Other One-Winner Rules

Plurality rules Majority rules

Condorcet completion Other rules

Plurality Rules

Single-vote plurality is the oldest and most often used voting system.  Each voter gets a single vote which he can give to one candidate.  The candidate who gets the most votes, a plurality, wins.  In multi-candidate races the winner often gets less than a majority, less than 50% of the votes.  It is often called first-past-the-post (FPTP).

Approval voting was first promoted in the 1970's.  It has since been adopted by several professional societies in the United States. It lets a voter give one vote to each candidate.  Brams (1979) suggests each voter cast an approval for one of the top two candidates and as many minor candidates as he rates above that one. The candidate with the most approvals wins.  Note that a majority is not required. Approval Voting is most useful for making non-competitive group decisions. For example, there is little incentive for tactical voting when a choir votes to schedule an extra rehearsal. Members briefly discuss suggested rehearsal times then raise their hands to show whether they are available as each time is offered. A voter may raise his hand for several time slots. The time with the most approvals wins.

Some arguments against AV:
The fundamental problem with AV is that it forces voters to choose from several strategies: Vote for the ones you really like. Vote for the ones you feel OK about. Vote for one of the two front runner and everyone you like better than that one. (The pre-election surveys strongly effect the outcome.) Voters must worry over this choice because the strategies they choose can change the results as the research below showed. Fundamentals of Voting Theory Illustrated with the 1992 Election or Why the Libertarian Party should favor Approval Voting, by Alexander Tabarrok (Dept. of Economics, Ball State University): available on-line

“Abstract: Different voting systems can lead to different election outcomes even when voter preferences are held constant. Using the 1992 election as an example, it is shown how the outcome of every positional vote system can be found. Similarly, every possible cumulative and approval vote outcome is shown. Multiple voting systems, like approval voting and cumulative voting have highly disturbing properties. Using the 1992 election as illustration, it is shown how a candidate who wins under each of the infinite number of positional vote systems, who wins every pairwise vote (i.e. is the Condorcet winner), and who has the most first place and least last place votes may nevertheless lose under approval or cumulative voting. Similarly, it is shown how a candidate who loses under each of the infinite number of positional systems, who loses every pairwise vote (i.e. is the Condorcet loser), and who has the least first place and most last place votes may nevertheless win under approval or cumulative voting.”

The major site advocating approval voting: Site

The system of "counts" created by Jean-Charles de Borda in 1781 gives a candidate points for each rank voted.  A first-rank vote gives points equal to the number of candidates minus one.  A second rank gets vote gives points equal to the number of candidates minus two and so on.  The candidate who gets the most points wins.
This would elect a winner with a high &ldqou;utility value.&rdqou;  But it is easily susceptible to tactical voting.  So it has little use in political situations including most private organizations.  It often causes controversy when by judges in sports competitions. Site

Majority Rules

The runoff system starts with a single-vote plurality election.  The two candidates with the most votes go on to a new campaign and a one-on-one election.  Instant Runoff Voting, IRV speeds the process by asking voters for ranked preferences so a runoff can be tallied without taking a second poll.  This saves money and increases turnout.  IRV usually elects the same person a runoff would; when they differ the runoff winner would lose a 1-on-1 election to the IRV winner.

The winner is usually the candidate who is popular with the core voters of the largest moderate party.

STV3Con, STV Then Condorcet: Condorcet's rule risks electing an unknown.  If the voters are polarized, they may give first choice to a favorite, last to her main rival, and rank many candidates in between -- not thinking how bad or bizarre an unknown might be in office.  This risk could be reduced by using STV to select the [3] candidates with most first- and second-place votes, then testing them 1 against 1 to elect a winner.

Unfortunately STV3 usually picks a triangle of reps, none of whom is near the center.  So a central candidate would be squeezed out by STV3 even if she is broadly (not intensely) popular.

STV5 is more likely to pick 1 candidate near the center -- who is then sure to win the Condorcet runoff.  The winning strategy is to be the 1 most popular with the central voters.  Breadth of support is not important; so this rule more like IRV than Condorcet.

STV7 might select 2 central candidates, and the broader 1 will win the Condorcet comparisons.  But if the top 2 candidates gather 80% of the first-place votes, the other 5 selected might be political weaklings and if 1 of them wins, would she have a mandate to govern, a solid base of support, or the respect of the legislature?

Perhaps a threshold rule should simply require each candidate to win 10 to 20 percent of the first-place votes in order to enter the Condorcet pairwise comparisons.  (The next chapter explains that rules for proportional representation commonly use lower thresholds because small parties are needed to create councils with full representation.)

Condorcet Completion Rules

The IRV page showed that in some elections more than one candidate can claim to win a majority -- the question then is who wins the strongest majorities.

Pairwise- or Condorcet-completion rules all give the same result in most elections.  They differ only when there is no Pairwise winner due to a voting cycle such as C>B>D>C.  Each completion rule is a way to resolve a voting cycle.  These may be evaluated on their ability to resist manipulation.

Duncan Black's 1958 rule elects the Condorcet winner if 1 exists; otherwise it elects the Borda winner.  It is the best completion rule for electing the "utility maximizing" option, if there is no manipulation.

Clyde Coombs' 1954 alternative vote, like Hare's, eliminates candidates until one gets a majority.  But it eliminates the candidate with the most last-place votes.

A. H. Copeland's 1950 rule gives a candidate 1 point for winning a pairwise contest against another candidate and -1 for losing. (In voting cycles, Copeland often produces ties - so it does not "complete" Condorcet.)

Mathematician Charles Ludwidge Dodgson (better known as author Lewis Carroll) proposed in 1876 to elect the Condorcet winner or, in the event of a cycle, the candidate who needs to change the fewest ballots to become the Condorcet winner.

John Kemeny's 1959 system determines how many rank pairs must be exchanged (flipped) on voters' ballots to make a candidate win by Condorcet's rule.  The candidate who requires the fewest changes wins.  The Kemeny distance between two preference orders is the number of adjacent pairwise switches needed to convert one preference order to the other.

The max-min system elects the candidate with the smallest pairwise loss.  (It is not the same as Dodgson.  A candidate may lose pairwise elections to two rivals by 5% each.  Her max-min score would be -5%.  But she might have to change 10% of the ballots to become Dodgson's winner.)

Nicholas Tideman's Ranked Pairs rule creates a complete ranking of the candidates from first to last.  Their ranks come from majority preferences between options: The biggest margin of victory in the Pairwise table is locked in, say C > B.  Then the second biggest victory is locked in, say B > D.  And then the third, as long as it does not create a voting cycle.  In this case, D > C would be ignored because that would say C>B>D>C.  The rule considers a big margin of votes (and voters) more certain and forceful than a small margin.Ranked Pairs

Markus Schulze's rule is similar to Tideman's but it takes only the candidates in the voting cycle, and then drops the weakest defeats (not strongest victories) one at a time until the voting cycle is resolved.

The last two rules are most resistant to manipulation.

Other Rules

In Samuel Merrill's 1988 standard-score system, voters rate candidates on a fixed scale, say 0 to 100.  It then makes each voter's ratings average zero (some ratings become negative).  It also "normalizes" the variation within a voter's ballot.  This keeps any one voter from spreading out his ratings to influence the election more than others voters.  ["...for each voter separately, replace her ratings r_i by their statistical standard scores, i.e., σ ∑

Books by Samuel Merrill and Phillip Straffin more fully explain these and less common single-winner rules.  Detailed definitions and discussions are also online at Lorrie Faith Crannor's site advocating Declared Strategy Voting rules and Blake Cretney's site for Tideman's Ranked Pairs rule and in ACE Project.  Prof. Han Dorussen includes examples with questions and answers in Rules of the Game for his course on Public Choice Theory.

Utility voting systems such as Clark's incentive revealing device or Hylland and Zeckhauser's influence point voting are not Condorcet completions rules, are easy to manipulable by "conspiracies", and are rather complicated so their ballots may confuse and burden voters.  They do not fit majority rule's one person / one vote.  Hopefully they can be adapted for groups seeking proportional outcomes for all parties but they were not designed for general elections.

A dozen plurality, majority and Condorcet voting rules have been researched by several authors to contrast and compare one-winner voting systems -- the topic of the next page.