Parliamentary Voting Procedures: Agenda Control, Manipulation, and - PowerPoint PPT Presentation

Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty ˚ Jiehua Chen 1 Dagstuhl Seminar: Computational Social Choice June 8th, 2015 Joint work with Robert Bredereck 1 , Rolf Niedermeier 1 , and Toby Walsh 2 1 TU Berlin, Germany 2 NICTA, Australia ˚ To be published in IJCAI’15. Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty ˚ Jiehua Chen (TU Berlin) 1/13

1956 Education Act in the USA Alternatives (b)ill : “Funding to primary and secondary schools”. (a)mended bill : “Funding, but not to segregated schools”. (s)tatus quo : “No bill”. Agenda L : b ą a ą s Two procedures Euro-Latin procedure: Anglo-American procedure: b beats t a , s u ? a beats b ? t a , s u a b b a beats t s u ? a beats s ? b beats s ? b a s s t s u b a a s s a s b Votes 100 voters: b s a s ą ą b 1 voter: a b s ą ą 99 voters: s a b ą ą a 1 voter: s b a ą ą Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty ˚ Jiehua Chen (TU Berlin) 2/13

Manipulation Manipulation Input: Election E “ p C , V q , s P C , and an agenda L for C . Task: Add as few voters as possible such that s wins under L . Example 101 : 99 100 voters: b s a ą ą s b 1 voter: a b s ą ą 99 voters: s a b 100 : 100 199 : 1 ą ą a Agenda L : b ą a ą s Fewest number of voters needed to make s win? • Anglo-American procedure? 1 voter: s ą a ą b . • Euro-Latin procedure? Already winner. Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty ˚ Jiehua Chen (TU Berlin) 3/13

Agenda Control Agenda Control Input: Election E “ p C , V q and s P C . Task: Find an agenda L for C such that s wins. a Example s b 1 voter: a b s d ą ą ą 1 voter: d a s b ą ą ą 1 voter: s b a d ą ą ą d Find an agenda such that s wins? • Anglo-American procedure? Not possible. • Euro-Latin procedure? L : a ą b ą d ą s. Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty ˚ Jiehua Chen (TU Berlin) 4/13

Possible/Necessary Winner Possible (or Necessary) Winner Input: Election E “ p C , V q with incomplete preferences, s P C , and a partial agenda B for C . Task: Decide whether s wins in an (or in every ) election completing E under an (or under every ) agenda completing B . Example 1 voter: a b s ą ą b 1 voter: b s a ą ą 1 voter: s t a , b u ą s a partial agenda : a ą b Can s possibly (or necessarily) win? • Anglo-American procedure? Possibly, but not necessarily. • Euro-Latin procedure? Not possible. Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty ˚ Jiehua Chen (TU Berlin) 5/13

Some Related Work Political studies “The theory of committees and elections”, Black, 1958 “Theory of voting”, Farquharson, 1969 “Insincere voting under the successive procedure”, Rasch, 2014 “A foundation for strategic agenda voting”, Apesteguia et al., 2014 Agenda Control “Graph-theoretical approaches to the theory of voting”, Miller, 1977 “Sophisticated voting outcomes and agenda control”, Banks, 1985 Manipulation “The computational difficulty of manipulating an election”, Bartholdi et al., 1989 “Single transferable vote resists strategic voting”, Bartholdi III & Orlin, 1991 “When are elections with few candidates hard to manipulate?”, Conitzer et al., 2007 Possible/Necessary Winner “Winner determination in voting trees with incomplete preferences and weighted votes”, Lang et al., 2012 “Incompleteness and incomparability in preference aggregation: Complexity results”, Pini et al., 2011 Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty ˚ Jiehua Chen (TU Berlin) 6/13

Related Procedure: Voting Tree Procedure Voting Tree procedure works on a binary tree : • leaves represent alternatives, • inner nodes represent the majority winner of their children, a c c b • root represents the winner. Anglo-American procedure: Each inner node has exactly one leaf and one non-leaf. c a b Known results for the Voting Tree: • Manipulation: Cubic-time algorithm [1]. • Weighted Possible Winner: NP-hard for three alternatives [2,3]. • Weighted Necessary Winner: coNP-hard for four alternatives [2,3]. [1] Conitzer et al., JACM, 2007 [2] Lang et el., AAMAS, 2012 [3] Pini et al., AI, 2011 Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty ˚ Jiehua Chen (TU Berlin) 7/13

Related Procedure: Single Transferable Voting Single Transferable Voting (STV): REPEAT Delete one alternative with lowest plurality score UNTIL Find an alternative with majority score (majority winner) Similar to Euro-Latin procedure: • Both try to find a majority winner. • But, STV doesn’t need an agenda. Known results for STV: Manipulation: NP-hard already for manipulation coalition size one. Bartholdi III et al., Soc. Choice Welf., 1991 Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty ˚ Jiehua Chen (TU Berlin) 8/13

Result Summary • n : the number of voters. • m : the number of alternatives. • k : manipulation coalition size. Problem Anglo-American Euro-Latin O pp k ` n q ¨ m 2 q Manipulation O pp k ` n q ¨ m q O p n ¨ m 2 ` m 3 q O p n ¨ m 2 q Agenda Control Possible Winner NP-hard NP-hard O p n ¨ m 3 q Necessary Winner coNP-hard Weighted Possible Winner NP-hard NP-hard O p n ¨ m 3 q Weighted Necessary Winner coNP-hard Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty ˚ Jiehua Chen (TU Berlin) 9/13

Empirical Study Reminder We have a poly-time algorithm of finding the min # voters for an alternative to manipulate. Question How many voters needed on average for the manipulation? Reminder We have a poly-time algorithm of finding an agenda for an alternative to win (if exists). Question How many alternatives can have successful agenda control? Plan Use the data from Preflib ˚ to find an answer. Background Preflib has 314 elections with complete preferences;135 profiles have odd number of voters. # alternatives ranges from 3 to 242; # voters ranges from 5 to 14081. ˚ “A library of preference data”, Mattei and Walsh, 2013 Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty ˚ Jiehua Chen (TU Berlin) 10/13

Experimental Result for Manipulation Goal Investigate the likelihood of successful manipulations for a given election p C , V q with m alternatives and n voters. Manipulation resistency ratio Ratio of “avg.” successful manipulation coalition size for a set X of agendas: ř ř c P C min # manipulators for c under L L P X | X | ¨ p m ´ 1 q ¨ p n ` 1 q Ratio of 0 . 6 � 0 . 6 ¨ p n ` 1 q voters needed. Experimental result Anglo-American Euro-Latin Measurement m ď 4 m ě 5 m ď 4 m ě 5 manipulation resistency ratio 0 . 442 0 . 933 0 . 474 0 . 949 2nd winner coalition ratio 0 . 221 0 . 440 0 . 286 0 . 530 smallest coalition ratio 0 . 220 0 . 386 0 . 262 0 . 388 Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty ˚ Jiehua Chen (TU Berlin) 11/13

Experimental Result for Agenda Control Goal Investigate the likelihood of successful agenda controls for a given election p C , V q with m alternatives and odd n voters. Control vulnerability ratio Ratio of “controllable” alternatives #(alternatives that may win under some agenda) ´ 1 m ´ 1 Ratio of 0 . 6 � 0 . 6 ¨ p m ´ 1 q alternatives are controllable. Experimental result Anglo-American Euro-Latin Measurement m ď 4 m ě 5 m ď 4 m ě 5 control vulerability ratio 0 . 000 0 . 035 0 . 157 0 . 081 Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty ˚ Jiehua Chen (TU Berlin) 12/13

Conclusion Result summary • The Anglo-American procedure tends to have a higher computational complexity than the Euro-Latin procedure. • Empirical study: In practice, manipulation by few voters is very rare and agenda control is almost impossible. Future research • Possible/Necessary Winner when voter preferences are restricted. • Other control problems. • Game-theoretical aspect when voters may vote strategically. Thanks for your attention! Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty ˚ Jiehua Chen (TU Berlin) 13/13