Limitations of spatial models of voting: Difference between revisions

The [[Spatial model of voting|spatial model]] is ubiquitous in theoretical study and simulations of voting methods. However, the dimension of this geometric embedding imposes fundamental restrictions on the allowed number of candidates which may be distinguished, as there is a finite number of regions possible for each possible ranking. The following article discusses this limitation and some implications.

 

 

With <math>n</math> candidates in an election, be it rated or ranked, there are <math>n!</math> possible rankings between candidates. These <math>n!</math> possible preferences indicate all the possible ''distinctions'' voters could ever possibly make between alternatives (no matter ''how'' those distinctions are made).

 

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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.

 

 

The table also informs us about the limitations of a voting method to really convey the information voters are using to classify the candidates. For a given number of candidates, after a given dimension any further dimensions will not add any extra resolution, as the ballot cannot express such information. This is why the rows in the table stop, as we would have maxed out the possible ballots.

The population as a whole can cast <math>\binom{n}{d}</math> ballots, which means the voting method "mixes" the information multiple voters expressed, as each voter is using different 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.

 

 

All of these observations can also inform us of how restrictive a voting simulation is. If we are simulating a 5 candidate election, we would need at least d=4 to ensure we account for all possible scenarios (realistic or not, which we can't really rule out with any distribution).

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, in order to account for such noise.

 

 

The immediate question is how this affects cardinal (rated) methods? The ordinal (ranked) information on the cardinal ballots can still be interpreted as above (provided there are at least <math>n</math> intervals in the scale), but there's more to it.

To indicate faction overlaps and degrees of preference, we would need at least three dimensions in this case, so this disagreement has "more room" to exist. Strictly speaking, this extra "overlap parameter" could exist for every pair of candidates, independently. So to simulate an arbitrary cardinal election in its full generality, we would need extra information on the <math>n(n-1)/2</math> pairs of candidates, giving the dimension required for full (theoretical) generality:

 

 

 

These extra "dimensions" may perhaps not necessarily be in the same geometric space, but as additional parameters in the model (i.e. the '''external factors''' discussed above, such as voters priorities and external circumstances weighting them).

Note, once more, that here we are interested in the ''opinion space''. A score ballot could be written as a point in a <math>n</math>-dimensional hypercube, but this space has the "axes" as the ''candidates'', not as the ''opinions'' of voters, which misses the point of the process of voting, as it is uncorrelated with the underlying opinion distribution of voters. Real-life or simulated voters are not defined by their ballots, but their opinions, which the ballots are intended to represent. It is the opinion differences that ''generate'' the ballots. The fact the minimum dimension for a complete opinion and ballot space for ranked methods are both on dimension <math>d=n-1</math> is a coincidence. For the rated case, this is unclear.