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There are many '''spatial models of voting'''. This article discusses some of them.
The '''spatial model of voting''' puts voters and candidates in a multi-dimensional space, where each dimension represents a single political issue,<ref name=":1">{{Cite journal|last=Davis|first=Otto A.|last2=Hinich|first2=Melvin J.|last3=Ordeshook|first3=Peter C.|date=1970-01-01|title=An Expository Development of a Mathematical Model of the Electoral Process|url=https://semanticscholar.org/paper/66661f9678dbe956e525e87a50b5b4ee6bf280f1|journal=The American Political Science Review|volume=64|issue=2|pages=426–448|doi=10.2307/1953842|jstor=1953842|quote=Since our model is multi-dimensional, we can incorporate all criteria which we normally associate with a citizen's voting decision process — issues, style, partisan identification, and the like.}}</ref><ref>{{Cite journal|last=Stoetzer|first=Lukas F.|last2=Zittlau|first2=Steffen|date=2015-07-01|title=Multidimensional Spatial Voting with Non-separable Preferences|url=http://pan.oxfordjournals.org/lookup/doi/10.1093/pan/mpv013|journal=Political Analysis|volume=23|issue=3|pages=415–428|doi=10.1093/pan/mpv013|issn=1047-1987|quote=The spatial model of voting is ''the'' work horse for theories and empirical models in many fields of political science research, such as the equilibrium analysis in mass elections ... the estimation of legislators’ ideal points ... and the study of voting behavior. ... Its generalization to the multidimensional policy space, the Weighted Euclidean Distance (WED) model ... forms the stable theoretical foundation upon which nearly all present variations, extensions, and applications of multidimensional spatial voting rest.|via=}}{{Dead link|date=November 2019|bot=InternetArchiveBot|fix-attempted=yes}}</ref> sub-component of an issue,<ref>If voter preferences have more than one peak along a dimension, it needs to be decomposed into multiple dimensions that each only have a single peak. "We can satisfy our assumption about the form of the loss function if we increase the dimensionality of the analysis — by decomposing one dimension into two or more"</ref> or candidate attribute,<ref>{{Cite journal|last=Tideman|first=T|last2=Plassmann|first2=Florenz|date=June 2008|title=The Source of Election Results: An Empirical Analysis of Statistical Models of Voter Behavior|url=https://www.researchgate.net/publication/228920943|quote=Assume that voters care about the “attributes” of candidates. These attributes form a multi-dimensional “attribute space.”|via=}}</ref> even including non-political properties of the candidates, such as perceived corruption, health, etc.<ref name=":1" /> Voters are then modeled as having an ''ideal point'' in this space, with a preference distance between themselves and each candidate (usually [[W:Euclidean distance|Euclidean distance]]), i.e. a voter may be closer to a candidate on gun control, but disagree on abortion. Voters are then modeled as voting for the candidates whose attributes or policy proposals are nearest to their ideal point (or [[Tactical voting|strategically voting]] to try to minimize their distance to the actual winner).<ref>{{Cite web|url=https://www.pitt.edu/~woon/courses/ps2703_Lec4.pdf|title=Introduction to spatial modeling|last=Woon|first=Jonathan|date=|website=University of Pittsburgh|url-status=live|archive-url=|archive-date=|access-date=}}</ref> Other models that follow the idea of “closeness” are called proximity models.<ref name=":0">{{cite journal | last1 = Rabinowitz | first1 = George | last2 = Macdonald | first2 = Stuart Elaine | title = A directional theory of issue voting | journal = [[American Political Science Review]] | volume = 83 | issue = 1 | pages = 93–121 | doi = 10.2307/1956436 | jstor = 1956436 | date = March 1989 | ref = harv }}</ref>{{Rp|93, 96}}▼
▲
The common one-dimensional political spectrum, or various two-dimensional [[W:Political compass|political compasses]], can then be considered [[W:Projection (mathematics)|projections]] of this multi-dimensional space onto a smaller number of dimensions.<ref name=":2">{{Cite journal|last=Alós-Ferrer|first=Carlos|last2=Granić|first2=Đura-Georg|date=2015-09-01|title=Political space representations with approval data|url=http://repub.eur.nl/pub/111247|journal=Electoral Studies|volume=39|pages=56–71|doi=10.1016/j.electstud.2015.04.003|quote=The analysis reveals that the underlying political landscapes ... are inherently multidimensional and cannot be reduced to a single left-right dimension, or even to a two-dimensional space. ... From this representation, lower-dimensional projections can be considered which help with the visualization of the political space as resulting from an aggregation of voters' preferences. ... Even though the method aims to obtain a representation with as few dimensions as possible, we still obtain representations with four dimensions or more.|hdl=1765/111247}}</ref> For example, a study of German voters found that at least four dimensions were required to adequately represent all political parties.<ref name=":2" />▼
== Projections ==
▲The common one-dimensional [[political spectrum]], or various two-dimensional [[W:Political compass|political compasses]], can then be considered [[W:Projection (mathematics)|projections]] of this multi-dimensional space onto a smaller number of dimensions.<ref name=":2">{{Cite journal|last=Alós-Ferrer|first=Carlos|last2=Granić|first2=Đura-Georg|date=2015-09-01|title=Political space representations with approval data|url=http://repub.eur.nl/pub/111247|journal=Electoral Studies|volume=39|pages=56–71|doi=10.1016/j.electstud.2015.04.003|quote=The analysis reveals that the underlying political landscapes ... are inherently multidimensional and cannot be reduced to a single left-right dimension, or even to a two-dimensional space. ... From this representation, lower-dimensional projections can be considered which help with the visualization of the political space as resulting from an aggregation of voters' preferences. ... Even though the method aims to obtain a representation with as few dimensions as possible, we still obtain representations with four dimensions or more.|hdl=1765/111247}}</ref> For example, a study of German voters found that at least four dimensions were required to adequately represent all political parties.<ref name=":2" />
The number of candidates and the dimensionality of the space impose [[Dimensional limitations of the spatial model|fundamental limitations]] on the information content of ballots, as well as the commensurability of their information content.
== Spatial modeling ==
:''see also: [[Political spectrum]]''
Spatial modeling attempts to show the perceptions and decisions of voters when issue voting strategies are used in elections.<ref>{{cite journal | last1 = Cho | first1 = Sungdai | last2 = Endersby | first2 = James W. | title = Issues, the spatial theory of voting, and British general elections: a comparison of proximity and directional models | journal = Public Choice | volume = 114 | issue = 3 | pages = 275–293 | doi = 10.1023/A:1022616323373 | jstor = 30025956 | date = March 2003 | ref = harv |url=|via=}}</ref>{{Rp|275}} Spatial modeling assumes that if someone’s issue preferences are placed on a hypothetical spatial field along with all possible candidates’ policy positions, the individual will vote for the candidate whose political stances are closest to their own.<ref name=":0" />{{Rp|94}}<ref>{{cite journal | last = McCullough | first = B. Claire | title = Effects of variables using panel data: a review of techniques | journal = Public Opinion Quarterly | volume = 42 | issue = 2 | pages = 199–220 | doi = 10.1086/268443 | date = Summer 1978 | ref = harv |url=|via=}}</ref>Spacial modeling puts voters and candidates in a multi-dimensional space, where each dimension represents a single political issue,<ref name=":1">{{Cite journal|last=Davis|first=Otto A.|last2=Hinich|first2=Melvin J.|last3=Ordeshook|first3=Peter C.|date=1970-01-01|title=An Expository Development of a Mathematical Model of the Electoral Process|url=https://semanticscholar.org/paper/66661f9678dbe956e525e87a50b5b4ee6bf280f1|journal=The American Political Science Review|volume=64|issue=2|pages=426–448|doi=10.2307/1953842|jstor=1953842|quote=Since our model is multi-dimensional, we can incorporate all criteria which we normally associate with a citizen's voting decision process — issues, style, partisan identification, and the like.}}</ref><ref>{{Cite journal|last=Stoetzer|first=Lukas F.|last2=Zittlau|first2=Steffen|date=2015-07-01|title=Multidimensional Spatial Voting with Non-separable Preferences|url=https://www.cambridge.org/core/journals/political-analysis/article/multidimensional-spatial-voting-with-nonseparable-preferences/112FA71B889588C52C011CE7CEBBDAF2|journal=Political Analysis|volume=23|issue=3|pages=415–428|doi=10.1093/pan/mpv013|issn=1047-1987|quote=The spatial model of voting is ''the'' work horse for theories and empirical models in many fields of political science research, such as the equilibrium analysis in mass elections ... the estimation of legislators’ ideal points ... and the study of voting behavior. ... Its generalization to the multidimensional policy space, the Weighted Euclidean Distance (WED) model ... forms the stable theoretical foundation upon which nearly all present variations, extensions, and applications of multidimensional spatial voting rest.|via=}}</ref> sub-component of an issue,<ref>If voter preferences have more than one peak along a dimension, it needs to be decomposed into multiple dimensions that each only have a single peak. "We can satisfy our assumption about the form of the loss function if we increase the dimensionality of the analysis — by decomposing one dimension into two or more"</ref> or candidate attribute,<ref>{{Cite journal|last=Tideman|first=T|last2=Plassmann|first2=Florenz|date=June 2008|title=The Source of Election Results: An Empirical Analysis of Statistical Models of Voter Behavior|url=https://www.researchgate.net/publication/228920943|quote=Assume that voters care about the “attributes” of candidates. These attributes form a multi-dimensional “attribute space.”|via=}}</ref> even including non-political properties of the candidates, such as perceived corruption, health, etc.<ref name=":1" /> Voters are then modeled as having an ''ideal point'' in this space, with a preference distance between themselves and each candidate (usually [[W:Euclidean distance|Euclidean distance]]), i.e. a voter may be closer to a candidate on gun control, but disagree on abortion. Voters are then modeled as voting for the candidates whose attributes or policy proposals are nearest to their ideal point (or [[Tactical voting|strategically voting]] to try to minimize their distance to the actual winner).<ref>{{Cite web|url=https://www.pitt.edu/~woon/courses/ps2703_Lec4.pdf|title=Introduction to spatial modeling|last=Woon|first=Jonathan|date=|website=University of Pittsburgh|url-status=live|archive-url=|archive-date=|access-date=}}</ref> Other models that follow the idea of “closeness” are called proximity models.<ref name=":0">{{cite journal | last1 = Rabinowitz | first1 = George | last2 = Macdonald | first2 = Stuart Elaine | title = A directional theory of issue voting | journal = American Political Science Review | volume = 83 | issue = 1 | pages = 93–121 | doi = 10.2307/1956436 | jstor = 1956436 | date = March 1989 | ref = harv |url=|via=}}</ref>{{Rp|93, 96}}
Mathematically (and spatially), a line on a political spectrum can be defined by:
* a '''dimension''' n, representing the number of independent issues under consideration. Voters are represented by points in V = [0,1]<sup>n</sup>.
* a '''voter density function''' v: V → ℜ
* a '''distance function''' d: V × V → ℜ that is positive definite and symmetric and satisfies the triangle inequality. Ballots are determined from the assumption that voters prefer candidates which are closer (according to this distance function) to them.
Ultimately, these are projections of [[Spatial model of voting|a multi-dimensional political space]] onto a space of fewer dimensions, to generalize and make discussion simpler.
== One-dimensional ==
{{main|Left-right political spectrum}}
{{wikipedia|Left-wing politics}}
{{wikipedia|Right-wing politics}}
A single-dimensional model envisions a horizontal line, with voters distributed along a single left-to-right axis. This is frequently referred to as the [[Left-right political spectrum|left–right political spectrum]], and is how many people classify political positions, ideologies and parties. The people on the ends are said to practice [[extremism]], and the intermediate stance is called [[centrism]]. On this type of political spectrum, left-wing politics and right-wing politics are often presented as opposed, although a particular individual or group may take a left-wing stance on one matter and a right-wing stance on another; and some stances may overlap and be considered either left-wing or right-wing depending on the ideology.<ref>{{cite journal|last=Milner|first=Helen|date=2004|title=Partisanship, Trade Policy, and Globalization: Is There a Left–Right Divide on Trade Policy|url=https://www.princeton.edu/~hmilner/forthcoming%20papers/ISQ_milner_judkins2004.PDF|journal=International Studies Quarterly|volume=48|issue=|pages=95–120|doi=10.1111/j.0020-8833.2004.00293.x|pmid=|accessdate=|via=}}</ref> In [[France]], where the terms originated, the left has been called "the party of movement" and the right "the party of order".<ref>Knapp & Wright, p. 10.</ref><ref>Adam Garfinkle, Telltale Hearts: The Origins and Impact of the Vietnam Antiwar Movement (1997). Palgrave Macmillan: p. 303.</ref><ref>"[http://www.merriam-webster.com/dictionary/left Left (adjective)]" and "[http://www.merriam-webster.com/dictionary/left?show=1&t=1325146819 Left (noun)]" (2011), ''Merriam-Webster Dictionary''.</ref><ref>Roger Broad, ''Labour's European Dilemmas: From Bevin to Blair'' (2001). Palgrave Macmillan: p. xxvi.</ref>
Using the formulas above: n=1, v(x)=1, and d(x,y)=|x-y|. The directions on this spectrum are normally referred to as left and right.
=== Horseshoe theory ===
:''Main article: [[horseshoe theory]]''
What is often called the "[[horseshoe theory]]" claims that the extreme authoritarian economic left (Communism) is adjacent or close to extreme authoritarian economic right (neo-reactionism/fascism). A classification that follows this thought must then place these two close by or next to each other: either by using dimensions where they naturally fit next to each other, or by making opinion space curved so that going in the direction of fascism leads to Communism.
== Two-dimensional ==
{{wikipedia|Nolan chart}}
{{wikipedia|The Poltical Compass}}
{{wikipedia|Pournelle chart}}
While the [[Spatial model of voting#Horseshoe theory|"horseshoe theory" noted above]] appears two-dimensional, it is obviously just a variation on the [[left-right political spectrum]], which is uni-dimensional.
There are many two-dimensional political spaces, many of which have enough credible citations to have articles on [[English Wikipedia]]. These include the following:
* The [[W:Nolan chart]]
* [[W:The Political Compass]]
* The [[W:Pournelle chart]]
The ''Nolan chart'' and ''the Political Compass'' are two popular examples, which can be seen as rotated versions of each other. The ''Pournelle chart'' is another variation with a different set of axes. Other two-dimensional models are described below.
=== Yee diagrams ===
{{Main|Yee diagram}}[[File:Yee diagram IrvSq2.png|thumb|A [[Yee diagram]] of [[Instant-Runoff Voting|instant-runoff voting (IRV)]] with four candidates, showing that the Yellow candidate has been squeezed out (due to "[[center squeeze]]") and cannot win.]]The "Yee diagram" (named after [[Ka-Ping Yee]]) is used to illustrate the behavior of election methods, given a fixed set of candidates in a [[Spatial model of voting|two-dimensional preference space]].<ref>{{Cite web|url=http://zesty.ca/voting/sim/|title=Voting Simulation Visualizations|last=Yee|first=Ka-Ping|date=2006-12-08|website=zesty.ca|url-status=live|archive-url=|archive-date=|access-date=2020-04-06}}</ref>
=== Three Telos Model ===
{{Main|Three Telos Model}}
[[File:Politics_map_triangle1.png|alt=|thumb]]
The "Three Telos Model" or "Triangle Political Map" is two-dimensional political model where voters tend to spread out in three directions. It describes political beliefs based on the core axiom of the philosophy, where the voter's depart from the center based on their core beliefs.
Each of the three colors (the "equality ''leftist''", the "freedom ''liberal"'' and the "tradition ''conservative''") have different criteria. The criteria are listed as:
* Justification
* Philosophical foundation
* Prestige idenifier
* Moral foundation<ref>Most people are sensitive to the fairness foundation</ref> (see [[Wikipedia:Moral foundations theory|Moral foundations theory]] on Wikipedia)
* Vision of nature <ref>[[W:A Conflict of Visions]]</ref><ref>{{Citation|title=Book sources|url=https://en.wikipedia.org/wiki/Special:BookSources/978-0688069124|work=Wikipedia|language=en|access-date=2021-01-14}}</ref><ref>{{Cite book|last=Sowell|first=Thomas|url=https://openlibrary.org/books/OL2724504M/A_conflict_of_visions|title=A conflict of visions|date=1987|publisher=W. Morrow|isbn=978-0-688-06912-4|edition=1st ed.|location=New York}}</ref> <ref>https://casnocha.com/2009/10/tragic-vs-utopian-view-of-human-nature.html</ref>
[[File:TelosTriangle.png|alt=|thumb|As in the two dimensional maps like the political compass, the differing ideologies can be put onto this map.]]
== Three or higher dimensions ==
Political opinion can be divided into essentially any number of dimensions. Some other examples include the 3-dimensional [https://sapplyvalues.github.io Sapply Compass], the 4-dimensional [https://8values.github.io/ 8values] space, and the [https://9axes.github.io/ 9Axes] space.
One study of German voters found that at least four dimensions were required to adequately represent all political parties.<ref name=":german">{{Cite journal|last1=Alós-Ferrer|first1=Carlos|last2=Granić|first2=Đura-Georg|date=2015-09-01|title=Political space representations with approval data|url=http://repub.eur.nl/pub/111247|journal=Electoral Studies|volume=39|pages=56–71|doi=10.1016/j.electstud.2015.04.003|quote=The analysis reveals that the underlying political landscapes ... are inherently multidimensional and cannot be reduced to a single left-right dimension, or even to a two-dimensional space. ... From this representation, lower-dimensional projections can be considered which help with the visualization of the political space as resulting from an aggregation of voters' preferences. ... Even though the method aims to obtain a representation with as few dimensions as possible, we still obtain representations with four dimensions or more.|hdl=1765/111247}}</ref>
There has been references to many other political compasses that are similar, orthogonal or even contradictive.
* https://www.reddit.com/r/Politicaltests/wiki/listoftests (archive [https://archive.md/wip/amTxo https://archive.md/amTxo])
* https://l-lists.com/en/lists/0isll4.html (archive https://archive.md/YATlp)
* https://www.reddit.com/r/PoliticalCompass/comments/flzinl/list_of_every_test_i_know_including_some_you/ (https://archive.md/MOS9q)
In the end, it is difficult to model the behaviors of human beings in such a way that they can be reduced to simple numbers and political spectra as lines on a graph.
== Limitations ==
{{Main|Limitations of spatial models of voting}}
While the spatial model is intended to be an approximate representation of real-life opinion distributions, the number of dimensions chosen for the geometric embedding impose 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.
Therefore, although any concrete spatial model is an approximation, and should not be taken as a faithful representation of an electorate, the ballots cast in an election are our definitive source of information from the electorate, and these still induce an effective opinion space. Understanding these dimensional limitations can inform us about how much relevant information is being potentially being collected (or discarded) in an election.
== See also==
*[[Statistics]]
*[[McKelvey–Schofield chaos theorem]]
*[[Dimensional limitations of the spatial model]]
*[[User:Lucasvb/An upgrade to the spatial model of voters|An upgrade to the spatial model of voters]]
==References==
<references />
[[Category:Voting theory]]
[[Category:Political spectrum]]
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