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Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty˚

Jiehua Chen1 Dagstuhl Seminar: Computational Social Choice June 8th, 2015

Joint work with Robert Bredereck1, Rolf Niedermeier1, and Toby Walsh2

1 TU Berlin, Germany 2 NICTA, Australia ˚To be published in IJCAI’15. Jiehua Chen (TU Berlin) Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty˚ 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 a beats b? a beats s? a a s s a b beats s? b b s s b Anglo-American procedure: b beats ta, su ? b b a beats tsu ? a a s tsu ta, su Euro-Latin procedure: Votes 100 voters: b ą s ą a 1 voter: a ą b ą s 99 voters: s ą a ą b 1 voter: s ą b ą a b s a

Jiehua Chen (TU Berlin) Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty˚ 2/13

Manipulation

Manipulation Input: Election E “ pC, 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 100 voters: b ą s ą a 1 voter: a ą b ą s 99 voters: s ą a ą b b s

101: 99

a

199: 1 100: 100

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.

Jiehua Chen (TU Berlin) Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty˚ 3/13

Agenda Control

Agenda Control Input: Election E “ pC, V q and s P C. Task: Find an agenda L for C such that s wins. Example 1 voter: a ą b ą s ą d 1 voter: d ą a ą s ą b 1 voter: s ą b ą a ą d s b a d Find an agenda such that s wins?

• Anglo-American procedure?

Not possible.

• Euro-Latin procedure?

L: a ą b ą d ą s.

Jiehua Chen (TU Berlin) Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty˚ 4/13

Possible/Necessary Winner

Possible (or Necessary) Winner Input: Election E “ pC, 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 ą b ą s 1 voter: b ą s ą a 1 voter: s ą ta , bu partial agenda: a ą b s a b Can s possibly (or necessarily) win?

• Anglo-American procedure?

Possibly, but not necessarily.

• Euro-Latin procedure?

Not possible.

Jiehua Chen (TU Berlin) Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty˚ 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

Jiehua Chen (TU Berlin) Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty˚ 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,
• root represents the winner.

a c c b Anglo-American procedure: Each inner node has exactly one leaf and one non-leaf. a b c 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

Jiehua Chen (TU Berlin) Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty˚ 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

Jiehua Chen (TU Berlin) Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty˚ 8/13

Result Summary

• n: the number of voters.
• m: the number of alternatives.
• k: manipulation coalition size.

Problem Anglo-American Euro-Latin Manipulation Oppk ` nq ¨ m2q Oppk ` nq ¨ mq Agenda Control Opn ¨ m2 ` m3q Opn ¨ m2q Possible Winner NP-hard NP-hard Necessary Winner coNP-hard Opn ¨ m3q Weighted Possible Winner NP-hard NP-hard Weighted Necessary Winner coNP-hard Opn ¨ m3q

Jiehua Chen (TU Berlin) Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty˚ 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

• dd number of voters.

# alternatives ranges from 3 to 242; # voters ranges from 5 to 14081.

˚“A library of preference data”, Mattei and Walsh, 2013

Jiehua Chen (TU Berlin) Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty˚ 10/13

Experimental Result for Manipulation

Goal Investigate the likelihood of successful manipulations for a given election pC, 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

LPX

ř

cPC min # manipulators for c under L

|X| ¨ pm ´ 1q ¨ pn ` 1q Ratio of 0.6 0.6 ¨ pn ` 1q voters needed. Experimental result Measurement Anglo-American Euro-Latin 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

Jiehua Chen (TU Berlin) Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty˚ 11/13

Experimental Result for Agenda Control

Goal Investigate the likelihood of successful agenda controls for a given election pC, 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 ¨ pm ´ 1q alternatives are controllable. Experimental result Measurement Anglo-American Euro-Latin m ď 4 m ě 5 m ď 4 m ě 5 control vulerability ratio 0.000 0.035 0.157 0.081

Jiehua Chen (TU Berlin) Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty˚ 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!

Jiehua Chen (TU Berlin) Parliamentary Voting Procedures: Agenda Control, Manipulation, and Uncertainty˚ 13/13