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, with the most notable example being the Tideman alternative method.

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%

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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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