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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Observe that the mean distances exactly match the coloring of the majority of consensus circle: if mean distance to the yellow candidate is lower than that of the purple candidate (the "voting"), the yellow dot is geometrically closer to the consensus ("magically" selected from the geometry of the problem). Note that the ''mean'' is used, not the median as one would naively expect. The median is inadequate under this scenario. (The reasons for this are a bit technical, so we omit it here.)
Under an actual cardinal voting scheme, the mapping of distances to the ballot scale are bounded by the limited ballot, confined to discrete steps, and may not be linear. This will reduce the resolution and distort the results away from this idealized scenario. But this example shows that under consensus, the cardinal formalism adequately captures a notion of "majority of consensus", which is a fundamental property of voters.
Moreover, even though voters can only express simple information about the candidates, the information given by all voters, taken together, has a direct connection to this "majority of consensus" notion.
What about the polarized case? In the case of polarization, factions are less accepting on one issue than in others. This effectively "shrinks" the consensus region along that axis, that is, voters are less accepting with people "outside the group" as perceived in that direction. The previous results still stand.
[[File:Majority of consensus polarization histograms.gif]]
As we can see, the histograms of cardinal information between any two candidates in an election can reveal to us whether between the two candidates, and their ideological leanings, there is a consensus or a polarization.
As before, the mean of these distributions (not the median) is capable of predicting which candidate is closer to the overall consensus.
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