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User:Lucasvb/Majority and consensus under ordinal and cardinal perspectives (view source)

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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 linelines isare the meanmedians of the distributions, that is, the ''mean distance'', which is also used to plot the dashed circles around the consensus for each candidate. The dashed gray distribution is the distance distribution relative to the consensus, with the red line the median distance, which defines the "majority of consensus circle".
 
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. In reality things are not so simple, but the goal here is to show that in principle the information is there. Also, since cardinal voting contains total comparative information (candidates are not judged in isolation), the best and worst candidates define a "yardstick" voters use to measure distance. If voters are to be taken as equally worthy in opinion, the aggregation of cardinal ballots represents taking the ballot to represent the "mean yardstick" of voters.
 
ObserveThe that'''mean''' (not the meanmedian) of the distances exactly matchmatches 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). See remarks at the end for explanation. Note that the ''mean'' is also used to define the consensus, 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. See the remarks at the end.)
 
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.
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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?
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, either the mean of these distributions (not theor median) is capable of predicting which candidate is closer to the overall consensus. This is a property independent of the distribution, and thus, it always approximates the "majority of consensus". However, the mean will generally be more accurate to predict proximity to the consensus.
 
== Final remarks ===
 
* A ranked preference is the answer to the question "which of these two candidates the voter feels it is closer to their interests?", so it gives an information about "distance". Thus, in the cardinal case, we are also showing continuous distance information. A more advanced model of voters would have to map distances to something like "utility", and then one would need to map utilities to cardinal ballots. This would introduce too many arbitrary steps. For our purposes, it is sufficient
 
* While the median is a better metric of "central tendency in response to outliers", that is only useful if we know ''where'' that median position is. This is not the information that is available to us with ranked ballots. All we have is "this side has 55% people, the other 45%" and so on. We have no information about "where" the line was drawn, and what the ideological distribution looks like at that location.
 
* The reason the mean and not the median is used in defining the consensus is related to the the role of consensus and polarization. Since we are trying to define the "majority of consensus", the contribution of polarizing issues to the "consensus" must be minimized, as they are not a consensus. Imagine the 1D case where there is maximum (50%+1,50%-1) polarization on an issue, and all voters on either side have very sharp-peaked equal beliefs. The "consensus", if defined as the '''median''' opinion, would lie entirely within one of the factions, and the "majority of consensus" would account only for that faction, completely ignoring the other. So this definition cannot capture the notion of a consensus under polarization.
 
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