VOTING TO ELECT A SINGLE CANDIDATESingle-Winner ElectionsBinary VotingKenneth May’s TheoremMay’s Conditions (Reformulated)May’s Conditions (cont.)StrategyproofnessThree or More CandidatesPreference ProfilesPreference Profile 1Plurality vs. Majority WinnerPlurality Plus Runoff (or Instant Runoff Voting [IRV])Approval VotingBorda Point VotingCondorcet VotingCondorcet WinnersCondorcet Voting (cont.)Problem: There may be no Condorcet WinnerPreference Profile 2Cyclical MajoritiesPreference Profile 3Condorcet ConsistencyParliamentary (Yes/No) VotingSingle-Peaked PreferencesSingle-Peaked Preferences (cont.)Slide 26Strategic Voting under Plurality RuleStrategic Voting under Plurality Rule (cont.)Strategic Voting under Runoff [or IRV]Strategic Voting Under Borda Point VotingStrategic Voting Under Borda (cont.)Borda Strategy: “Turkey Raising”“Turkey Raising” (cont.)Spoiler EffectsSpoiler Effects (cont.)Spoiler Effects under IRVTwo Variants of IRVTwo Variants of IRV (cont.)Slide 39Spoiler effects under Plurality Runoff (and 1st IRV variant)Slide 41Would Runoff/IRV Lead to a Proliferation of Presidential Candidates?Spoiler Effects under 2nd IRV Variant (AV)Negative Responsiveness under Plurality Runoff (or IRV)Negative Responsiveness under Plurality Runoff or IRV (cont.)IRV and the “No Show Paradox”“Clone” CandidatesClone Candidates (cont.)Plurality Runoff [or IRV] and ClonesApproval Voting and ClonesDouble-Vote List-PR and ClonesBorda Point Voting and ClonesBorda Point Voting and Clones (cont.)Overall ConclusionVOTING TO ELECT A SINGLE CANDIDATESingle-Winner ElectionsSingle-Winner Elections•Election of a [unitary] executive office.•Election of a representative from a single-member district [SMD]•Election of slate Presidential electors running on a (“winner-take-all”) general ticket/slate.Binary Voting•Just two options (so one winner and just one loser)–Single-winner elections with just two candidates•“straight fight” [British terminology] –Yes/No voting•referendums•parliamentary votingKenneth May’s Theorem•Choices for voters in straight fight between A and B–vote for A –vote for B–abstain (if indifferent)•Possible voting outcomes–A is the winner–B is the winner–deadlock (no winner, “hung electorate”)May’s Conditions (Reformulated)•Anonymity (of voters)•Neutrality (between options A and B)•Resoluteness (no deadlock)•Preliminary Theorem: No voting system can be simultaneously Anonymous, Neutral, and Resolute.•All voting systems considered here are Anonymous and Neutral.May’s Conditions (cont.)•Almost Resoluteness: any deadlock is broken by any voter changing his vote.•Non-Negative Responsiveness: votes don’t count negatively.•May’s Theorem: Anonymity + Neutrality + Almost Resoluteness + Non-Negative Responsiveness <===> Simple (Relative) Majority RuleStrategyproofness•Moreover, in a straight fight SMR is strategyproof–That is, no voter can ever improve the outcome with respect to his or her true preferences by misreporting those preferences on a ballot.–Voters will never regret voting “sincerely” or “honestly.”–Sincere voting is Nash Equilibrium.•However, in this respect SMR is not unusual; in fact all non-negatively responsive voting systems are strategy-proof in a straight fight.Three or More Candidates•Once the number k of options/candidates exceeds two, all sorts of problems arise.–Different (“reasonable looking”) voting procedures, all satisfying “May-like” conditions and equivalent to SMR in a straight fight, may produce different winners from the same “preference profile.”–No voting procedure is strategyproof.–No voting procedure is spoilerproof.•This is essentially “Arrow’s Theorem.”•One rationale for a two-party system is that it typically produces (something very close to) straight fights and therefore precludes these problems.Preference ProfilesWith more than two candidates (k > 2), voters’ preferences cannot be specified simply by listing their most preferred (top-ranked) candidates; rather we must specify voters’ full preference ordering over all candidates (like the ice cream flavors).A collection of preference orderings, one for each voter is called a preference profile.We use British party labels to identify three candidates — Labour, Liberal, and Conservative — one of whom is to be elected. While there are six possible orderings of three candidates, we first consider a simple profile in which only three of these orderings are present and we indicate the popularity of each.We assume all have strict orderings (no indifference) and there are no ties.Preference Profile 1# of voters 46 20 341st pref. Labour LiberalConserv.2nd pref. LiberalConserv. Liberal3rd pref. Conserv. Labour LabourSimple Plurality Voting (SPV) [“first past the post” or FPTP] Plurality RankingCandidates Votes Received (= First Preferences)Labour 46 votes (winner)Conservative 34 votesLiberal 20 votesAssuming “sincere” voting, SPV takes account of first preferences only.Plurality vs. Majority Winner•Given a preference profile with k > 2:–A plurality winner is a candidate who has more first preferences than any other candidate.–A plurality winner always exists [in the absence of ties].–A majority winner is a candidate who is the first preference of a majority of voters.–Every majority winner is also a plurality winner, but the reverse is not true.–There may not be a majority winner.Plurality Plus Runoff (or Instant Runoff Voting [IRV]) •If there is no majority winner, there is a runoff between the top two candidates in the plurality ordering.•Under IRV, Profile 1 would produce a runoff between Labour and Conservative [first and second in the plurality ranking], which Conservative would win.Approval Voting•Voters can vote for (“approve of”) more than one candidate.•The candidate with the most approval wins.•Approval voting outcomes are indeterminate. Presumably a voter casts an approval vote for A and not for B only if the voter’s preference ordering ranks A over B [“no skipping”].•In Profile 1–Labour wins if all voters cast only one vote.–Conservative wins if only voters in the middle bloc cast two votes–Liberal wins if all voters in the right bloc cast two votes or if all voters cast two votes.•S. Brams and P. Fishburn, Approval VotingBorda Point Voting•Votes rank the candidates on the