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Random Tie Breaking

Random Tie Breaking. Toby Walsh NICTA and UNSW. Random Tie Breaking. Haris Aziz, Serge Gaspers, Nick Mattei , Nina Narodytska , Toby Walsh NICTA and UNSW. Ties matter. Manipulators can only change result if election is close!. Ties matter.

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Random Tie Breaking

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  1. Random Tie Breaking Toby Walsh NICTA and UNSW

  2. Random Tie Breaking Haris Aziz, Serge Gaspers, Nick Mattei, Nina Narodytska, Toby Walsh NICTA and UNSW

  3. Ties matter • Manipulators can only change result if election is close!

  4. Ties matter • Manipulators can only change result if election is close! • How we deal with ties often matters critically • Typical assumption is ties broken in favour of the manipulators

  5. Ties matter • Manipulators can only change result if election is close! • How we deal with ties often matters critically • Typical assumption is ties broken in favour of the manipulators • In real elections, ties broken randomly, by the chair, by age ….

  6. Ties matter • Manipulators can only change result if election is close! • How we deal with ties often matters critically • Typical assumption is ties broken in favour of the manipulators • In real elections, ties broken randomly, by the chair, by age …. • Tie breaking can itself be a source of computational complexity • 2nd order Copeland, • Copeland with weighted votes: polynomial to manipulate if ties are scored 1, but NP-hard if ties are scored 0 [Faliszewski, Hemaspaandrea, Schnoor 08]

  7. Unique and co-winner problems • Unique winner manipulation problem • Equivalent to tie-breaking against manipulator • Can we construct a strategic vote so given candidate is the unique winner of the election? • Co-winner manipulation problem • Equivalent to tie-breaking in favour of the manipulator • Can we construct a strategic vote so given candidate is one of the co-winners of the election?

  8. Tie-breaking in practice • Random candidate • E.g. UK general elections • Random vote • E.g. Schulze voting breaks ties according to order of candidates in a random vote • By the chair

  9. Tie-breaking with a random candidate • See [Obraztsova, Elkind, Hazon AAMAS 2011], [Obratzsova, Elkind IJCAI 2011] • Agents assign utilities to candidates • Look to maximize expected utility of result • Can get a large way though with a simple model of just asking if a given candidate can win with probability > p? • Equivalent to u(a)=1, u(b)=0 for all other candidates • Avoids the difficult problem of having to assign utilities!

  10. Tie-breaking with a random candidate • Several common rules have been shown to be (in)tractable • THM: When tie-breaking with a random candidate, all scoring rules (including Borda) are polynomial to manipulate, as are plurality with runoff and Bucklin • THM: When tie-breaking with a random candidate, Copeland and Maximin are NP-hard to manipulate [Obraztsova & Elkind 2011]

  11. Tie-breaking with a random vote • In case of a tie, pick a vote uniformly at random • Order candidates according to this vote • In some forthcoming work, we’ve shown that this has different computational properties to tie-breaking with a random candidate • In practice, it seems harder • Indeed, it is often proposed as a barrier to manipulation • Suppose you vote strategic to get a preferred candidate to win, but then your strategic vote may actually make them loose!

  12. Tie-breaking with a random vote • Candidates can have quite different probabilities of winning than tie-breaking with a random candidate • Suppose we use Borda scoring • Half voters vote a>b>c • Half voters vote c>b>a

  13. Tie-breaking with a random vote • Candidates can have quite different probabilities of winning than tie-breaking with a random candidate • Suppose we use Borda scoring • Half voters vote a>b>c • Half voters vote c>b>a • Tie-breaking with a random vote • a or c win with probability 1/2

  14. Tie-breaking with a random vote • Candidates can have quite different probabilities of winning than tie-breaking with a random candidate • Suppose we use Borda scoring • Half voters vote a>b>c • Half voters vote c>b>a • Tie-breaking with a random vote • a or c win with probability 1/2 • Tie-breaking with a random candidate • a, b, or c win with probability 1/3

  15. Tie-breaking with a random vote • Formally incomparable to tie-breaking with a random candidate

  16. Tie-breaking with a random vote • Formally incomparable to tie-breaking with a random candidate • THM: exists voting rule, such thatthemanipulation problem when tie-breaking with a random candidate is polynomial but tie-breaking with a random vote is NP-complete, and vice versa

  17. Tie-breaking with a random vote • Formally incomparable to tie-breaking with a random candidate • THM: exists voting rule, such thatthemanipulation problem when tie-breaking with a random candidate is polynomial, but tie-breaking with a random vote is NP-complete, and vice versa • Proof: Consider Borda voting, and a single manipulator, then tie-breaking with a random candidate is polynomial [Obraztsova, Elkind, and Hazon 2011]

  18. Tie-breaking with a random vote • Formally incomparable to tie-breaking with a random candidate • THM: exists voting rule, such thatthemanipulation problem when tie-breaking with a random candidate is polynomial, but tie-breaking with a random vote is NP-complete, and vice versa • Proof: Consider Borda voting, and a single manipulator, then tie-breaking with a random candidate is polynomial [Obraztsova, Elkind, and Hazon 2011]. But when tie-breaking with a random vote, manipulation is NP-complete [forthcoming 2013]

  19. Tie-breaking with a random vote • Tie-breaking with a random vote is incomparable to the unique and co-winner manipulation problems

  20. Tie-breaking with a random vote • Tie-breaking with a random vote is incomparable to the unique and co-winner manipulation problems • THM: exists voting rule, such that the co-winner and unique winner manipulation problems are polynomial, but tie-breaking with a random vote is NP-complete, and vice versa • Contrast this with tie-breaking with a random candidate • If unique winner or co-winner manipulation problems are NP-hard then tie-breaking with a random candidate is also

  21. Random vote versus Random candidate

  22. Random vote versus Random candidate

  23. Random vote versus Random candidate How you break ties impacts on the computational complexity!

  24. Control by breaking ties • Somewhat related problem • If I am chair, how do I control the result by breaking ties? • Tie-breaking only once (between co-winners), this is trivial • Pick the person you want to win • Tie-breaking even just twice, control can be NP-hard!

  25. Control by breaking ties • Control by tie-breaking with two stage rules • THM: Exists a two stage rule combining veto and plurality for which control by tie-breaking is NP-hard • Proof: Consider rule that eliminates half the candidates using veto, then elects the plurality winner

  26. Control by breaking ties • Control by tie-breaking with two stage rules • THM: Exists a two stage rule combining veto and plurality for which control by tie-breaking is NP-hard • Proof: Consider rule that eliminates half the candidates using veto, then elects the plurality winner • Control by tie-breaking with multi-stage rules • THM: Control by tie-breaking with STV, Baldwin and Coombs is NP-hard

  27. Control by breaking ties • Control by tie-breaking with two stage rules • THM: Exists a two stage rule combining veto and plurality for which control by tie-breaking is NP-hard • Proof: Consider rule that eliminates half the candidates using veto, then elects the plurality winner • Control by tie-breaking with multi-stage rules • THM: Control by tie-breaking with STV, Baldwin and Coombs is NP-hard • THM: Control by tie-breaking with Nanson is polynomial

  28. Control by breaking ties

  29. Control by breaking ties • Incomparable to the manipulation problem when breaking ties with a random candidate, or in a fixed order • THM: Exists voting rule such that control by tie-breaking is polynomial but manipulation problem breaking ties at random/in a fixed order is NP-complete, and vice versa • E.g. control by breaking ties for Copeland is polynomial, but manipulation when breaking ties at random is NP-hard

  30. Conclusions • Ties do matter • Breaking ties with a random vote somewhat more computationally challenging than with a random candidate • For two and multi-stage rules, it can be NP-hard for the chair to control result by breaking ties • Of course, these are all worst case observations and we need to consider the difficulty of breaking ties in practice/on average/…

  31. Questions? • PS I’m recruiting PhD students and a PostDoc

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