Round-robin voting

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Round-robin voting (also called paired/pairwise comparison or tournamentvoting) refers to a set of ranked voting systems that elect winners by comparing all candidates in a round-robin tournament. Every candidate is matched up against every other candidate, where their point total is equal to the number of votes they receive; the method then selects a winner based on the results of these paired matchups.

Contents

Round-robin methods are one of the four major categories of single-winner electoral methods, along with multi-stage methods (including instant-runoff voting and Baldwin's method), positional methods (including plurality and Borda), and graded methods (including score and STAR voting).

While most methods satisfying the Condorcet criterion are pairwise-counting methods, some are not. A handful of sequential-loser methods satisfy the Condorcet criterion, as do many Condorcet-hybrid methods.

Summary

In paired voting, each voter ranks candidates from first to last (or rates them on a scale); candidates not ranked by voters are given the lowest rank or score. [1]

For each pair of candidates (as in a round-robin tournament), we count how many votes rank each candidate over the other candidate. Thus each pair will have two totals, the size of its majority and the size of its minority. [2]

Pairwise counting

In the pairwise-counting procedure, we compare each pair of candidates (as in a round-robin tournament), counting how many voters rank each candidate over the other. [3]

Pairwise counts are often displayed in a pairwise comparison [4] or outranking matrix. [5] In these matrices, each row represents candidate as a 'runner,' while each column represents each candidate as an 'opponent'. The cells at the intersection of rows and columns each show the result of a particular pairwise comparison. Cells comparing a candidate to themselves are left blank. [6] [7]

Imagine there is an election between four candidates: A, B, C, and D. The first matrix below records the preferences expressed on a single ballot paper, in which the voter's preferences are (B, C, A, D); that is, the voter ranked B first, C second, A third, and D fourth. In the matrix a '1' indicates that the runner is preferred over the 'opponent', while a '0' indicates that the runner is defeated. [6] [4]

Alternatively, the margin matrix can be used for most methods. The margin matrix considers only the difference in the vote shares of the two candidates, making it antisymmetric (i.e. the top half is the negative of the bottom half).

Example

Tennessee map for voting example.svg

Suppose that Tennessee is holding an election on the location of its capital. The population is concentrated around four major cities. All voters want the capital to be as close to them as possible. The options are:

The preferences of each region's voters are:

42% of voters
Far-West		26% of voters
Center		15% of voters
Center-East		17% of voters
Far-East
  1. Memphis
  2. Nashville
  3. Chattanooga
  4. Knoxville
  1. Nashville
  2. Chattanooga
  3. Knoxville
  4. Memphis
  1. Chattanooga
  2. Knoxville
  3. Nashville
  4. Memphis
  1. Knoxville
  2. Chattanooga
  3. Nashville
  4. Memphis

A pairwise-comparison matrix can be constructed as:

A
B
MemphisNashvilleChattanoogaKnoxville
Memphis[A] 58%

[B] 42%

[A] 58%

[B] 42%

[A] 58%

[B] 42%

Nashville[A] 42%

[B] 58%

[A] 32%

[B] 68%

[A] 32%

[B] 68%

Chattanooga[A] 42%

[B] 58%

[A] 68%

[B] 32%

[A] 17%

[B] 83%

Knoxville[A] 42%

[B] 58%

[A] 68%

[B] 32%

[A] 83%

[B] 17%

Copeland score:0-3-03-0-02-1-01-2-0
Minimax score:58%42%68%83%

Related Research Articles

<span class="mw-page-title-main">Condorcet method</span> Pairwise-comparison electoral system

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. The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking.

<span class="mw-page-title-main">Copeland's method</span> Single-winner ranked vote system

Copeland's method, also called Llull's method or round-robin voting, is a ranked-choice voting system based on scoring pairwise wins and losses.

The Smith set, also known as the top cycle, is a concept from the theory of electoral systems that generalizes the Condorcet winner to cases where no such winner exists, by allowing cycles of candidates to be treated jointly as if they were a single Condorcet winner. Named after John H. Smith, the Smith set is the smallest non-empty set of candidates in a particular election, such that each member defeats every candidate outside the set in a pairwise election. The Smith set provides one standard of optimal choice for an election outcome. Voting systems that always elect a candidate from the Smith set pass the Smith criterion.

The independence of irrelevant alternatives (IIA), also known as binary independence or the independence axiom, is an axiom of decision theory and the social sciences that describes a necessary condition for rational behavior. The axiom says that adding "pointless" (rejected) options should not affect behavior. This is sometimes explained with a short story by philosopher Sidney Morgenbesser:

Morgenbesser, ordering dessert, is told by a waitress that he can choose between blueberry or apple pie. He orders apple. Soon the waitress comes back and explains cherry pie is also an option. Morgenbesser replies "In that case, I'll have blueberry."

Ranked pairs, sometimes called the Tideman method, is a tournament-style system of ranked-choice voting first proposed by Nicolaus Tideman in 1987.

The Schulze method is an electoral system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. The method can also be used to create a sorted list of winners. The Schulze method is also known as Schwartz Sequential dropping (SSD), cloneproof Schwartz sequential dropping (CSSD), the beatpath method, beatpath winner, path voting, and path winner. The Schulze method is a Condorcet method, which means that if there is a candidate who is preferred by a majority over every other candidate in pairwise comparisons, then this candidate will be the winner when the Schulze method is applied.

An electoral system satisfies the Condorcet winner criterion if it always chooses the Condorcet winner when one exists. The Condorcet winner is the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, although Condorcet winners do not exist in all cases. It is sometimes simply referred to as the "Condorcet criterion", though it is very different from the "Condorcet loser criterion". Any voting method conforming to the Condorcet winner criterion is known as a Condorcet method. For a set of candidates, the Condorcet winner is always the same regardless of the voting system in question, and can be discovered by using pairwise counting on voters' ranked preferences.

The Smith criterion is a voting system criterion that formalizes the concept of a majority rule. A voting system satisfies the Smith criterion if it always elects a candidate from the Smith set, which generalizes the idea of a "Condorcet winner" to cases where there may be cycles or ties, by allowing for several who together can be thought of as being "Condorcet winners." A Smith method will always elect a candidate from the Smith set.

The participation criterion, also called vote or population monotonicity, is a voting system criterion that says that a candidate should never lose an election because they have "too much support." It says that adding voters who support A over B should not cause A to lose the election to B.

A voting system is consistent if combining two sets of votes that both elect A over B always results in a combined electorate that ranks A over B. This property is sometimes called join-consistency or separability.

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 voting systems, the Minimax Condorcet method is a single-winner ranked-choice voting method that always elects the majority (Condorcet) winner. Minimax compares all candidates against each other in a round-robin tournament, then ranks candidates by their worst election result. The candidate with the largest (maximum) margin of victory in their worst (minimum) matchup is declared the winner.

Reversal symmetry is a voting system criterion which requires that if candidate A is the unique winner, and each voter's individual preferences are inverted, then A must not be elected. Methods that satisfy reversal symmetry include Borda count, ranked pairs, Kemeny–Young method, and Schulze method. Methods that fail include Bucklin voting, instant-runoff voting and Condorcet methods that fail the Condorcet loser criterion such as Minimax.

CPO-STV, or the Comparison of Pairs of Outcomes by the Single Transferable Vote, is a ranked voting system designed to achieve proportional representation. It is a more sophisticated variant of the Single Transferable Vote (STV) system, designed to overcome some of that system's perceived shortcomings. It does this by incorporating some of the features of Condorcet's method, a voting system designed for single-winner elections, into STV. As in other forms of STV, in a CPO-STV election more than one candidate is elected and voters must rank candidates in order of preference. As of February 2021, it has not been used for a public election.

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.

In voting systems theory, the independence of clones criterion measures an election method's robustness to strategic nomination. Nicolaus Tideman was the first to formulate this criterion, which states that the winner must not change due to the addition of a non-winning candidate who is similar to a candidate already present. It is a relative criterion: it states how changing an election should or shouldn't affect the outcome.

The Borda count is a family of positional voting rules which gives each candidate, for each ballot, a number of points corresponding to the number of candidates ranked lower. In the original variant, the lowest-ranked candidate gets 0 points, the next-lowest gets 1 point, etc., and the highest-ranked candidate gets n − 1 points, where n is the number of candidates. Once all votes have been counted, the option or candidate with the most points is the winner. The Borda count is intended to elect broadly acceptable options or candidates, rather than those preferred by a majority, and so is often described as a consensus-based voting system rather than a majoritarian one.

<span class="mw-page-title-main">Ranked voting</span> Family of electoral systems

The term ranked voting, also known as preferential voting or ranked-choice voting, pertains to any voting system where voters indicate a rank to order candidates or options—in a sequence from first, second, third, and onwards—on their ballots. Ranked voting systems vary based on the ballot marking process, how preferences are tabulated and counted, the number of seats available for election, and whether voters are allowed to rank candidates equally.

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.

A major branch of social choice theory is devoted to the comparison of electoral systems, closely related to 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 behave given a collection of individual preferences.

References

  1. ↑ Darlington, Richard B. (2018). "Are Condorcet and minimax voting systems the best?". arXiv: 1807.01366 [physics.soc-ph]. CC [Condorcet] systems typically allow tied ranks. If a voter fails to rank a candidate, they are typically presumed to rank them below anyone whom they did rank explicitly.
  2. ↑ Hazewinkel, Michiel (2007-11-23). Encyclopaedia of Mathematics, Supplement III. Springer Science & Business Media. ISBN   978-0-306-48373-8. Briefly, one can say candidate Adefeats candidate B if a majority of the voters prefer A to B. With only two candidates [...] barring ties [...] one of the two candidates will defeat the other.
  3. ↑ Hazewinkel, Michiel (2007-11-23). Encyclopaedia of Mathematics, Supplement III. Springer Science & Business Media. ISBN   978-0-306-48373-8. Briefly, one can say candidate Adefeats candidate B if a majority of the voters prefer A to B. With only two candidates [...] barring ties [...] one of the two candidates will defeat the other.
  4. 1 2 Mackie, Gerry. (2003). Democracy defended. Cambridge, UK: Cambridge University Press. p. 6. ISBN   0511062648. OCLC   252507400.
  5. ↑ Nurmi, Hannu (2012), "On the Relevance of Theoretical Results to Voting System Choice", in Felsenthal, Dan S.; Machover, Moshé (eds.), Electoral Systems, Studies in Choice and Welfare, Springer Berlin Heidelberg, pp. 255–274, doi:10.1007/978-3-642-20441-8_10, ISBN   9783642204401, S2CID   12562825
  6. 1 2 Young, H. P. (1988). "Condorcet's Theory of Voting" (PDF). American Political Science Review. 82 (4): 1231–1244. doi:10.2307/1961757. ISSN   0003-0554. JSTOR   1961757. S2CID   14908863. Archived (PDF) from the original on 2018-12-22.
  7. ↑ Hogben, G. (1913). "Preferential Voting in Single-member Constituencies, with Special Reference to the Counting of Votes". Transactions and Proceedings of the Royal Society of New Zealand. 46: 304–308.
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