Difference between revisions of "Score voting"

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=== Connection to Condorcet methods ===
[[File:Condorcet utilitarianism compare and contrast example.png|thumb|2052x2052px]]
Usually, Score voting is computed by adding the scores on each voter's ballot to find the candidate with the most points, who wins. But one can also do it (in a theoretical, and more difficult manner) by, for each pair of candidates, subtracting the score of the lower-scored candidate from the higher-scored candidate, and putting this in a [[Pairwise counting|pairwise counting]] table. The candidate who gets more points in their matchups against all other candidates wins. Example: <blockquote>2: A:5 B:4 C:1
Score can be thought of as a [[Condorcet method]] where a voter may only put up to 1 vote (i.e. the maximum number of points allowed) in between any pair of candidates in a [[beatpath]]. That is, a strategic voter whose preference is A>B>C can maximally contribute to A getting more points than B or to B getting more points than C, but not both. A rated ballot A:5 B:4 C:0 with max score of 5 is treated as "A is 1 point better than B, B is 4 points better than C, and A is 5 points better than C", whereas in Condorcet all three [[Pairwise counting|pairwise comparisons]] are treated as "more-preferred candidate is 1 vote i.e. 5 points better than less-preferred candidates." Both Score and Condorcet elect the candidate who can get more points/votes than any other opponent in one-on-one comparisons, though in Condorcet such a candidate may [[Condorcet paradox|not always]] exist.
 
1: C:5 B:3 A:2 </blockquote>The scores are A 12 B 11 C 7.
 
The pairwise matrix:
{| class="wikitable"
|+Score voting matchups
!
!A
!B
!C
|-
|A
| ---
|2 (+1 Win)
|8 (+5 Win)
|-
|B
|1 (-1 Loss)
| ---
|6 (+4 Win)
|-
|C
|3 (-5 Loss)
|2 (-4 Loss)
| ---
|}
A gets more points than B or C (2 voters gave A 1 more point than B, with 1 voter giving B 1 more point than A, so A>B. 2 voters gave A 4 points more than C, with 1 voter giving C 3 points more than A, so A>C), so A wins. B is 2nd place because B beats C, and C loses all of their matchups, so they're in last place. Note that the Score voting [[Order of finish|order of finish]] can be constructed using a [[Condorcet ranking]] from this matrix. Some information is captured with this counting approach that is not normally captured. Note that to handle write-ins, one would have to ensure that a voter who gave an on-ballot candidate a score of, say, 3, would be counted as scoring that candidate 3 points higher than any candidates the voter didn't personally write in, which could complicate things.
 
Score can be thought of as a [[Condorcet method]] where a voter may only put up to 1 vote (i.e. the maximum number of points allowed) in between any pair of candidates in a [[beatpath]]. That is, a strategic voter whose preference is A>B>C can maximally contribute to A getting more points than B or to B getting more points than C, but not both. A rated ballot A:5 B:4 C:0 with max score of 5 is treated as "A is 1 point better than B, B is 4 points better than C, and A is 5 points better than C", whereas in Condorcet all three [[Pairwise counting|pairwise comparisons]] are treated as "more-preferred candidate is 1 vote i.e. 5 points better than less-preferred candidates." Both Score and Condorcet elect the candidate who can get more points/votes than any other opponent in one-on-one comparisons, though in Condorcet such a candidate may [[Condorcet paradox|not always]] exist. See [[Self-referential Smith-efficient Condorcet method]].
 
Score's satisfaction of the above-mentioned property (max of 1 vote of differentiation in a beatpath) is one of the reasons it nominally passes Independence of Irrelevant Alternatives where Condorcet methods don't, as the only time those methods fail it is when no [[Beats-all winner|beats-all winner]] exists, and forcing Condorcet methods to satisfy that property ensures a beats-or-ties-all winner will exist.
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