# VOTING TO ELECT A SINGLE CANDIDATE Single-Winner Elections.

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VOTING TO ELECT A SINGLE CANDIDATE Single-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 voting

Kenneth 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 Rule

Strategyproofness 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 Profiles With 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 34 1st pref.LabourLiberalConserv. 2nd pref.LiberalConserv.Liberal 3rd pref.Conserv.LabourLabour Simple Plurality Voting (SPV) [“first past the post” or FPTP] Plurality Ranking Candidates Votes Received (= First Preferences) Labour46 votes (winner) Conservative34 votes Liberal20 votes Assuming “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 Voting

Borda Point Voting Votes rank the candidates on the ballot. Candidates are awarded three points for each ballot on which they are ranked first, two points for each ballot on which they are ranked second, and one point for each ballot on which they are ranked third (if k = 3). –In general, m points, m-1 points, etc., when there are m candidates. Borda Ranking for Profile 1: Liberal220 points (winner) Labour192 points Conservative188 points

Condorcet Voting Votes rank the candidates on the ballot. Examine all pairs of candidates and see who wins each straight fight. For Profile 1: –Liberal vs. Conservative: Liberal wins by 66-34 –Conservative vs. Labour: Conservative wins by 54-46 –Liberal vs. Labour: Liberal wins by 54-46 Assemble the Majority (or Condorcet) Ranking –1 st pref.Liberal (Condorcet Winner) –2 nd pref.Conservative –3 rd pref.Labour (Condorcet Loser) Precisely opposite of Plurality Ranking (based on 1 st prefs. only) and different from Borda Ranking (also based on full rankings)

Condorcet Winners A Condorcet winner is a candidate who can beat every other candidate in a straight fight. A Condorcet loser is a candidate who is beaten by every other candidate in a straight fight. A majority winner is always a Condorcet winner, but the reverse is not true. A plurality winner may not be a Condorcet winner. A Condorcet winner may not be a plurality winner. –Indeed, a Condorcet winner may have the fewest first preferences (e.g., Liberal in Profile 1).

Condorcet Voting (cont.) Moreover, Condorcet Voting appears to be: –Strategyproof, because it is built on Simple Majority Rule, which is strategyproof; and –Spoilerproof, because (for example), regard- less of whether Nader is a candidate or not, Bush is the Condorcet winner only if he beats Gore in a straight fight. So what about “Arrow’s Theorem”?

Problem: There may be no Condorcet Winner This assertion seems puzzling because every ranking has a highest ranked element. But there may be no Majority Ranking.

Preference Profile 2 # of voters 46 20 34 1 st pref.LabourLiberalConserv. 2 nd pref.LiberalConserv.Labour 3 rd pref.Conserv.LabourLiberal First preferences are unchanged from Profile 1, so the plurality winner is unchanged. As before, there is no majority winner. Conservative remains the Plurality Runoff winner. Labour becomes the Borda point winner. Approval voting remains indeterminate.

Cyclical Majorities Re-examine the straight fights: Liberal vs. ConservativeLiberal wins by 66-34 Conservative vs. LabourConservative wins by 54-46 Labour vs. LiberalLabour wins by 80-20 There is no majority ranking, no Condorcet Winner, and no Condorcet Loser. Instead we have a cyclical majority that behaves like the “even stranger” ice cream customer. In general, Condorcet Winners may not exist and Condorcet Voting is not a proper voting system, so Arrow’s Theorem stands.

Preference Profile 3 # of voters 35 33 32 1 st pref.BCD 2 nd pref.AAA 3 rd pref.CD B 4 th pref.DBC Example of a socially divided society (e.g., Shiites, Sunnis, and Kurds in Iraq); Candidate A is the “alliance” (cross-community) candidate. Candidate A is the Condorcet Winner even though A has no first preferences. Candidate A is the Condorcet Winner even though the is a majority cycle among the other candidates. Side point: If A, B, C, and D are parties, the “Alliance Party” A wins no seats under list-PR.

Condorcet Consistency A voting rule is Condorcet consistent if, given sincere voting, it always selects the Condorcet winner when one exists. Previous examples showed that, given Profile 1, Liberal may fail to win under each of the other voting rules discussed, so none of them is Condorcet consistent. Condorcet voting is obviously Condorcet consistent but, since it is does not always select a winner, it cannot be deemed a full-fledged voting rule.

Parliamentary (Yes/No) Voting There are options A, B, C, D, …. Voting is by a sequence of binary votes taken in some fixed order (say alphabetical): –(1)A vs. B –(2)winner of (1) vs. C –(3)and so forth If there is a Condorcet Winner, it can never be knocked out, so CW wins regardless of the voting order. If there is a majority cycle and no CW, winner is determined by the voting order (later entry is advantageous).

Single-Peaked Preferences To say that majority cycles may exist is not to say that they typically are present. Indeed, if preferences are structured in a simple way by ideology (or otherwise), cycles cannot occur. In British politics, the three major parties are generally perceived to be ideologically ranked from left to right in the following manner: More leftwing:Labour Relatively centrist: Liberal More rightwing:Conservative

Single-Peaked Preferences (cont.) “Admissible” Orderings“Inadmissible” Leftwingers Centrists Rightwingers Orderings 1 st pref.LabLibLibConConLab 2 nd pref.LibLabConLibLabCon 3 rd pref.ConConLabLabLibLib Why the name “single-peaked”? In diagrams below, L = leftist, C = centrist, and R = rightist

Single-Peaked Preferences (cont.) Note the strength of the “centrist” (Liberal) candidate in the admissible orderings. –While it may be that few voters most prefer the centrist, no one likes the centrist least. –The centrist candidate must be the Condorcet Winner unless a majority of voters have the leftwing or rightwing ordering. –In other words, the centrist candidate fails to be the Condorcet Winner only if one of the “extreme” candidates is a majority winner. In the general case, if all voters can be ranked from most leftwing to most rightwing with respect to their first preferences, –no cyclical majority occurs; –some position on the ideological spectrum is the Condorcet Winner; and –that CW position corresponds to the first preference of the median voter, such that no more than half the voters are more leftwing and no more than half are more rightwing (Duncan Black, Theory of Committees and Elections. The Hotelling-Downs theory of electoral competition (to be discussed later) states that two competing vote-seeking parties or candidates achieve equilibrium only when both adopt the position that corresponds to the first preference of the median voter.

Strategic Voting under Plurality Rule Any voting rule with three or more candidates may give some voters incentives to vote otherwise than sincerely. Consider Profile 1 again. # of voters 46 20 34 1st pref.LabourLiberalConserv. 2nd pref.LiberalConserv.Liberal 3rd pref.Conserv.LabourLabour –Labour wins under Plurality Voting if voters are sincere. –But a majority of 54 voters prefer both other candidates to Labour. –If they all vote for the same other candidate, that candidate wins, an outcome they all prefer to a Labour victory. –But doing this requires some voters among the 54 to vote “insincerely,” i.e., for their second preferences. –Thus simple Plurality Voting can encourage what the British call tactical voting and most political scientists call strategic voting, i.e., non-sincere voting.

Strategic Voting under Plurality Rule (cont.) But how will the 54 voter majority coordinate their votes? Will they all vote for Liberal or for Conservative? –While all 54 voters prefer to see Labour defeated, they disagree as to how to defeat him, i.e., by voting Conservative or by voting Liberal. –It is generally believed that, in practice, tactical voting in Britain mostly leads Liberal supporters to shift their votes “tactically” to their second- preference (Labour or Conservative) candidate, because they typically observe pre-election polls showing Liberal trailing well behind both other candidates, and they therefore conclude that a Liberal vote is “wasted” and that they should vote for the one of the two leading (non-Liberal) candidates that they prefer. –This can happen even though Liberal is the Condorcet winner, reflecting the fact that polls (almost always) ask only about first preferences and Liberal's great strength lies in second preferences. –If Liberal supporters find Labour and Conservative to be equally objectionable, they have no incentive to vote tactically. –If pre-election polls show something close to a tie for second place (or a three-way tie), tactical voting becomes far more conjectural.

Strategic Voting under Runoff [or IRV] Preference Profile 1 # of voters 46 20 34 1st pref.LabourLiberalConserv. 2nd pref.LiberalConserv.Liberal 3rd pref.Conserv.LabourLabour Under Plurality Runoff [or IRV], the 46 voters who most prefer Labour would do better by ranking Liberal first, as this assures a Liberal victory (without a runoff), which outcome they prefer to the Conservative victory that otherwise results. Moreover, other voters have no countermoves available. No strategic vote is available to 34 voters whose first preference is Conservative, and the 20 voters whose first preference is Liberal are already getting their first prefer- ence.

Strategic Voting Under Borda Point Voting Given Profile 1, no voters can change their Borda score ballots in a way that improves the outcome for them. Profile 2 # of voters 46 20 34 1 st pref.LabourLiberalConserv. 2 nd pref.LiberalConserv.Labour 3 rd pref.Conserv.LabourLiberal Given Profile 2, if the bloc of 20 ranks Conservative first and the bloc of 34 ranks Labour third, then Conservative gets the most Borda points (208 vs. 200 for Liberal and 192 for Labour), an outcome all 54 such voters prefer to victory by the sincere Borda winner Labour.

Strategic Voting Under Borda (cont.) Preference Profile 3 [4 in Handout] 46 54 1 st pref.LabourConservative 2 nd pref.Liberal Labour 3 rd pref.Conservative Liberal Labour wins if voting is sincere (demonstrating that Borda Point Voting can deny victory to a majority winner). But the 54 Conservative-preferring voters can elect Conservative if they (“insincerely”) push Labour down to third place on their ballots. In turn, the 46 Labour-preferring voters can counteract this by pushing Liberal to the top of their ballots (the resulting Liberal victory being preferable to Conservative). Note that if strategic manipulation stops at this point (though it need not), Liberal is elected even though everyone prefers Labor to Liberal. And things can get worse.

Borda Strategy: “Turkey Raising” There are three candidates: a more or less reasonable Democrat D, a more or less reasonable Republican R, and a real “turkey” T. Everyone one ranks T last, except two deranged T supporters. The profile is: 50 voters48 voters 1 voter 1 voter D R T T R D D R T T R D If everyone votes sincerely, the Borda Point totals are D=249, R=247, and T=104. Anticipating this defeat, Republican voters caucus and notice an interesting feature of Borda Point Voting — it can pay voters to engage in “turkey raising,” i.e., to strategically raise the “turkey” in their ballot rankings, so as to push the rival “serious” candidate down in their rankings and increase the point spread between the two.

“Turkey Raising” (cont.) Suppose the Republicans strategically modify all their ballots so as to produce the following ballot profile: 50 voters48 voters 1 voter 1 voter D R T T R T D R T D R D The point totals would then be D=201, R=247, and T=152. Democrats also notice this feature of Borda Voting and, concerned that Republicans may engage in turkey raising, they engage in some turkey raising of their own in order to counteract the anticipated Republican stratagem. So the final ballot profile is: 50 voters48 voters 1 voter 1 voter D R T T T T D R R D R D The final point scores are D = 201, R = 197, and T = 202. May the best turkey win!

Spoiler Effects A sincere electorate using Plurality Voting may behave exactly the confused ice cream customer. Preference Profile 1 # of voters 46 20 34 1st pref.LabourLiberalConserv. 2nd pref.LiberalConserv.Liberal 3rd pref.Conserv.LabourLabour Conservative beats Labour in a straight fight, but Labour wins if Liberal enters the election. So Liberal is a spoiler to Conservative. Likewise, Liberal beats Labour in a straight fight, but Labour wins if Conservative enters the election. So Conservative is also a spoiler to Liberal. Note that the effect of the spoiler’s entry is to elect the last preference of the spoiler’s supporters, so the spoiler’s entry is self-defeating. Note that, in this profile, Labour cannot be a spoiler, because if Labour enters what had been a Liberal-Conservative straight fight, Labour wins.

Spoiler Effects (cont.) Spoiler effects may be mitigated by strategic voting. Duverger’s Law: Single-Winner elections with Simple Plurality Voting tend to produce two-party systems and straight fights Strategic voting to mitigate spoiler effects is one factor that drives Duverger’s Law. –“Don’t waste your vote on a hopeless third candidate.” Candidate “entry deterrence” is the other, probably more important, factor that drives Duverger’s Law. –“Don’t run as a third candidate, you will spoil the chances of the preferable of the two major candidates.” Multi-Winner elections [no elected executives] with any variant of Proportional Representation [PR] sustains multiparty systems. [Other half of Duverger’s Law]

Spoiler Effects under IRV Plurality Runoff/IRV is sometimes advocated on the grounds that it precludes the spoiler effects that characterize Simple Plurality. Plurality Runoff/IRV is an improvement over Simple Plurality in this respect. –Under IRV, a single third candidate (such as Nader) with little first-preference support cannot act as a spoiler in what is essentially a straight fight between two major candidates, because the runoff will become precisely that straight fight. However, if there are many candidates and/or first preferences are dispersed, Plurality Runoff/IRV is subject to spoiler effects.

Two Variants of IRV There are (at least) variants of IRV. –They are equivalent if there are just three candidates but are distinct (and may produce different winners) if there are more than three candidates. The first variant mimics Plurality Runoff. –If there is no majority winner, all candidates except the leading and second-place candidate in the plurality ranking are simultaneously eliminated. –The ballots of all eliminated candidates are trans- ferred to one or other surviving candidate on the basis of lower preferences. –The surviving candidate with the most (original plus transferred) ballots is elected.

Two Variants of IRV (cont.) The second variant is distinct from plurality runoff. –If there is no majority winner, the candidate with the fewest first preference ballots is eliminated. –The ballots of the eliminated candidate are trans- ferred to one of the surviving candidate on the basis of second preferences. –If there is still no majority winner, the surviving candi- date with the fewest (original plus transferred) ballots is eliminated and the ballots are transferred on the basis of second or lower preferences. –And so forth until there is a majority winner (which necessarily occurs once all candidates but two have been eliminated.

Two Variants of IRV (cont.) The second variant of IRV is also called the Alternative Vote. –It is the single-winner special case of a multi-winner voting system called the Single Transferable Vote (STV). –STV is used in small multi-member districts (MMDs). –It produces approximately proportional representation of groups among the winners, even if those groups are not political parties. contrast with List-PR –STV is used to elect: the Irish Dail (Parliament) the Australian House of Representative the Cambridge MA City Council

Spoiler effects under Plurality Runoff (and 1st IRV variant) When there are many candidates, the Plurality Runoff and the first variant of IRV are subject to spoiler effects in precisely the same manner as Simple Plurality Voting, with respect to the question of which two candidates will survive into the runoff. This is illustrated by the 2002 French Presidential election (“Two Round” or Plurality Runoff).

Left splinter candidates “spoiled” Jospin’s chance to get into the second round runoff against Chirac. However, if the second [AV] variant of IRV had been in used: –ballots for left splinter candidates would have (presumably) transferred to Jospin; –few ballot would have transferred to Le Pen; and –Jospin would have gotten into the runoff and might have won.

Would Runoff/IRV Lead to a Proliferation of Presidential Candidates?

Spoiler Effects under 2 nd IRV Variant (AV) However, spoiler effects also exist under the second IRV variant. This is illustrated by Preference Profile 1 with just three candidates (in which case the two variants are equivalent): # of voters 46 20 34 1st pref.LabourLiberalConserv. 2nd pref.LiberalConserv.Liberal 3rd pref.Conserv.LabourLabour Liberal wins a straight fight with Conservative, but Liberal does not even make it into the runoff if Labour enters the field. So Labour is a spoiler to Liberal under IRV. This is not a distinctive flaw in Plurality Runoff (or IRV) because, as previously noted, the problem is unavoid- able with three or more candidates.

Negative Responsiveness under Plurality Runoff (or IRV) Plurality Runoff (or IRV) has another flaw that is distinctive (and avoidable). Plurality Runoff (or IRV) can respond negatively when a candidate’s position in a preference (or ballot) profile becomes more favorable — put otherwise, it can punish a candidate for gaining more support. –This sometimes called “monotonicity failure.” –It is a violation of May’s Non-Negative Responsive- ness condition

Negative Responsiveness under Plurality Runoff or IRV (cont.) Original Preference Profile 4 Revised Preference Profile 4 35 10 25 30 35 10 25 30 A B B C A A B C B A C A B B C A C C A B C C A B Prior to the election, a poll indicates that voting intentions those shown in Original Preference Profile 4. Candidate A is pleased with this information, because it projects that A and B will go into the runoff, which A will win. However, A doesn’t want to take any chances and urges his supporters to go out and drum up still more support for A. A’s supporters are successful, transforming the original profile into the revised profile, in which 10 voters who previously preferred B to A now prefer A to B. (No other preferences have changed.) As a result of this successful campaign, A loses the election.

IRV and the “No Show Paradox” Here is a related peculiarity of Plurality Runoff [or IRV]. Preference Profile 5 564[2] BCA[A] CBB[B] AAC[C] The preference profile is as shown above, but the two individuals with the bracketed preference orderings fail to vote. –The election outcome is determined by the remaining 15 voters. Candidates B and C are paired in a runoff, which B wins. –This is somewhat disappointing for the two individuals who failed to vote, in that their second preference won. –They regret their failure to get to the polls and wonder whether their first preference A might have won if they had not failed to vote. –It can be checked that, if they had gotten to the polls and voted according to their preferences, C would have won (so the outcome would have been worse, not better, for them).

“Clone” Candidates When I was 12 years old I was nominated to be treasurer of my class at school. A girl named Michelle was also nominated. I relished the prospect of being treasurer, so I made a quick calculation and nomi- nated Michelle’s best friend, Charlotte. In the ensuing election, I received 13 votes, Michelle received 12, and Charlotte received 11, so I became treasurer.” T.N. Tideman Consider the following preference profile, in which a Republican minority is united behind a single candidate R but the Democratic majority is split between the two “clone” candidates D1 and D2. Preference Profile 6 Democrats Republicans 35%25%25%15% D1 D2 R R D2 D1 D1 D2 R R D2 D1

Clone Candidates (cont.) Clones are candidates who very similar in terms of their ideology, issue positions, etc. and are therefore adjacent in all voters’ preference rankings. Simple Plurality is notorious for penalizing clone candi- dates. –In this case, the Republican candidate wins due to the Demo- cratic split, even though R is the Condorcet loser and would be beaten by both D1 and D2 in straight fights. –D1 and D2 are spoilers against each other. It is precisely the expectation of such outcomes under Simple Plurality voting that leads to party formation and party discipline. –The Democrats [and the girls] have a huge incentive to hold a prior nominating convention or primary to choose between D1 and D2 and then send just one of the two clones forward against the Republican. –Given the preference profile above, D1 would win the nomination and then the general election.

Plurality Runoff [or IRV] and Clones Are there voting rules that can reduce, eliminate, or even reverse the self-defeating effect of run- ning clone candidates? Given Profile 6, Plurality Runoff (instant or other- wise) solves the clone problem. –The first-round election functions as the (Democratic) “primary” (except that Republicans also vote in this primary). –The runoff functions as the general election in which the Democratic majority gets its way. If there are four or more candidates, Plurality Runoff [or IRV] does not treat clones so well and, as we have seen, it is subject to other problems in addition.

Approval Voting and Clones Brams and Fishburn advocate Approval Voting as a desirable a voting rule that (among other things) does not punish clones. –In Profile 6, presumably (almost all) Democrats would vote for both D1 and D2, one of whom would be elected. –By not penalizing clones, AV does not encourage party formation or party unity. –For this reason, many political scientists are more inclined to support AV for primary elections and non- partisan elections than for partisan general elections.

Double-Vote List-PR and Clones A variation of one type of party-list PR (Propor- tional Representation) system is another voting method that does not penalize clones who have the same party affiliation. –In a district that elects m candidates, each voter votes for m candidate, but these vote counts in two ways: first, as a party vote to determine which party wins the election; and second, as a candidate vote to determine which candidate(s) of the winning party is elected. In the profile above (with m = 1), D1 would be elected.

Borda Point Voting and Clones Perhaps surprisingly, Borda Point Voting actually rewards the running of clones. Suppose that there are two candidates and Republicans are again in the minority. Preference Profile 7A 60 voters 40 voters D R1 R1 D With just two candidates, the Borda point rule is identical to Plurality Voting (and SMR), so the Republican candidate R1 loses.

Borda Point Voting and Clones (cont.) But, if Borda Voting is in use, the Republicans can reverse this outcome by nominating an additional clone candidate R2 whom everyone sees as identical to R1 with respect to issues and ideology but inferior with respect to (let’s say) personal qualities. Preference Profile 8A 60 voters 40 voters D R1 R1 R2 R2 D Democrats can counteract this Republican stratagem by strategi- cally ranking R2 above R1. Alternatively, they can counteract it by running their own clone. Borda voting is highly susceptible to strategic maneuvers of this sort (which, moreover, have the effect of expanding the candidate field rather than winnowing it down in the manner of Plurality Rule). Also recall “turkey raising” under Borda

Overall Conclusion Two-candidate elections (“straight fights”) are clean and simple. Elections with three or more candidates are complex and inevitably quirky.

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