Score voting: Difference between revisions

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=== Connection to Condorcet methods ===

[[File:Condorcet utilitarianism compare and contrast example.png|thumb|2052x2052px]]

==== Computing the result using pairwise counting ====

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

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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.

==== Limits on strength of transitive pairwise preference ====

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 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 [[Transitivity|transitive]] [[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."

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.

==== Head-to-head winner ====

When the Score winner is the Condorcet winner, and all voters expressed all of their ranked preferences with the scores, then this means that each voter could exaggerate their preference in each head-to-head matchup from the weak pairwise preference they expressed in Score to a maximal pairwise preference and obtain the same result if the head-to-head matchups are used to find the Condorcet winner in both circumstances. The same holds for when the Score ranking is equivalent to the [[Condorcet ranking]].

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]]. When the Score winner is the Condorcet winner, and all voters expressed all of their ranked preferences with the scores, then this means that each voter could exaggerate their preference in each head-to-head matchup from the weak pairwise preference they expressed in Score to a maximal pairwise preference and obtain the same result if the head-to-head matchups are used to find the Condorcet winner in both circumstances. The same holds for when the Score ranking is equivalent to the [[Condorcet ranking]].

==== Connection to pairwise counting ====

When using the [[Rated pairwise preference ballot#Rated or ranked preference]] implementation with [[Pairwise counting#Negative vote-counting approach]], the score for a candidate can be interpreted as a partial ballot marking that candidate i.e. a voter with rated preferences giving a candidate a 3/5 would be considered to give them 0.6 votes in every matchup, whereas a voter who has ranked preferences would instead give 1 vote.