Ranked voting: Difference between revisions
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{{Wikipedia|Ranked voting}}
:''This article is about voting systems that use ranked ballots, which can also include voting systems that use <nowiki>[[w:Level of measurement#Interval scale|interval scale]]</nowiki> ballots, i.e. [[cardinal voting systems]]''
[[Image:Preferential ballot.svg|thumb|Sample ballot of ranked voting using written numbers]]
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== Notes ==
One criticism that can be made of ranked voting is that it creates a logical contradiction: if a voter ranks X>Y>Z, then the strength of their preference for X>Z must be stronger than their preference for X>Y or Y>Z, yet all 3 preferences are generally treated as equally strong in most ranked methods.
[[Approval voting]] (and some [[Rated method|rated methods]] in general) can be thought of as a ranked method with constraints placed that fully resolve this contradiction: if an Approval ballot is thought of as a voter ranking one set of candidates equally 1st and above all others, then when a voter ranks an approved candidate above a disapproved candidate, they can't further indicate a preference between the disapproved candidates, thus ensuring that the strength of preference in each matchup is consistent with the strength in other matchups i.e. if they approve only X, then the strength of X>Y will be the same as X>Z, since the full preference is treated as X>Y=Z. In other words, from a [[pairwise counting]] perspective, if the voter gives 1 vote to X>Y, then they must give 0 votes to Y>Z, and when the number of votes given in both matchups is added up, this equals the 1 vote the voter gave to X>Z. If the voter's ranked preference is visualized as a [[beat-or-tie path]] from their 1st choice to their last choice, then the strength of their preference between any pair of candidates in the path will equal the strength of preference of each matchup between each pair of candidates starting from the first candidate of the pair vs the candidate sequentially after them, the candidate sequentially after them vs the candidate sequentially after the candidate sequentially after them, etc. all the way until the candidate sequentially before the second candidate in the pair vs the second candidate in the pair, added up. Another example would be a voter who approves A, B, and C, and disapproves of D, E, and F; this voter's Approval preference can be represented as A=B=C>D=E=F. If, for example, the B vs E matchup is analyzed, this voter is considered as giving 1 vote to B>E; this is equal to adding up the strength of their preference in the B vs C matchup (0 votes, because they ranked the two equally) plus their strength of preference in the C vs D matchup (1 vote, because they ranked C>D) plus the preference in the D vs E matchup (0 votes, because D=E). So, by starting at B and going sequentially one pair at a time down the beat-or-tie path that is the voter's ranked preference until you reach E, you can see that the strength of B>E is equal to the strength of the intervening matchups added up.
[[Score voting]] takes this a step further by allowing voters to vary their degree of approval; in some sense, this can be seen in the ranked context by first using the [[KP transform]] and then converting the resulting Approval ballots into ranked ballots as mentioned above. This allows voters to essentially "vote against themselves" in certain matchups or otherwise split their ballot up in such a way that only a fraction of it shows a preference between certain candidates, while the rest of the ballot is treated as indifferent between those candidates i.e. a voter giving 100% support to A, 70% to B, and 10% for C is treated as 10% of an A=B=C voter, 60% of an A=B voter, and 30% of an A voter, thus allowing them to have, for example, only 60% of their ballot showing preference for B>C, rather than 100%. Again, the same "the strength of X>Z is equal to X>Y plus Y>Z" beat-or-tie path consistency is achieved here; if analyzing the A vs C matchup, the voter gives 90% of their ballot to A>C and 10% to A=C, so they are in effect giving 0.9 votes to A>C. This equals the strength of the A vs B matchup (0.3 votes for A>B, since the voter gives 30% of their ballot to A>B and 70% to A=B) plus the B vs C matchup (60% or 0.6 votes for B>C, as mentioned above).
== References ==
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