Limitations of spatial models of voting: Difference between revisions
Limitations of spatial models of voting (view source)
Revision as of 06:02, 4 February 2023
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== Mathematics of a spatial model ==
[[File:Maximum Voronoi regions 2D.svg|thumb|For d=2 dimensions and n=3 candidates (ABC), there is a region in the space for each of the 3! = 6 possible rankings between the candidates, so no information is lost: all possible opinion distributions and ballots can exist
In a <math>d</math>-dimensional spatial model for voter behavior, in which voters judge candidates in terms of proximity using <math>d</math> separate attributes (no matter ''how'' such attributes are used), there is a fundamental mathematical limit for how many ballots can possibly occur, in any arbitrary distribution of voters and candidates. (Equalities or partial rankings do not matter in this analysis, as they can be included in the same space with minimal adjustment.)
[[File:Voronoi regions 2D 4 candidates.svg|thumb|With a fourth candidate there are 4! = 24 possible rankings, but it's impossible to partition the space (under Euclidean metric) into more than 18 regions, one example as shown here. Therefore, many of the rankings cannot occur under this 2-dimensional model. For 3 dimensions, we can construct all of the 24 required regions for the ballots.]]
This restriction is less about the existence of an actual "Euclidean space of opinions" in the abstract (i.e. the accuracy of our chosen ''models''), but instead, about how candidates could ''ever'' be classified in terms of a finite set of attributes by voters. ''Any'' comparison voters are actually doing between any two candidates must occur in at least ''one'' attribute between them, that can be used to classify the voter's preference one way or another. This dimension <math>d</math> quantifies how many such attributes must exist in order for us to observe a given set of ballots. Thus, this is a very real and fundamental limitation of any realistic and operational description of voter behavior.
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