Limitations of spatial models of voting: Difference between revisions
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Note, however, that '''''opinion space''' is distinct from '''''ballot space'''. Opinion space is what contains the actual distribution of voters and candidates, and this may have any number of dimensions, voters and candidates. In contrast, ballot space is the [[Space_of_possible_elections|space of possible ballots that voters can cast]], which confines them to express their opinions in a particular way. One can think of ballot-casting as a function that takes a voter's opinion and that of the candidates (plus additional external factors), and produces a ballot: <math>\text{ballot} = f(\text{voter opinion}| \text{distribution of candidates}, \text{external factors})</math>. This article refers to the limitations of this function, that is, how much information about ''opinion space'' can in principle survive inside ''ballot space''. |
Note, however, that '''''opinion space''' is distinct from '''''ballot space'''. Opinion space is what contains the actual distribution of voters and candidates, and this may have any number of dimensions, voters and candidates. In contrast, ballot space is the [[Space_of_possible_elections|space of possible ballots that voters can cast]], which confines them to express their opinions in a particular way. One can think of ballot-casting as a function that takes a voter's opinion and that of the candidates (plus additional external factors), and produces a ballot: <math>\text{ballot} = f(\text{voter opinion}| \text{distribution of candidates}, \text{external factors})</math>. This article refers to the limitations of this function, that is, how much information about ''opinion space'' can in principle survive inside ''ballot space''. |
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To address this problem, a specific metric space has to be chosen. For the Euclidean case, this dimensional dependence was already addressed by Tideman in 1977<ref>[https://www.sciencedirect.com/science/article/pii/0097316577900772 Stirling numbers and a geometric, structure from voting theory (Good and Tideman, 1977)]</ref> (see also <ref>[https://link.springer.com/content/pdf/10.1007/s00454-001-0073-4.pdf Perpendicular Dissections of Space, Thomas Zaslavsky]</ref>, and the similar idea of Vapnik–Chervonenkis dimension<ref>[https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_dimension Vapnik–Chervonenkis dimension]</ref>.). Unfortunately, this result and its fundamental implications to the field of voting theory have gone underappreciated. |
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With these mathematical results, it is possible to infer the minimum dimensions of any real life ranked election or ballot scenario, and maybe even infer whether enough candidate diversity was present. For <math>n</math> candidates and <math>d</math> dimensions, the following table shows the absolute maximum number of ballots that ''any'' distribution of voters and candidates could possibly generate if voters are using those <math>d</math> dimensions to classify the candidates. |
With these mathematical results, it is possible to infer the minimum dimensions of any real life ranked election or ballot scenario, and maybe even infer whether enough candidate diversity was present. For <math>n</math> candidates and <math>d</math> dimensions, the following table shows the absolute maximum number of ballots that ''any'' distribution of voters and candidates could possibly generate if voters are using those <math>d</math> dimensions to classify the candidates. |
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