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A voting system is a process that allows a group of people to express their desires about a number of options, and then selects one or more of those options, typically in a way meant to satisfy many of the voters. Voting is best known for its use in elections and is often seen as the defining feature of democracy, where the options are candidates for public office, and the preferences of the citizens determine who gets to hold those offices. In addition, voting can be used to award prizes, to select between different plans of action, or for a computer program to decide on the best solution to a complex problem.

Specifically, a voting system is a well-defined method (an algorithm) that determines a winning result given a set of votes. The process must be formally defined to be considered a voting system; the rules that specify how the votes will be counted must be known beforehand. This can be contrasted with consensus decision making, another process for selecting an option based on people's preferences which, unlike a voting system, does not specify how to precisely determine the winning option.

The study of formally-defined voting systems is called voting theory, which can be seen as a subfield of both political science and mathematics.

Majority rule

Most voting systems are based on the concept of majority rule, or the principle that a group of more than half of the voters should be able to get the outcome they want. Given the simplicity of majority rule, those who are unfamiliar with voting theory are often surprised that such a variety of voting systems exists.

If every election had only two choices, in fact, the winner could always be determined using majority rule alone. However, when there are three or more options, there may not be a single option that is preferred by a majority. The goal of most voting systems is to give a sufficiently fair way to choose the winner in such a situation. Different voting systems arise from different approaches to this goal.

Aspects of voting systems

Each voting system specifies the ballot, which defines the set of allowable votes, and the method, an algorithm for determining the outcome from those votes. This outcome may be a single winner, or may involve multiple winners, such as in the election of a legislative body. The voting system may also specify how to divide the voters into groups (constituencies) whose votes are counted independently.

The real-world implementation of an election is generally not considered part of the voting system. For example, though a voting system specifies the ballot abstractly, it does not specify whether the actual ballot takes the form of a piece of paper, a punch card, or a computer display, to give a few examples. It also does not specify whether or how votes are kept secret, or how to verify that the votes are counted accurately. These are aspects of the broader topic of elections and electoral systems.

The ballot

Different voting systems have different forms for allowing the individual to express their votes. In ranked ballot or "preference" voting systems, like Instant-runoff voting, the Borda count, or a Condorcet method, voters order the list of options from most to least preferred. In range voting, voters rate each option separately on a scale. In plurality voting (also known as "first-past-the-post"), voters select only one option, while in approval voting, they can select as many as they want. In voting systems that allow "plumping", like cumulative voting, voters may vote for the same candidate multiple times.

Some voting systems include additional choices on the ballot, such as write-in candidates or a none of the above option.

Constituencies

Often the purpose of an election is to choose a legislative body made of multiple winners. This can be done by running a single election and choosing the winners from the same pool of votes, or by dividing up the voters into constituencies that have different options and elect different winners.

Some countries, like Israel, fill their entire parliament using a single multiple-winner district (constituency), while others, like the Republic of Ireland or Belgium, break up their national elections into smaller, multiple-winner districts, and yet others, like the United States or the United Kingdom, hold only single-winner elections. Some systems, like the Additional member system, embed smaller districts within larger ones.

Single-winner methods

Single-winner systems can be classified based on their ballot type. Binary voting systems are those in which a voter either votes or doesn't vote for a given candidate. In ranked voting systems, each voter ranks the candidates in order of preference. In rated voting systems, voters give a score to each candidate.

Binary voting methods

The most prevalent single-winner voting method, by far, is plurality (also called "first-past-the-post", "relative majority", or "winner-take-all"), in which each voter votes for one choice, and the choice that receives the most votes wins, even if it receives less than a majority of votes.

Approval voting is another binary voting method, where voters may vote for as many candidates as they like. The choice that receives the most approval votes wins.

Runoff methods hold multiple rounds of plurality voting to ensure that the winner is elected by a majority. Top-two runoff voting, the second most common method used in elections, holds a runoff election between the top two options if there is no majority. In elimination runoff elections, the weakest candidate is eliminated until there is a majority. In an exhaustive runoff election, no candidates are eliminated, so voting is simply repeated until there is a majority.

Random ballot is a method in which each voter votes for one option, and a single ballot is selected at random to determine the winner. This is mostly used as a tiebreaker for other methods.

Ranked voting methods

Main article: Preferential voting

Also known as preferential voting methods, these methods allow each voter to rank the candidates in order of preference. Often it is not necessary to rank all the candidates: unranked candidates are usually considered to be tied for last place. Some of these methods also allow voters to give multiple candidates the same ranking.

The most common ranked voting method is instant-runoff voting (IRV), also known as "alternative vote" or simply "preferential voting", which uses voters' preferences to simulate an elimination runoff election without multiple voting events. As the votes are tallied, the option with the fewest first-choice votes is eliminated. In successive rounds of counting, the preferred choice still available from each of the ballots is tallied. The least preferred option is eliminated in each round of counting until there is a majority winner. All ballots are considered in every round of counting.

The Borda count is a simple ranked voting method in which the options receive points based on their position on each ballot. A class of similar methods are called positional voting systems.

Other ranked methods include Coombs' method, Supplementary voting, and Bucklin voting, as well as the specific kinds of ranked methods listed below.

Condorcet methods

Main article: Condorcet voting

Condorcet methods, or pairwise methods, are a class of ranked voting methods that meet the Condorcet criterion. These methods compare every option pairwise with every other option, and an option that defeats every other option is the winner. An option defeats another option if a majority of voters rank it higher on their ballot than the other option.

These methods are often referred to collectively as the Condorcet method, because the Condorcet criterion ensures that they all give the same result in most elections. The differences occur in situations where no option is undefeated, meaning that there exists a cycle of options that defeat each other. Considering the Condorcet method to be the abstract method that does not resolve these cycles, specific versions of Condorcet are called Condorcet completion methods.

A simple version of Condorcet is Minimax. If no option is undefeated, the option that is defeated by the fewest votes in its worst defeat wins. Another simple method is Copeland's method, in which the winner is the option that wins the most pairwise contests.

The Schulze method (also known as "cloneproof Schwartz sequential dropping" or the "beatpath method") and Ranked Pairs are two recently-designed Condorcet methods that satisfy a large number of voting method criteria.

Rated voting methods

Rated ballots allow even more flexibility than ranked ballots, but few methods are designed to use them. Each voter gives a score to each option; the allowable scores could be numeric (for example, from 0 to 100) or could be "grades" like A/B/C/D/F.

In range voting, voters give numeric ratings to each option, and the option with the highest total score wins. Approval voting can be seen as an instance of range voting where the allowable ratings are 0 and 1. Cumulative voting restricts the range differently by requiring the points on a ballot to add up to a certain total.

Rated ballots may also be used for ranked voting methods, as long as the ranked method allows tied rankings. Some ranked methods assume that all the rankings on a ballot are distinct, but many voters would be likely to give multiple candidates the same rating on a rated ballot.

Multiple-winner methods

Seats won by each party in the 2005 German federal election, an example of a proportional voting system.

A vote with multiple winners, such as the election of a legislature, has different practical effects than a single-winner vote. Often, participants in the voting system are more concerned with the overall composition of the legislature than exactly which candidates get elected. For this reason, many multiple-winner systems aim for proportional representation, which means that if a given party (or any other political grouping) gets X% of the vote, it should also get approximately X% of the seats in the legislature. Not all multiple-winner voting systems are proportional.

Non-proportional and semi-proportional methods

Many multiple-winner voting methods are simple extensions of single-winner methods, without an explicit goal of producing a proportional result. Bloc voting, or plurality-at-large, has each voter vote for N options and selects the top N as the winners. Due to its propensity for landslide victories won by a single winning slate of candidates, bloc voting is non-proportional. Two similar plurality-based methods with multiple winners are the single non-transferable vote method, where the voter votes for only one option, and cumulative voting, where the voter distributes a fixed number of points between the options of his choice. Unlike bloc voting, elections using the Single Non-transferable Vote or cumulative voting can acheive proportionality when voters use proper strategy via tactical voting or strategic nomination.

Because they encourage proportional results without guaranteeing them, the Single Non-transferable Vote and cumulative voting methods are classified as semi-proportional. Other methods that can be seen as semi-proportional are mixed methods, which combine the results of a plurality election and a party-list election (described below). Parallel voting is an example of a mixed method because it is only proportional for a subset of the winners.

Proportional methods

Main article: Proportional representation

Truly proportional methods make some guarantee of proportionality by making each winning option represent approximately the same number of voters. This number is called a quota. For example, if the quota is 1000 voters, then each elected candidate reflects the opinions of 1000 voters, within a margin of error.

Most proportional systems in use are based on party-list proportional representation, in which voters vote for parties instead of for individual candidates. For each quota of votes a party receives, one of their candidates wins a seat on the legislature. The methods differ in how the quota is determined, or, equivalently, how the proportions of votes are rounded off to match the number of seats.

The methods of seat allocation can be grouped overall into highest averages methods and largest remainder methods. Largest remainder methods set a particular quota based on the number of voters, while highest averages methods, such as the Sainte-Laguë method and the d'Hondt method, determine the quota indirectly by dividing the number of votes the parties receive by a sequence of numbers.

Independently of the method used to assign seats, party-list systems can be open list or closed list. In an open list system, voters decide which candidates within a party win the seats. In a closed list system, the seats are assigned to candidates in a fixed order that the party chooses. The Mixed Member Proportional system is a mixed method that only uses a party list for a subset of the winners, filling other seats with the winners of regional elections, thus having features of open list and closed list systems.

In contrast to party-list systems, single transferable vote is a proportional representation system in which voters rank individual candidates in order of preference. Unlike party-list systems, STV does not depend on the candidates being grouped into political parties. Votes are transferred between candidates in a manner similar to instant runoff voting, but in addition to transferring votes from candidates who are eliminated, votes are also transferred from candidates who already have a quota.

Criteria in evaluating voting systems

Main article: Voting system criteria

In the real world, attitudes toward voting systems are highly influenced by the systems' impact on groups that one supports or opposes. This can make the objective comparison of voting systems difficult. In order to compare systems fairly and independently of political ideologies, voting theorists use voting system criteria, which define potentially desirable properties of voting systems mathematically.

It is impossible for one voting system to pass all criteria in common use. For example, Arrow's impossibility theorem demonstrates that several desirable features of voting systems are mutually contradictory. For this reason, someone implementing a voting system has to decide which criteria are important for the election.

Using criteria to compare systems does not make the comparison completely objective. For example, it is relatively easy to devise a criterion that is met by one's favorite voting method, and by very few other methods. Then one can make a biased argument in favor of the criterion, instead of directly in favor of the method. No one can be the ultimate authority on which criteria should be considered, but the following are some criteria that are accepted and considered to be desirable by many voting theorists:

The following table shows which of the above criteria are met by several single-winner systems, ranked approximately in order of how commonly they are used.

Majority Monotonicity Consistency Participation Condorcet Condorcet loser IA independence clone independence
Plurality Yes Yes Yes Yes No No No No (vote-splitting)
2 round runoff Yes No No No No Yes No No (vote-splitting)
IRV Yes No No No No Yes No Yes
Approval No Yes Yes Yes No No Yes Ambiguous
Range voting No Yes Yes Yes No No Yes Ambiguous
Borda No Yes Yes Yes No Yes No No (teaming)
Minimax Yes Yes No No Yes No No No (vote-splitting)
Schulze Yes Yes No No Yes Yes No (see local IIA note) Yes
Ranked Pairs Yes Yes No No Yes Yes No (see local IIA note) Yes

In addition to the above criteria, voting systems are also judged with criteria that are not mathematically precise but are still important, such as simplicity, speed of vote-counting, the potential for fraud or disputed results, the opportunity for strategic voting, and (for multiple-winner methods) the degree of proportionality.

History

Founders of voting theory

File:Borda182x244.jpg
Jean-Charles de Borda, an early voting theorist.
The Marquis de Condorcet, another early voting theorist.

Although voting has been used in democracy since at least the time of ancient Greece, voting theory became an object of academic study around the time of the French Revolution.[1] Jean-Charles de Borda proposed the Borda count in 1770 as a method for electing members to the French Academy of Sciences. His system was opposed by the Marquis de Condorcet, who proposed instead the method of pairwise comparison that he had devised. Implementations of this method are known as Condorcet methods. He also wrote about the Condorcet paradox, which he called the intransitivity of majority preferences.[2]

While Condorcet and Borda are usually credited as the founders of voting theory, recent research has shown that the philosopher Ramon Llull discovered both the Borda count and a pairwise method that satisfied the Condorcet criterion in the 13th century. The manuscripts in which he described these methods had been lost to history until they were rediscovered in 2001.[3]

Apportionment and proportional representation

In the 18th century, the impetus for research into fair apportionment methods came, in fact, from the United States Constitution, which mandated that seats in the House of Representatives had to be allocated among the states proportionally to their population, but did not specify how to do so.[4] A variety of methods were proposed by statesmen such as Alexander Hamilton, Thomas Jefferson, and Daniel Webster.

Some of the apportionment methods discovered in the United States were rediscovered in Europe in the 19th century, as seat allocation methods for party-list proportional representation. The result is that many apportionment methods have two names: for instance, Jefferson's method is the same as the d'Hondt method, and Webster's method is the Sainte-Laguë method.[5]

Around the same time, the Single Transferable Vote system was devised by Carl Andrae in Denmark in 1855, and also in England by Thomas Hare in 1857. Their discoveries may or may not have been independent. STV elections were first held in Denmark in 1856, and in Tasmania in 1896 after its use was promoted by Andrew Inglis Clark.

Party-list proportional representation was first implemented to elect European legislatures in the early 20th century, with Belgium implementing it first in 1900. Since then, proportional and semi-proportional methods have come to be used in almost all democratic countries, with most exceptions being former British colonies.[6]

The single-winner revival

Perhaps influenced by the rapid development of multiple-winner voting methods, theorists began to publish new findings about single-winner methods in the late 19th century.

Around 1870, W. R. Ware proposed applying STV to single-winner elections, yielding instant runoff voting.[7] Also around that time, Charles Dodgson, better known as Lewis Carroll, published pamphlets on voting theory that introduced matrices as a way of analyzing the results of Condorcet elections (although this, too, was done first by Ramon Llull), and proposed the Condorcet method known as Dodgson's method. Edward J. Nanson proposed Nanson's method, which combined the Borda count with instant runoff voting to form a Condorcet method.

Influence of game theory

After John von Neumann and others developed the mathematical field of game theory in the 1940s, new mathematical tools were available to analyze voting systems and strategic voting. This led to significant new results that changed the field of voting theory.[1]

The use of mathematical criteria to evaluate voting systems was introduced when Kenneth Arrow showed in Arrow's impossibility theorem that certain criteria were mutually contradictory, demonstrating the inherent limitations of voting theorems. Arrow's theorem is easily the single most cited result in voting theory, and it inspired further significant results such as the Gibbard-Satterthwaite theorem, which showed that strategic voting is unavoidable in certain common circumstances.

The use of game theory to analyze voting systems also led to discoveries about the emergent strategic effects of certain systems. Duverger's Law is a prominent example of such a result, showing that plurality voting often leads to a two-party system.

New research into the game theory aspects of voting led Steven Brams and Peter Fishburn to formally define and promote the use of approval voting in 1977. Approval voting had been used before then, but it had not been named or considered as an object of academic study, particularly because it violated the assumption made by most research that single-winner methods were based on preference rankings.

Recent developments in voting theory

Voting theory has come to focus on voting system criteria almost as much as it does on particular voting systems. Now, any description of a benefit or weakness in a voting system is expected to be backed up by a mathematically-defined criterion. Recent research in voting theory has largely involved devising new criteria and new methods devised to meet certain criteria.

One prominent current voting theorist is Nicolaus Tideman, who formalized concepts such as strategic nomination and the spoiler effect in the independence of clones criterion. He also devised the ranked pairs method to be a Condorcet method that is not susceptible to strategic nomination.

Donald G. Saari has brought renewed interest to the Borda count with the books he has published since 2001. He created geometric models of positional voting systems, and uses these models to promote the use of the Borda count.

The advent of the Internet has increased the interest in voting systems. Unlike many other mathematical fields, voting theory is generally accessible enough to non-experts that new results can be discovered by amateurs, and frequently are. As such, many recent discoveries in voting theory come not from published papers, but from informal discussions among hobbyists on online forums and mailing lists.

See also

References

General references

  • . ISBN 0333801628. {{cite book}}: Missing or empty |title= (help); Unknown parameter |Author= ignored (|author= suggested) (help); Unknown parameter |Publisher= ignored (|publisher= suggested) (help); Unknown parameter |Title= ignored (|title= suggested) (help); Unknown parameter |Year= ignored (|year= suggested) (help)
  • Cretney, Blake (October 3). "Election Methods Resource". condorcet.org. {{cite web}}: Check date values in: |date= and |year= / |date= mismatch (help)
  • Cranor, Lorrie (October 3). "Vote Aggregation Methods". Declared-Strategy Voting: An Instrument for Group Decision-Making. {{cite web}}: Check date values in: |date= and |year= / |date= mismatch (help)

Cited sources

  1. ^ a J. J. O'Connor and E. F. Robertson (October 12). "The history of voting". The MacTutor History of Mathematics Archive. {{cite web}}: Check date values in: |date= and |year= / |date= mismatch (help)
  2. ^ J. J. O'Connor and E. F. Robertson (October 12). "Marie Jean Antoine Nicolas de Caritat Condorcet". The MacTutor History of Mathematics Archive. {{cite web}}: Check date values in: |date= and |year= / |date= mismatch (help)
  3. ^ Template:Journal reference url
  4. ^ Joseph Malkevitch (October 13). "Apportionment". AMS Feature Columns. {{cite web}}: Check date values in: |date= and |year= / |date= mismatch (help)
  5. ^ Joseph Malkevitch (October 13). "Apportionment II". AMS Feature Columns. {{cite web}}: Check date values in: |date= and |year= / |date= mismatch (help)
  6. ^ "Proportional Voting Around the World". FairVote.org. October 13. {{cite web}}: Check date values in: |date= and |year= / |date= mismatch (help)
  7. ^ "The History of IRV". FairVote.org. November 9. {{cite web}}: Check date values in: |date= and |year= / |date= mismatch (help)

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