Weighted positional method

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

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

• $a_{\mathrm {Borda} }=(m-1,m-2,\dots ,0)$
• $a_{\mathrm {Plurality} }=(1,0,\dots ,0)$
• $a_{\mathrm {Antiplurality} }=(1,\dots ,1,0)$

where $m$ is the number of candidates.

Criterion compliances

Every weighted positional method except for First past the post fails 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. 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

Consider an election with three candidates. The method's $a$ vector can be normalized to one of $a=(1,\alpha ,0),\,a=(0,\alpha ,1)$ or $a=(0,0,0)$.

In the latter two cases, the method trivially fails unanimity and thus also majority. So normalize $a$ so that $a=(1,\alpha ,0)$ with $\alpha \neq 0$ since the method is not First past the post.

If $\alpha >0$, construct the following election:

• x: A>B>C
• y: B>C>A

with $x={\frac {1}{\min(1,\alpha )}},\,y=x-1$.

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 $\alpha <0$, construct the following election:

• x: A>C>B
• y: B>A>C

with $x={\frac {\alpha -1}{\alpha }},\,y=x-1$. 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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