User:Lucasvb/Majority and consensus under ordinal and cardinal perspectives: Difference between revisions
User:Lucasvb/Majority and consensus under ordinal and cardinal perspectives (view source)
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We consider the smallest region around the consensus which contains a majority of voters within it. As opposed to the "majority of preference", this is a ''true'' majority, a property of the voters that is independent of candidates and whatever factions they create. In this diagram, this majority of consensus is denoted as a red circle around the consensus.
As voters are not being forced to take sides, and may support both candidates simultaneously to various degrees, there is no immediate notion of "factions" or "consensus within a faction", unless such a distinction exists in the voters themselves.
In practice, however, we do not have direct access to this geometric picture. We are confined by the information presented in ballots, which related directly to the candidates in question, as was the case with ranked ballots.▼
▲In practice, however, we do not have direct access to this geometric picture. We are confined by the information presented in ballots
Can we recover the spirit of this "majority of consensus"? It turns out yes, we can.
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In the diagram, the candidate closest to the consensus is being "magically" picked as the "winner", coloring the interior of the circle. There is no "voting" taking place! It is a completely geometric property being depicted, representing the candidate closest to the consensus. This candidate would be the closest to represent the "majority of consensus", by definition.
At the bottom, we have a distribution of distances from voters to the candidates, one distribution per candidate. This is what voters would be intuitively measuring during an election, and attempting to convey in their ballots. The vertical line is the mean of the distributions, that is, the ''mean distance'', which is also used to plot the dashed circles around the consensus for each candidate.
This is analogous to voters voting in a continuous cardinal scale, from 0 (candidate has exactly the same beliefs as the voter) to infinity (candidate is completely incomprehensible to the voter), mapping distance perfectly to this scale.
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