Limitations of spatial models of voting: Difference between revisions
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==How many ballots could voters ''actually'' cast?== |
==How many ballots could voters ''actually'' cast?== |
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| ⚫ | 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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| ⚫ | [[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 6 regions. Therefore, most of the rankings cannot occur under this 2-dimensional model. For 3 dimensions, we can construct the 24 required regions.]] |
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| ⚫ | This is true independently of any abstract mathematical model of reality of human behavior, as it is a constraint of the ballots themselves and their information content. In reality, these distinctions are based on some internal attributes and judgements voters have about the world and the candidates, and this is the information voters want ballots to convey. This is what voting methods attempt to ''represent'' from voters. |
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| ⚫ | 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 |
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| ⚫ | But due to several limitations, not not all these ballots can ''actually get cast'' in an election. In practice, we only observe a few preference orders, indicating that there's a lot of correlation between voters and between candidates, or putting it in another way, that the "space" of attributes relevant in the election is smaller than the one expressible by the ballots. This is important to consider when developing a mathematical structure to abstractly discuss real voting methods and voter behavior. |
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In other words, while a spatial model attempts to reverse engineer real-life behaviors and construct a model of the information underlying an election, the ballots themselves, be it from real life elections or computer simulations, can only capture some of the information. |
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| ⚫ | But due to several limitations, not not all these ballots can ''actually get cast''. In practice, we only observe a few preference orders, indicating that there's a lot of correlation between voters and between candidates, or putting it in another way, that the "space" of attributes relevant in the election is smaller. This is important to consider when developing a mathematical structure to abstractly discuss voting methods and voter behavior. |
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== 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 6 regions. Therefore, most of the rankings cannot occur under this 2-dimensional model. For 3 dimensions, we can construct 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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