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| Social choice and electoral systems |
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 Mathematicsportal |
The Borda count electoral system can be combined with an instant-runoff procedure to create hybrid election methods that are called Nanson method and Baldwin method (also called Total Vote Runoff or TVR). Both methods are designed to satisfy the Condorcet criterion, and allow for incomplete ballots and equal rankings.
The Nanson method is based on the original work of the mathematician Edward J. Nanson in 1882. [1]
Nanson's method eliminates those choices from a Borda count tally that are at or below the average Borda count score, then the ballots are retallied as if the remaining candidates were exclusively on the ballot. This process is repeated if necessary until a single winner remains.
If a Condorcet winner exists, they will be elected. If not, (there is a Condorcet cycle) then the preference with the smallest majority will be eliminated. [1] : 214
Nanson's method can be adapted to handle incomplete ballots (including "plumping") and equal rankings ("bracketing"), though he describes two different methods to handle these cases: a theoretically correct method involving fractions of a vote, and a practical method involving whole numbers (which has the side effect of diminishing the voting power of voters who plump or bracket). [1] : 231, 235 This then allows the use of Approval-style voting for uninformed voters who merely wish to approve of some candidates and disapprove of others. [1] : 236
The method can be adapted to multi-winner elections by removing the name of a winner from the ballots and re-calculating, though this just elects the highest-ranked n candidates and does not result in proportional representation. [1] : 240
Schwartz in 1986 studied a slight variant of Nanson's rule, in which candidates less than but not equal to the average Borda count score are eliminated in each round. [2]
Candidates are voted for on ranked ballots as in the Borda count. Then, the points are tallied in a series of rounds. In each round, the candidate with the fewest points is eliminated, and the points are re-tallied as if that candidate were not on the ballot.
This method actually predates Nanson's, who notes it was already in use by the Trinity College Dialectic Society. [1] : 217
It was systematized by Joseph M. Baldwin [3] in 1926, who incorporated a more efficient matrix tabulation [4] and extended it to support incomplete ballots and equal rankings, by counting fractional points in such cases.
The two methods have been confused with each other in some literature. [2]
This system has been proposed for use in the United States under the name "Total Vote Runoff", by Edward B. Foley and Eric Maskin, as a way to fix problems with the instant-runoff method in U.S. jurisdictions that use it. [5] [6] [7] [8] [9]
The Nanson method and the Baldwin method satisfy the Condorcet criterion. [2] Because Borda always gives any existing Condorcet winner more than the average Borda points, the Condorcet winner will never be eliminated.
They do not satisfy the independence of irrelevant alternatives criterion, the monotonicity criterion, the participation criterion, the consistency criterion and the independence of clones criterion, while they do satisfy the majority criterion, the mutual majority criterion, the Condorcet loser criterion and the Smith criterion. The Nanson method satisfies and the Baldwin method violates reversal symmetry. [10]
Both the Nanson and the Baldwin methods can be run in polynomial time to obtain a single winner. For the Baldwin method, however, at each stage, there might be several candidates with lowest Borda score. In fact, it is NP-complete to decide whether a given candidate is a Baldwin winner, i.e., whether there exists an elimination sequence that leaves a given candidate uneliminated. [11]
Both methods are computationally more difficult to manipulate than Borda's method. [12]
Nanson's method was used in city elections in the U.S. town of Marquette, Michigan in the 1920s. [13] It was formerly used by the Anglican Diocese of Melbourne and in the election of members of the University Council of the University of Adelaide. It was used by the University of Melbourne until 1983.
A Condorcet method is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, whenever there is such a candidate. A candidate with this property, the pairwise champion or beats-all winner, is formally called the Condorcet winner or Pairwise Majority Rule Winner (PMRW). The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking.
Coombs' method is a ranked voting system. Like instant-runoff (IRV-RCV), Coombs' method is a sequential-loser method, where the last-place finisher according to one method is eliminated in each round. However, unlike in instant-runoff, each round has electors voting against their least-favorite candidate; the candidate ranked last by the most voters is eliminated.
The Copeland or Llull method is a ranked-choice voting system based on counting each candidate's pairwise wins and losses.
Bucklin voting is a class of voting methods that can be used for single-member and multi-member districts. As in highest median rules like the majority judgment, the Bucklin winner will be one of the candidates with the highest median ranking or rating. It is named after its original promoter, the Georgist politician James W. Bucklin of Grand Junction, Colorado, and is also known as the Grand Junction system.
A Condorcet winner is a candidate who would receive the support of more than half of the electorate in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the Condorcet winner criterion. The Condorcet winner criterion extends the principle of majority rule to elections with multiple candidates.
In social choice, a no-show paradox is a pathology in some voting rules, where a candidate loses an election as a result of having too many supporters. More formally, a no-show paradox occurs when adding voters who prefer Alice to Bob causes Alice to lose the election to Bob. Voting systems without the no-show paradox are said to satisfy the participation criterion.
The majority criterion is a winner-takes-all voting system criterion that says that, if only one candidate is ranked first by over 50% of voters, that candidate must win.
In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion.
In social choice theory, the best-is-worst paradox occurs when a voting rule declares the same candidate to be both the best and worst possible winner. The worst candidate can be identified by reversing each voter's ballot, then applying the voting rule to the reversed ballots find a new "anti-winner".
The Kemeny–Young method is an electoral system that uses ranked ballots and pairwise comparison counts to identify the most popular choices in an election. It is a Condorcet method because if there is a Condorcet winner, it will always be ranked as the most popular choice.
Later-no-harm is a property of some ranked-choice voting systems, first described by Douglas Woodall. In later-no-harm systems, increasing the rating or rank of a candidate ranked below the winner of an election cannot cause a higher-ranked candidate to lose.
In social choice theory, the independence of (irrelevant) clones criterion says that adding a clone, i.e. a new candidate very similar to an already-existing candidate, should not spoil the results. It can be considered a weak form of the independence of irrelevant alternatives (IIA) criterion that nevertheless is failed by a number of voting rules. A method that passes the criterion is said to be clone independent.
The Borda method or order of merit is a positional voting rule which gives each candidate a number of points equal to the number of candidates ranked below them: the lowest-ranked candidate gets 0 points, the second-lowest gets 1 point, and so on. Once all votes have been counted, the option or candidate with the most points is the winner.
Instant-runoff voting (IRV) is a winner-takes-all multi-round elimination voting system that uses ranked voting to simulate a series of runoff elections, where the last-place finisher according to a plurality vote is eliminated in each round and the votes supporting the eliminated choice are transferred to their next available preference until one of the options reaches a majority of the remaining votes. Its purpose is to elect the candidate in single-member districts with majority support even when there are more than two candidates. IRV is most closely related to two-round runoff election.
An electoral or voting system is a set of rules used to determine the results of an election. Electoral systems are used in politics to elect governments, while non-political elections may take place in business, non-profit organisations and informal organisations. These rules govern all aspects of the voting process: when elections occur, who is allowed to vote, who can stand as a candidate, how ballots are marked and cast, how the ballots are counted, how votes translate into the election outcome, limits on campaign spending, and other factors that can affect the result. Political electoral systems are defined by constitutions and electoral laws, are typically conducted by election commissions, and can use multiple types of elections for different offices.
Ranked voting is any voting system that uses voters' rankings of candidates to choose a single winner or multiple winners. More formally, a ranked system is one that depends only on which of two candidates is preferred by a voter, and as such does not incorporate any information about intensity of preferences. Ranked voting systems vary dramatically in how preferences are tabulated and counted, which gives them very different properties.
There are a number of different criteria which can be used for voting systems in an election, including the following
The later-no-help criterion is a voting system criterion formulated by Douglas Woodall. The criterion is satisfied if, in any election, a voter giving an additional ranking or positive rating to a less-preferred candidate can not cause a more-preferred candidate to win. Voting systems that fail the later-no-help criterion are vulnerable to the tactical voting strategy called mischief voting, which can deny victory to a sincere Condorcet winner.
Homogeneity is a common property for voting systems. The property is satisfied if, in any election, the result depends only on the proportion of ballots of each possible type. That is, if every ballot is replicated the same number of times, then the result should not change.
A major branch of social choice theory is devoted to the comparison of electoral systems, otherwise known as social choice functions. Viewed from the perspective of political science, electoral systems are rules for conducting elections and determining winners from the ballots cast. From the perspective of economics, mathematics, and philosophy, a social choice function is a mathematical function that determines how a society should make choices, given a collection of individual preferences.
In each case where on a voting paper no preference is expressed as between two candidates, half a preference is to be credited to each of the two candidates … For each paper where any number, p, of candidates are placed equal with a preference ranking as first, 1/p is to be credited to each of the candidates so placed.
the way Alaska uses ranked-choice voting also caused the defeat of Begich, whom most Alaska voters preferred to Democrat Mary Peltola … A candidate popular only with the party's base would be eliminated early in a Total Vote Runoff, leaving a more broadly popular Republican to compete against a Democrat.
a small but significant adjustment to the "instant runoff" method … equivalent to a candidate's Borda score, and eliminating sequentially the candidate with the lowest total votes
Begich and Peltola each get half a vote by being tied for second place on this ballot
In this respect, TVR differs from Baldwin's method, which without checking whether any candidate has more than 50% of first-place votes would immediately recalculate Borda scores
