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 21:52, 20 February 2021
, 3 years ago→Final remarks
Line 152:
* 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, and the distributions would look coarser in resolution. This would introduce too many arbitrary steps and wouldn't illustrate anything important. For our purposes, the distance is 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. One could have hoped to estimate this by imagining a "line" between the two candidates, and placing a point along this line that represents the ratios of the votes received by either side (the "consensus between the candidates"), but this would still have a "sideways" bias away from the consensus.
* 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.
Line 158:
* 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 median is in a sense more "neutral" to the underlying polarization structure.
* 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.
* The cardinal method closest to applying this notion of "majority of consensus" is likely [[Majority Judgement]], but as per above, it will still bias towards majority factions, so even though it approximates the consensus, it ultimately sides with the dominant faction.▼
▲* The cardinal method closest to applying this notion of "majority of consensus" is likely [[Majority Judgement]], but as per above, it will still bias towards majority factions, so even though it approximates the consensus
</div>
| |||