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Online Manipulation and Control in Sequential Voting

Online Manipulation and Control in Sequential Voting. Lane A. Hemaspaandra. Edith Hemaspaandra. Jörg Rothe. Standard Election on Location of COMSOC-14. s imultaneously express their preferences. Voters. Vince. Ulle. Jeff. Edith E. Piotr. Felix. Lena. Edith H. Lane. 1 2

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Online Manipulation and Control in Sequential Voting

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  1. Online Manipulation andControl in SequentialVoting Lane A. Hemaspaandra Edith Hemaspaandra Jörg Rothe

  2. Standard Electionon Location ofCOMSOC-14 simultaneously express theirpreferences Voters Vince Ulle Jeff Edith E. Piotr Felix Lena Edith H. Lane 1 2 3 4 5 6

  3. Standard Electionon Location ofCOMSOC-14 Manipulator Vince Ulle Jeff Edith E. Piotr Felix Lena Edith H. Lane 1 2 3 4 5 6

  4. SequentialElectionon Location ofCOMSOC-14 sequentiallyexpress theirpreferences Voters Vince Ulle Jeff Edith E. Piotr Felix Lena Edith H. Lane 1 2 3 4 5 6

  5. SequentialElectionon Location ofCOMSOC-14 Vince Ulle Jeff Edith E. Piotr Felix Lena Edith H. Lane 1 2 3 4 5 6

  6. SequentialElectionon Location ofCOMSOC-14 Vince Ulle Jeff Edith E. Piotr Felix Lena Edith H. Lane 1 2 3 4 5 6

  7. SequentialElectionon Location ofCOMSOC-14 Manipulator Vince Ulle Jeff Edith E. Piotr Felix Lena Edith H. Lane 1 2 3 4 5 6

  8. SequentialElectionon Location ofCOMSOC-14 Manipulator Vince Ulle Jeff Edith E. Piotr Felix Lena Edith H. Lane 1 2 3 4 5 6

  9. SequentialElectionon Location ofCOMSOC-14 Manipulator Vince Ulle Jeff Edith E. Piotr Felix Lena Edith H. Lane 1 2 3 4 5 6

  10. SequentialElectionon Location ofCOMSOC-14 Manipulator Vince Ulle Jeff Edith E. Piotr Felix Lena Edith H. Lane 1 2 3 4 5 6

  11. SequentialElectionon Location ofCOMSOC-14 Manipulator Nonmanipulator Vince Ulle Jeff Edith E. Piotr Felix Lena Edith H. Lane 1 2 3 4 5 6 It‘spossible, and Lena and Edith E. just don´tknowwhentheyvote!

  12. SequentialElectionon Location ofCOMSOC-14 Manipulator Nonmanipulator Vince Ulle Jeff Edith E. Piotr Felix Lena Edith H. Lane 1 2 3 4 5 6 Lena and Edith E. exerted a destructive online manipulation.

  13. Magnifying Glass Moment Manipulator Vince Ulle Jeff Edith E. Piotr Felix Lena Edith H. Lane 1 2 3 4 5 6 • Online Manipulation Setting (OMS) • is a tuple , where • Cis a setofcandidates, • uis a distinguishedvoter, • is an „electionsnapshotforCandu,“ • σisthepreferenceorderofu´scoalition, and • dis a distinguishedcandidate. Whatisu´s „best“ votetocastnow?

  14. Online Manipulation Problems • online-E-Unweighted-Coalitional-Manipulation (online-E-UCM), for a votingsystem E. Given: An OMS asdescribedabove. Question: Doesthereexistsomevotethatucancast (assumingsupportfromthemanipulators after u) such thatno matter whatvotesare castbythenonmanipulators after u, there existssome Ewinnerc such that ? • Inonline-E-Weighted-Coalitional-Manipulation (online-E-WCM), eachvotercomeswith a weight. • In online-E-UCM[k]andonline-E-WCM[k], thenumberofmanipulatorsfromu onwardisatmostk. • online-E-DUCM, online-E-DWCM,online-E-DUCM[k],andonline-E-DWCM[k]arethedestructivevariants.

  15. Related Work andHowItDiffers • Xia andConitzer (AAAI 2010) and, relatedly, DesmedtandElkind (EC 2010) studythe „Stackelbergvotinggame,“ whichiscalled „roll-callvotinggame“ bySloth (GEB 1993): • Votersvote in orderand • Preferencesarecommonknowledge: Everyoneknowseveryoneelse´spreferences, everyoneknowsthateveryoneknowseveryone else´spreferences, and so on out toinfinity. Theiranalysisisfundamentallygame-theoretic: with such completeknowledgethereispreciselyone (subgameperfect Nash) equilibrium, whichcanbecomputedfromthe back end forward. Forboundednumberofmanipulators, manipulationis in P.

  16. Related Work andHowItDiffers • Tennenholtz (EC 2004) studies „dynamicvoting,“ but focuses on axiomsandvotingrulesratherthan on coalitionsandmanipulation. • ParkesandProcaccia (2011) studysequentialdecision-makingfordynamicallyvaryingpreferences, usingMarkovDecisionProcesses. • Also somewhatrelatedto, but quite different from, ourworkisthestudyofpossibleandnecessarywinnersinitiatedbyKonczakand Lang (2005). • Bycontrast, inspiredby „online algorithms“ ourwork • focuses on complexitytheoryratherthangametheory, • is in a partial-information model, • is not aboutaxiomsandisnonprobabilistic, and • involvesnumbersofquantifiersthatcangrowwiththeinput.

  17. Easy Observations Proposition: • Foreachvotingsystem E such thatthe (unweighted) winnerproblemissolvable in polynomial time, E-UCMreducestoonline-E-UCM. • ForeachvotingsystemE such thattheweightedwinnerproblemissolvable in polynomial time, E-WCMreducestoonline-E-WCM. • ForeachvotingsystemE such thatthewinnerproblemissolvable in polynomial time, E-DUCMreducestoonline-E-DUCM. • ForeachvotingsystemE such thattheweightedwinnerproblemissolvable in polynomial time, E-DWCMreducestoonline-E-DWCM.

  18. General Results Theorem: • Foreachvotingsystem E whosewinnerproblemcanbesolved in polynomial time (oreven in polynomialspace), online-E-UCMis in PSPACE. • Foreachvotingsystem E whoseweightedwinnerproblemcanbesolved in polynomial time (oreven in polynomialspace), online-E-WCMis in PSPACE. • Thereis a votingsystemE with a polynomial-time winnerproblemsuch thatonline-E-UCMisPSPACE-complete. • Thereis a votingsystemE with a polynomial-time weightedwinnerproblemsuch thatonline-E-WCMisPSPACE-complete.

  19. General Results (k Manipulators) Theorem: Fix any • Foreachvotingsystem E whosewinnerproblemcanbesolved in polynomial time, online-E-UCM[k]is in , the2k-th levelofthepolynomialhierarchy. • Foreachvotingsystem E whoseweightedwinnerproblemcanbesolved in polynomial time, online-E-WCMis in . • Thereis a votingsystemE with a polynomial-time winnerproblemsuch thatonline-E-UCM[k]is-complete. • Thereis a votingsystemE with a polynomial-time weightedwinnerproblemsuch thatonline-E-WCM[k]is -complete.

  20. Voting Systems PositionalScoring Rules (formcandidates) • definedbyscoringvectorwith • eachvotergivespointstothecandidate on positioni • winners: all candidateswithmaximum score Borda:PluralityVoting: k-Approval ((m-k)-Veto): Veto (Anti-Plurality):

  21. ResultsforSpecific Natural Voting Systems Theorem: • online-plurality-WCM (andthus also online-plurality-UCM) is in P. • online-plurality-DWCM(andthus also online-plurality-DUCM) is in P. This resultis in ourstandard, thenonunique-winner, model. The nextresultis on problems in theunique-winnermodel. Theorem: • online-plurality-DWCM-UWis NP-hard, evenwhenrestrictedtoonlytwo (orthree, four, etc.) candidates. • online-plurality-WCM-UWiscoNP-hard, evenwhenrestrictedtoonlytwo (orthree, four, etc.) candidates.

  22. Results for Specific Natural Voting Systems Proof: 1.ReductionfromPartition: Given a nonemptysequenceof positive integerswithforsome integer doesthereexist a set such that ? • Letandletbe a Partitioninstance. • Construct an online-plurality-DWCM-UW instance such thatVcontains voterswhovote in thatorder: • Eachvotesforandhasweight • Eachis a manipulatorofweight

  23. Results for Specific Natural Voting Systems • is a yes-instanceofPartitionifandonlyif is a yes-instanceof online-plurality-DWCM-UW. • Ifis inPartition, themanipulators • cangivepointstobothandand • zeropointstotheothercandidates. So andaretiedforthemostpointsandthereisnouniquewinner. • Conversely, theonlywaytoavoidhaving a uniquewinneristohave a tieforthemostpoints: • Onlyandcantie (all others´ scoresare different modulo ), andtheytieonlywithpoints. So is in Partition.

  24. Results for Specific Natural Voting Systems Theorem: • Foreachscoringrule , online- -WCM • is in P if , and • is NP-hardotherwise. • Foreachk, online- k -approval-UCM andonline-k -veto-UCM are in P. • online-3candidate-veto-WCMis P -complete. • online-veto-WCMisin P . NP[1] NP

  25. UncertaintyAboutthe Order of Future Voters • Schedule-Robust-online-E-UCM, fora votingsystemE. Given: An „OMS“ asbefore, exceptnow • wedon´tfocus on onemanipulatoru, but at a moment in time, • wedon´tknowtheorderofthevoters still tocome. Question: Can our manipulative coalitionensurethatdor someonelikedmorethand w.r.t. σ will win, regardlessofwhatordertheremainingvoters vote in? Theorem: • Foreachvotingsystem E whosewinnerproblemis in P,Schedule-Robust-online-E-UCMis in . • Thereis a votingsystemE, whosewinnerproblemis in P,such thatSchedule-Robust-online-E-UCMis -complete.

  26. References • E. Hemaspaandra , L. Hemaspaandra, and J. Rothe: The Complexityof Online Manipulation ofSequentialElections. Toappear in Proceedingsof COMSOC-2012. Also relatedarethepapers: • E. Hemaspaandra , L. Hemaspaandra, and J. Rothe: Online VoterControl in SequentialElections. Toappear in ProceedingsofECAI-2012. • E. Hemaspaandra , L. Hemaspaandra, and J. Rothe: Controlling Candidate-SequentialElections. Toappear in ProceedingsofECAI-2012.

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