# Weighted positional method

A **weighted positional method** is a preferential voting method that assigns points to the candidate a voter ranks in *i*th place. It then sums the scores and the candidate with the greatest score wins.

These methods can be characterized by the vector. For instance, the Borda count, the First Past the Post electoral system, and Antiplurality are all weighted positional methods, and their vectors are:

where is the number of candidates.

## Criterion compliances[edit | edit source]

Every weighted positional method passes the participation criterion, the consistency criterion, and mono-add-top; and are summable with order k=1.^{[1]}

Every weighted positional method except for First past the post fails the later-no-harm criterion^{[1]} and the majority criterion. Since First past the post fails the Condorcet criterion, and Condorcet implies majority, every weighted positional method fails Condorcet.

The Borda count is the only weighted positional method that never ranks the Condorcet winner last.^{[2]} It follows that the only Condorcet-compliant runoff method that eliminates one loser at a time and is based on a weighted positional method is Baldwin (Borda-elimination).

### Majority criterion[edit | edit source]

Consider an election with three candidates. The method's vector can be normalized to one of or .

In the latter two cases, the method trivially fails unanimity and thus also majority. So normalize so that with since the method is not First past the post.

If , construct the following election:

- x: A>B>C
- y: B>C>A

with .

A and B will be tied even though A has a majority of the first preferences, thus constituting a violation of the majority criterion.

On the other hand, if , construct the following election:

- x: A>C>B
- y: B>A>C

with . Again A and B will be tied even though A has a majority of the first preferences.

With more work, the examples can be generalized to any number of candidates greater then 3 by assuming every voter ranks all the other candidates in the same order.

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## References[edit | edit source]

- ↑
^{a}^{b}Smith, Warren D. (2007-06-12). "Descriptions of single-winner voting systems" (PDF). p. 28. Retrieved 2020-02-17. - ↑ Smith, John H. (1973). "Aggregation of Preferences with Variable Electorate".
*Econometrica*.**41**(6): 1027–1041. doi:10.2307/1914033. ISSN 0012-9682.