# Difference between revisions of "Consistency criterion"

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− | The '''consistency criterion''' requires that any candidate who wins both of two separate sets of voters must also win the combined electorate. Although it is not a logical requirement, generally speaking systems which satisfy the consistency criterion also satisfy the [[participation criterion]]. The reverse implication is also common, though slightly less so. | + | The '''consistency criterion''' requires that any candidate who wins both of two separate sets of voters must also win the combined electorate. In this definition, "winning" also includes being tied for first place: if a candidate X ties with some other candidate in the first electorate, and wins outright in the second, X must win outright in the combined electorate. |

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+ | Although it is not a logical requirement, generally speaking systems which satisfy the consistency criterion also satisfy the [[participation criterion]]. The reverse implication is also common, though slightly less so. | ||

A [[voting system]] is '''consistent''' if, when the electorate is divided arbitrarily into two parts and separate elections in each part result in the same alternative being selected, an election of the entire electorate also selects that alternative. If a voting system is not consistent then it may be manipulated through the establishment of strategically configured election districts. | A [[voting system]] is '''consistent''' if, when the electorate is divided arbitrarily into two parts and separate elections in each part result in the same alternative being selected, an election of the entire electorate also selects that alternative. If a voting system is not consistent then it may be manipulated through the establishment of strategically configured election districts. | ||

− | ==Complying | + | ==Complying methods== |

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+ | A strict [[preferential voting]] method is "consistent if and only if it is a scoring function"<ref name="Young 1975 pp. 824–838">{{cite journal | last=Young | first=H. P. | title=Social Choice Scoring Functions | journal=SIAM Journal on Applied Mathematics | publisher=Society for Industrial & Applied Mathematics (SIAM) | volume=28 | issue=4 | year=1975 | issn=0036-1399 | doi=10.1137/0128067 | pages=824–838 |url=http://www.eecs.harvard.edu/cs286r/courses/fall11/papers/Young75.pdf}}</ref>, i.e. a [[weighted positional method]] or a combination of these where one or more weighted positional methods are used in sequence to break the ties of another. If the preferential voting method admits weak preference orders (rankings with equal-rank or truncation), it must reduce to a scoring function when no voters make use of equal-rank or truncation. | ||

+ | <!-- It would be nice to find a reference that handles weak preference orderings, but I haven't been able to find one. --> | ||

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+ | [[Plurality voting]], and the [[Borda count]] are weighted positional methods and thus pass the consistency criterion. [[Condorcet method | Condorcet methods]], [[Majority Choice Approval]], and [[IRV]] fail. | ||

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+ | In addition, [[Approval voting]] and [[Score voting]] are consistent. If X is the winner in the first district, then that means that X's score is greater than or equal to any other Y, and the same for the second district. Then summing the districts' scores can not make any Y's sum exceed X's sum. | ||

− | + | ==References== | |

[[Category:Voting system criteria]] | [[Category:Voting system criteria]] |

## Revision as of 15:19, 8 February 2020

The **consistency criterion** requires that any candidate who wins both of two separate sets of voters must also win the combined electorate. In this definition, "winning" also includes being tied for first place: if a candidate X ties with some other candidate in the first electorate, and wins outright in the second, X must win outright in the combined electorate.

Although it is not a logical requirement, generally speaking systems which satisfy the consistency criterion also satisfy the participation criterion. The reverse implication is also common, though slightly less so.

A voting system is **consistent** if, when the electorate is divided arbitrarily into two parts and separate elections in each part result in the same alternative being selected, an election of the entire electorate also selects that alternative. If a voting system is not consistent then it may be manipulated through the establishment of strategically configured election districts.

## Complying methods

A strict preferential voting method is "consistent if and only if it is a scoring function"^{[1]}, i.e. a weighted positional method or a combination of these where one or more weighted positional methods are used in sequence to break the ties of another. If the preferential voting method admits weak preference orders (rankings with equal-rank or truncation), it must reduce to a scoring function when no voters make use of equal-rank or truncation.

Plurality voting, and the Borda count are weighted positional methods and thus pass the consistency criterion. Condorcet methods, Majority Choice Approval, and IRV fail.

In addition, Approval voting and Score voting are consistent. If X is the winner in the first district, then that means that X's score is greater than or equal to any other Y, and the same for the second district. Then summing the districts' scores can not make any Y's sum exceed X's sum.

## References

- ↑ Young, H. P. (1975). "Social Choice Scoring Functions" (PDF).
*SIAM Journal on Applied Mathematics*. Society for Industrial & Applied Mathematics (SIAM).**28**(4): 824–838. doi:10.1137/0128067. ISSN 0036-1399.