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"[[Spatial models of voting]]" are ubiquitous in theoretical study and simulations of voting methods. This article describes many '''limitations of spatial models of voting'''.
{{rename|from=Dimensional limitations of the spatial model|to=Limitations of the spatial model|date=January 2023}}
 
[[Spatial model of voting|Spatial models]] are ubiquitous in theoretical study and simulations of voting methods. In thesespatial models of agent behavior, agents (e.g. voters, candidates) are placed in an abstract geometric space, usually Euclidean, in which each dimension denotes some ideological alignment or opinion on an issue. The behavior of agents is modeled by how "close" (under some appropriate metric) they are to other agents in this space. In the context of voting, voters are modelled as ranking candidates depending on their proximity within this space.
 
However, models based too strictly on geometric representations have challenges representing both voters and candidates. This article describes some of the challenges.
However, the number of dimensions chosen for this geometric embedding imposes fundamental restrictions on the allowed number of candidates which may be effectively distinguished by the voters using ballots, as there is only a finite number of regions possible for each possible ranking assignment of candidates. Conversely, an insufficient number of candidates in a ballot (either by a small number of candidates or arbitrarily restricting the ballot) will also fundamentally restrict the effective opinion space voters can express, as the effective dimensionality is inherently reduced.▼
 
== Number of dimensions ==
The following article discusses this limitation and some implications, both in theory and practice. The specific numerical results below assume an Euclidean space and Euclidean distances, but similar qualitative arguments apply to any spatial model and chosen metric, as well as the actual real-life behavior of voters (although quantifying it is impossible).▼
▲However, theThe number of dimensions chosen for this geometric embedding imposes fundamental restrictions on the allowed number of candidates. There is a limited number of dimensions whichthat may be effectively distinguished by the voters using ballots, as there is only a finite number of regions possible for each possible ranking assignment of candidates. Conversely, an insufficient number of candidates in a ballot (either by a small number of candidates or arbitrarily restricting the ballot) will also fundamentally restrict the effective opinion space voters can express, as the effective dimensionality is inherently reduced.
 
▲The following article discusses this limitation and some implications, both in theory and practice. The specific numerical results below assume an Euclidean space and Euclidean distances, but similar qualitative arguments apply to any spatial model and chosen metric, as well as the actual real-life behavior of voters (although quantifying it is impossible).
 
==How many ballots could voters ''actually'' cast?==
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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. 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.]]
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, e.g., any ballot with D ranked last, in the image. 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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These observations also have important implications on specific voting methods. An [[Instant-runoff Voting]] election limited to top-three rankings fundamentally limits what sort of ideological distributions can be conveyed, no matter how many candidates are running.
 
From the table above, we see that if every voter is forced to rank only 3 candidates, then every voter can only express information about at most two relevant issues in their ballot<ref>There's at least one extra dimension, because a voter has to classify which are the "top three" candidates, so there has to be a "line" separating these three candidates from everyone else.</ref>, as more issues cannot ever classify the 3 ranked candidates morefurther. Even if they are inherently ranking the candidates based on many other things, this information cannot fit into the ballot and information is fundamentally being lost. It is functionally equivalent to a scenario where voters are forced to use only two attributes to judge their candidates.
 
If ranked ballots are constrained to <math>k</math> out of <math>n</math> candidates, the population, as a whole, can only cast <math>\frac{n!}{(n-k)!}</math> ballots, which means the voting method "mixes" the information multiple voters expressed, as each voter is using a different subset of 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.
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