Limitations of spatial models of voting: Difference between revisions
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== Mathematics of a spatial model == |
== Mathematics of a spatial model == |
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[[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. With a fourth candidate there are 4! = 24 possible rankings, but it's impossible to partition the space (under Euclidean metric) into more than |
[[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. 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. Therefore, many of the rankings cannot occur under this 2-dimensional model. For 3 dimensions, we can construct all of the 24 required regions.]] |
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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.) |
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.) |
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