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)
Revision as of 02:52, 22 February 2021
, 3 years ago→Final remarks
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* 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.
* The "majority of consensus" reproduces the intuitive notion of majority, and it is well-captured by the median distance. However, the median is mathematically less capable of minimizing the distance to the consensus, as defined by the mean opinion as just explained. In the animations above, if one pays attention it can be seen that the smallest median distance does not correlate precisely with the color of the circle, "magically picked" by directly picking the candidate closer to the consensus. This is because the median still biases the results in favor of the dominant faction, as can be observed by how quickly the median lines move across the distance distributions in the polarized case. The
* The mean is more optimal than the median as it minimizes the sum of squares of Euclidean distances, and thus the direct Euclidean distance to any point, whereas the geometric median minimizes the simple sum of distances. The sum of squares can be understood as a weighted sum, where each distance is weighted by a factor proportional to the distance itself, penalizing points which stray too far away from the consensus more.
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