c 4 | c 1 c 5 c 4 c 1 | ← − v 7 = v 8 = v 7 = v 8 = c 1 c 5 | c 4 c 1 c 5 c 4 | ← − v 9 = c 5 | c 1 c 4 v 10 = c 5 | c 1 c 4 Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Approval Voting Approval Voting and its Versions SP-AV Results Example for Re-Enforcing the Conventions Example v 1 = v 2 = c 4 | c 1 c 5 v 3 = c 4 | c 5 c 1 The result of the second stage: v 4 = c 1 c 4 | c 5 score ( c 1 ) = 3 v 5 = v 6 = c 5 c 4 | c 1 score ( c 4 ) = 6 v 7 = v 8 = c 1 c 5 | c 4 score ( c 5 ) = 6 v 9 = c 5 | c 1 c 4 v 10 = c 5 | c 1 c 4 Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting

Introduction Previous Results Approval Voting and its Versions Results Results Summary Plurality and Approval Theorem Plurality AV Control by Constr. Destr. Constr. Destr. Adding an Unlimited Number of Candidates R R I V Adding a Limited Number of Candidates R R I V Deleting Candidates R R V I Partition of Candidates TE: R TE: R TE: V TE: I TP: R TP: R TP: I TP: I Run-off Partition of Candidates TE: R TE: R TE: V TE: I TP: R TP: R TP: I TP: I Adding Voters V V R V Deleting Voters V V R V Partition of Voters TE: R TE: R TE: R TE: V TP: V TP: V TP: R TP: V Erd´ elyi, Nowak, Rothe Sincere-Strategy Preference-Based Approval Voting