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Limitations of spatial models of voting: Difference between revisions

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Limitations of spatial models of voting (view source)

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{| class="wikitable"
! !! d=1 !! d=2 !! d=3 !! d=4 !! d=5 !! d=6 !! d=7 !! d=8 !! d=9 !!
|-
| '''''n=2''' ''|| 2|| || || || || || || ||
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For example, suppose a real-life Condorcet election with 6 candidates is observed. There are 720 possible rankings. If in practice we observe 250 unique ballots, then we can know with certainty that to interpret/simulate this election in terms of an ideological space we would need <math>d \ge 3</math> dimensions, because for (n=6, d=2) there are only 101 possible ballots, and we observed more than that. In other words, these candidates are being compared by voters in ''at least'' three separate attributes, otherwise these 250 ballots couldn't exist, as there would be not enough attributes to construct these observed preferences.
 
If we now look at the column <math>d = 3</math>, we can see that we could (in principle) achieve full representation of all voters on 3 issues with only 4 highly distinguished candidates. Given that we only observed 250/720 = 35% of all possible 6-candidate ballots, and about 250/326 = 77% of the maximum (n=6,d=3) ballots, it seems that we did have a lot of similarity between candidates. This suggests that there's probably significant degrees of indifference in the population that are not coming through the ballots, and are instead being expressed as strict preferences due to the limitations of the voting method.
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The practical effect of this is effectively forcing each voter to "collapse" their ideological space to at most a certain number of dimensions, which is the dimension in which their ballots saturate.
 
To interpret this, we consult the table once more. If there are d=4 important issues voters are using to judge candidates, then we require ''at least'' 5 candidates to potentially allow voters to account for all possible political positions in an election. This is how when only n=2 candidates exist, any further dimension or attribute will not lead to more resolution than for d=1. In other words, there is a collapse of the entire ideological spectrumspace in one dimension for each voter. This is, effectively, the problem of two-party domination and single-issue voting.
 
These observations also have important implications on specific voting methods. An IRV election limited to top-three rankings fundamentally limits what sort of ideological distributions can be conveyed, no matter how many candidates are running.
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From the table above, we see that if every voter is forced to rank only 3 candidates, then every voter can only express information about at most two relevant issues in their ballot<ref>There's at least one extra dimension, because a voter has to classify which are the "top three" candidates, so there has to be a "line" separating these three candidates from everyone else.</ref>, as more issues cannot ever classify the 3 ranked candidates more. Even if they are inherently ranking the candidates based on many other things, this information cannot fit into the ballot and information is fundamentally being lost. It is functionally equivalent to a scenario where voters are forced to use only two attributes to judge their candidates.
 
TheIf ranked ballots are constrained to <math>k</math> out of <math>n</math> candidates, the population, as a whole, can only cast <math>\binomfrac{n!}{d(n-k)!}</math> ballots, which means the voting method "mixes" the information multiple voters expressed, as each voter is using a different subset of attributes in their ballots. Thus, there are no guarantees all the voters are expressing information about the same issues in their ballots, and the ballots cease to be informationally commensurable, even in principle. In effect, we are left to simply hope that their priorities are, on average, similar, as to restore commensurability.
 
==Simulations ==
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If we only run simulations in d=2, with n=6 candidates we would be ignoring (1-101/720) ~ 86% of the possible opinions at any given time, possibly more in practice. Considering that [[Space_of_possible_elections|the space of possible elections]] depends heavily on the number of ballots, this means such a simulation would be ignoring the ''vast'' majority of the possible 6 candidate elections that could exist, and almost certainly would exclude very relevant and realistic scenarios for real elections, as well as very real opinion distributions.
 
In real life scenarios, there's also noise and uncertainty, which can allow for more ballots to appear in practice. But this does not fundamentally affect the above observations. It is always possible to threshold the ballots to infer the minimum "latent space" voters are using to judge candidates, inby orderignoring toballots accountwho forare too infrequent in suchthe noiseanalysis.
 
==Cardinal/rated ballots==
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