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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To illustrate, let us imagine two issues and the distribution of voters regarding those issues. We will first consider two consensus issues, so the entire population is in strong agreement here.
In an election, there would be many candidates, and voters would cast ranked ballots giving preference information between any two of them. We will look at two such candidates out of many (the others will be hidden, as preference is strictly pairwise information). So keep that in mind, this is not an election with only two candidates.
We will consider two candidates, each forming a "faction" defined by the preferences people express. How does the ordinal/ranked framework react to this situation?▼
▲We will consider the two candidates
[[File:Ranking centroids.gif]]
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Note that while there is plenty of consensus, the ranked preferences "slices" the population in various ways (here, we assume a voter sides with whatever candidate is ideologically closer). Thus, rankings are inherently factionalist, and any "majority" created (shaded background) represents a distorted and artificial picture of the true opinions of the population under such a scenario.
Additionally, each of the two artificial factions will perceive its own "factional consensus" (moving crosses), which will be far away from the other. This happens even though both groups actually have a greater underlying consensus, which remains unchanged (black static cross, center). You can think of these crosses as "the ideological picture" the ranked preferences are painting to us. As you can see, it is a very distorted picture.
Thus, under ordinalism or ranked preferences, "
Notice how fringe candidates (when the dots move towards the edge) can easily radicalize their minority faction, creating a highly distorted faction consensus near the fringe. In real life, complete allegiance to a faction, and support for political candidates, usually creates an echo chamber effect. These people will be more likely to side and engage with other "like-minded people", according to this faction that was established. But as we can see from the above diagram, even if the population as a whole shares a lot of consensus and agreement, a fringe candidate can generate the illusion of a faction having its own fringe consensus. Furthermore, if this occurs, it will be worse when a consensus candidate actually exits, as that pushes the dividing line further towards the fringe.
What if we had a mixture of polarization and consensus?
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Ordinal voting methods all appeal to "majority rule" in one form or another. But given the fundamental limitations of the ordinal "majority", which is dominated by the candidates, not the voters, one should consider alternative justifications for this criterion.
In conclusion, since people have multiple attributes which can be used to classify them, in many attributes they will be a majority and in many others a minority. How are we supposed to claim that any one of these possible divisions of the population has a greater claim to power over the other? More importantly, what processes are defining this partition, and how legitimate are they?
== Cardinalism and the "majority of consensus" ==
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As we have seen, the notion of a "majority" as an inherent property of the voters is hard to establish using the ordinal formalism. The candidates have too much influence in what it actually conveys.
Under a cardinal framework, however, the concept is more subtle. Taking the consensus as the blueprint of voter cohesion, we can informally define a "'''''majority of consensus'''''" as the group of 50%+1 voters which lie closest to '''''all''''' of the existing consensuses. A more natural notion of majority can be defined in terms of the spatial model of voters.
As before, we will consider an election with many candidates. Voters would be casting cardinal ballots which inherently carry comparative information between the many candidates. We then look at what information would be available between two candidates, if we look at the scores given to both by the voters. Once more, this is not an election with two candidates, but a picture of the electorate two candidates in an election provide to us.
[[File:Majority of consensus histograms.gif]]
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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
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.
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?
[[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 there is a consensus or a polarization.
As before, either the mean or 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.
== Conclusion ==
* There is no such thing as "''the'' majority", as it is usually promoted in democracy and ranked method advocacy. It is not a property of the voters that we are "trying to find out" through the voting process.
* The existence of multiple issues implies the existence of multiple majorities and minorities, which will generally be incompatible. What is the legitimacy of giving power/representation to any one of them?
* A voting method cannot "guarantee a majority" in any meaningful or representative way.
* Ranking encourages factionalism and creates artificial polarization where there is none. This distorts our picture of the true ideological distribution of voters and factions, and voters will respond to this by becoming even more factionalist.
* If the goal of democracy is to represent the population as a whole, with all its agreements and disagreements, ranked methods are sub-optimal. If the goal of democracy is to promote the ideals of the dominant faction, established largely arbitrarily on the spot, then ranked methods suit this goal.
* More generally, forcing voters to take sides destroys consensus and agreements. The corollary of this is that [[Instant-Runoff Voting]] is anti-consensus.
* Condorcet methods are designed to make the best use of limited ranking information to find 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, 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.
* 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 mean 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.
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