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CMU CS 15892 - Lecture

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Avoiding manipulation in elections through computational complexityIntroductionVotingSlide 4Manipulation in votingSlide 6Some well-known protocolsSoftware agents may manipulate moreComputational manipulation problemPrior researchUniversal voting protocol tweaks to make manipulation hard“Tweaks” for protocolsAdding a preround to the protocolPreround example (with Borda)Matching first, or vote collection first?Could also interleave…Main result: how hard is manipulation?NP-hard with preround matching firstStill not done…Conclusions on tweaksHardness of manipulation with few candidatesWhat if there are few candidates?Manipulation complexity with few candidatesWhy study weighted coalitional manipulation?Constructive manipulation now becomes:Destructive manipulationSome other well-known protocolsA simple example of hardnessWhat does it mean for a protocol to be easy to manipulate?Example: Maximin with 3 candidates is easy to manipulate constructivelyWhy do we care about the exact number of candidates required for hardness?Results for constructive manipulationResults for destructive manipulationWorst-case hardness…Thank you for your attention!Avoiding manipulation in elections through computational complexityVincent Conitzer Computer Science DepartmentCarnegie Mellon UniversityGuest lecture 15-892Introduction. [Bartholdi, Tovey, Trick 1989][Bartholdi, Orlin 1991]Voting•In multiagent systems, agents may have conflicting preferences•Based on reported preferences, a preference aggregator often must choose one candidate from the possible outcomes–Deciding on a leader/coordinator/representative–Joint plans–Allocations of tasks/resources–…•Voting is the most general preference aggregation method–Applicable to any preference aggregation setting–No side paymentsVoting“A > B > C”“A > C > B”“B > A > C”Winner (probably A here)Voter 1Voter 2Voter 3VOTING PROTOCOLManipulation in voting•A voter is said to manipulate when it does not rank the candidates according to its true preferences•Example: not ranking your most preferred candidate first because that candidate has no chance anyway•Why is manipulation bad?–Protocol is designed to maximize (some measure of) social welfare with respect to the reported preferences•Manipulation will cause a suboptimal outcome to be chosen–Also: if the protocol actually relies on manipulation to choose the right outcome…–… then there exists another nonmanipulable protocol that will lead to the same outcome (Revelation Principle)Manipulation in voting“Gore”“Bush” “Gore”“Bush”Nader > Gore > Bush• Voting truthfully (for Nader) might let Bush win, certainly will not get Nader to win• So, better rank Gore first“Gore”MANIPULATION!!!Some well-known protocols•Plurality: candidate with the most votes wins•Borda: candidate gets m-1 points for each vote, but also m-2 points for each second place in a vote, m-3 for each third place, …•Maximin:–From the complete rankings, for each pair of candidates, we can deduce how each voter would have voted with only these two candidates–This defines (m choose 2) “pairwise elections”–In Maximin, winner is the candidate with the best score in her worst pairwise•Single Transferable Vote (STV)–Each round, candidate with fewest votes drops out–When your candidate drops out, your vote transfers to your next most preferred (remaining) candidate–Continue until one candidate remains•Note: now our voter can safely vote for Nader, then let the vote transfer to Gore•Still manipulable in other casesSeminal result (Gibbard-Satterthwaite): all nondictatorial voting protocols with >2 candidates are manipulable!Software agents may manipulate more•Human voters may not manipulate because:–Do not consider the option of manipulation–Insufficient understanding of the manipulability of the protocol–Manipulation algorithms may be too tedious to run by hand•For software agents, voting algorithms must be coded explicitly–Rational strategic algorithms are preferred–The (strategic) voting algorithm needs to be coded only once–Software agents are good at running algorithmsKey idea: use computational complexity as a barrier to manipulation!Computational manipulation problem •The simplest version of the manipulation problem (defined relative to a protocol):•CONSTRUCTIVE-MANIPULATION. We are given the (unweighted) votes of the other candidates, and a candidate c. We are asked if we can cast our (single) vote to make c win.•E.g. for the Borda protocol:•Voter 1 votes A > B > C; Voter 2, B > A > C; Voter 3, C > A > B•Borda scores are now: A: 4, B: 3, C: 2•Can we make B win?•Answer: YES. Vote B > C > A (Borda scores: A: 4, B: 5, C: 3)•Utility-theoretically, the special case where the manipulator has utility 1 for c and 0 for everyone elsePrior research•Theorem. CONSTRUCTIVE-MANIPULATION is NP-complete for the second-order Copeland protocol. [Bartholdi, Tovey, Trick 1989]•Theorem. CONSTRUCTIVE-MANIPULATION is NP-complete for the STV protocol. [Bartholdi, Orlin 1991]•All the other protocols are easy to manipulate (in P)Universal voting protocol tweaks to make manipulation hard[Conitzer, Sandholm IJCAI-2003]“Tweaks” for protocols•Hardness of manipulation is one factor in choosing a voting protocol, but…•Many existing protocols have other nice properties–Each attempts to maximize a certain notion of welfare•It would be nice to be able to tweak protocols:–Change the protocol slightly so that•Hardness of manipulation is increased (significantly)•(Most of) the original protocol’s properties still hold•It would also be nice to have a single, universal tweak for all (or many) protocols•A preround turns out to be such a tweak!•And it introduces hardness far beyond previous resultsAdding a preround to the protocol•A preround proceeds as follows:–Pair the candidates–Each candidate faces its opponent in a pairwise election–The winners proceed to the original protocolOriginal protocolPreround example (with Borda)Voter 1: A>B>C>D>E>FVoter 2: D>E>F>A>B>CVoter 3: F>D>B>E>C>AA gets 2 pointsF gets 3 pointsD gets 4 points and wins!Voter 1: A>D>FVoter 2: D>F>AVoter 3: F>D>AA vs B: A ranked higher by 1,2C vs F: F ranked higher by 2,3D vs E: D ranked higher by allMatch A with BMatch C with FMatch D with E STEP 1:A. Collect votes and B. Match candidates (no order required) STEP 2:Determine winners of preround


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