Other One-Winner Rules

Plurality rules Majority rules

Condorcet completion Other rules

This list starts with methods that are common and appear simple at first but which are complex in the strategies voters need to worry about.  The list of Condorcet-completion rules ends with three which are hard to manipulate and so relatively strategy free and reliably central.  They are also inclusive in the sense that all voters have equal power in the final tally and there are few wasted votes.

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.  Several professional societies in the United States have used it briefly. 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: Strategic voters will be given a significant advantage over voters who have less information.

Approval Voting has a very basic problem for voters: Marking a second choice might help it defeat your first choice. FairVote has analyzed this problem. short version, full version, pdf version.

A further problem with AV is that it forces voters to choose from several strategies:
Play it safe, protect your first choice and mark only your first choice.
Vote for all the ones you really like.
Vote for all the ones you feel okay about.
Vote for all the ones you think are above average.
Vote for one of the two front runners and everyone you like better than that one. Thus pre-election surveys would strongly effect the outcome.

Voters must worry over this choice because the strategies they choose can change the results. As the research below showed, they could elect the Condorcet loser; that is the one candidate who loses 1-against-1 to every other candidate.

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. [emphasis added]”

Electing a Condorcet loser by Approval Voting is not likely, but the same is true of many potential flaws in other voting rules. To know the real-world risks of such rare events, we need data from thousands of simulations that recreate the “structure of the election-generating universe.” And any voting rule has to prove itself in the real world: It must give good results from repeated cycles of competitive voting with diverse voters.

The main advocates are here.

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 one-on-one election to the IRV winner.

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

This is a good, realistic reform and a big step toward both multi winner Single Transferable Vote for Proportional Representation and perhaps a Condorcet-Hare hybrid.

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.

In practice, the approval, Borda and IRV rules usually elect the Condorcet winner, as the next page shows. But to elect 100% of the Condorcet winners, we need a Condorcet-completion rule.

Condorcet Completion Rules

The Pairwise- or Condorcet-completion rules all give the same result in most elections.  They differ only when there is no pair-wise winner due to a voting-cycle such as C tops B, B tops D, D tops C (C>B>D>C).  Each completion rule is a way to resolve a voting cycle.  Later pages compare them on their abilities to elect the most central member of a voting cycle and 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.

A. H. Copeland’s 1950 rule gives a candidate 1 point for win¬ning 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 Minimax system elects the can¬didate 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 Minimax score would be -5%.  But she might have to change 10% of the ballots to become Dodgson’s winner.)

Markus Schulze's method takes the candidates in the voting cycle and finds the stongest path from one candidate to another. For example, A>C>B is a path for A over B, even if B topped A 1 against 1, (B>A). The strength of a path is the strength of its weakest link. It elects the candidate who's weakest path is stronger than every other candidate's weakest path.

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

Tideman's Condorcet-Hare hybrid elects the Condorcet winner if there is one. If there is a voting-cycle tie, it eliminates all candidates outside the Smith Set. Then it eliminates the candidate who now ranks at the top of the fewest ballots. These two steps repeat until just one candidate remains.

Those last three rules are most resistant to manipulation. In the 1980s, Chamberlin et al published research about the ease and frequency of manipulation. Some of their results are given on the next page. In the 2010s, James Green-Armytage has added significantly to this research by including newer voting rules.

Other Rules

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.

Range voting or scoring is similar to Borda but allows a voter to skip ranks in order to vary the “distances” between candidates.

This would elect a winner with a high “utility value.”  But it is easily manipulated by tactical voting.  So it has little use in competitive situations including most private organizations.  Even when used with visible votes by judges in sports competitions, it has caused controversies that delegitimized the results.

“Rankings, it appears, may be easier to obtain than ratings, and they have the added advantage of not seeming to be more than they are. Rating are often treated as if they interpersonally comparable interval scales for which a response of four by one person means the same thing as a response of four by another person, and the difference between two and three means the same thing as the difference between three and four. These things may be true, but there is no reason to suppose that they must be.”
— Henry E. Brady, “Dimensional Analysis of Ranking Data” American Journal of Political Science, Vol. 34, No. 4, November 1990, Pp. 1017-48 Site

In 1988, Samuel Merrill published the standard-score system.  It avoids the obvious problems in plain score voting.  It lets 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 reduces the chance for a voter to exagerate and spread out her ratings to influence the election more than other voters.

“...for each voter separately, replace her ratings r_i by their statistical standard scores, i.e., σ ∑

∑_i = (r_i - μ)/σ

where μ and σ are the mean and standard deviation of the voter’s ratings.” ...

“The standard-score system, because of the complexity of its decision rule, should be recommended only for a mathematically knowledgeable electorate.”
(Merrill, pages 101 and 103) )

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.

Some unusual voting systems are the incentive revealing devices by Clark or by Tullock and Tideman, or the influence point voting by Hylland and Zeckhauser or the various insurance bidding schemes.  Most were not designed for public general elections.  They do not meet the principle of “one person / one vote.”  They are easy to manipulate by “conspiracies”, and are rather complicated for voters.  These are not Condorcet completions rules nor are they proportional rules.  Most give more voting power to people who have more money, so they might be adapted by groups seeking economic outcomes.

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.