# Spatial models of voting

There are many **spatial models of voting**. This article discusses some of them.

Each of these models puts voters and candidates in a multi-dimensional space, where each dimension represents a single political issue,^{[1]}^{[2]} sub-component of an issue,^{[3]} or candidate attribute,^{[4]} even including non-political properties of the candidates, such as perceived corruption, health, etc.^{[1]} Voters are then modeled as having an *ideal point* in this space, with a preference distance between themselves and each candidate (usually 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 strategically voting to try to minimize their distance to the actual winner).^{[5]} Other models that follow the idea of “closeness” are called proximity models.^{[6]}^{:93, 96}

## Projections

The common one-dimensional political spectrum, or various two-dimensional political compasses, can then be considered projections of this multi-dimensional space onto a smaller number of dimensions.^{[7]} For example, a study of German voters found that at least four dimensions were required to adequately represent all political parties.^{[7]}

The number of candidates and the dimensionality of the space impose 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.^{[8]}^{: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.^{[6]}^{:94}^{[9]}Spacial modeling puts voters and candidates in a multi-dimensional space, where each dimension represents a single political issue,^{[1]}^{[10]} sub-component of an issue,^{[11]} or candidate attribute,^{[12]} even including non-political properties of the candidates, such as perceived corruption, health, etc.^{[1]} Voters are then modeled as having an *ideal point* in this space, with a preference distance between themselves and each candidate (usually 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 strategically voting to try to minimize their distance to the actual winner).^{[13]} Other models that follow the idea of “closeness” are called proximity models.^{[6]}^{: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]^{n}. - 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 a multi-dimensional political space onto a space of fewer dimensions, to generalize and make discussion simpler.

## One-dimensional

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, 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.^{[14]} In France, where the terms originated, the left has been called "the party of movement" and the right "the party of order".^{[15]}^{[16]}^{[17]}^{[18]}

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

While the "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 *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

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 two-dimensional preference space.^{[19]}

### Three Telos Model

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
^{[20]}(see Moral foundations theory on Wikipedia) - Vision of nature
^{[21]}^{[22]}^{[23]}^{[24]}

## Three or higher dimensions

Political opinion can be divided into essentially any number of dimensions. Some other examples include the 3-dimensional Sapply Compass, the 4-dimensional 8values space, and the 9Axes space.

One study of German voters found that at least four dimensions were required to adequately represent all political parties.^{[25]}

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/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

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
- An upgrade to the spatial model of voters

## References

- ↑
^{a}^{b}^{c}^{d}Davis, Otto A.; Hinich, Melvin J.; Ordeshook, Peter C. (1970-01-01). "An Expository Development of a Mathematical Model of the Electoral Process".*The American Political Science Review*.**64**(2): 426–448. doi:10.2307/1953842. JSTOR 1953842.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.

- ↑ Stoetzer, Lukas F.; Zittlau, Steffen (2015-07-01). "Multidimensional Spatial Voting with Non-separable Preferences".
*Political Analysis*.**23**(3): 415–428. doi:10.1093/pan/mpv013. ISSN 1047-1987.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. - ↑ 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"
- ↑ Tideman, T; Plassmann, Florenz (June 2008). "The Source of Election Results: An Empirical Analysis of Statistical Models of Voter Behavior".
Assume that voters care about the “attributes” of candidates. These attributes form a multi-dimensional “attribute space.”

Cite journal requires`|journal=`

(help) - ↑ Woon, Jonathan. "Introduction to spatial modeling" (PDF).
*University of Pittsburgh*. - ↑
^{a}^{b}^{c}Rabinowitz, George; Macdonald, Stuart Elaine (March 1989). "A directional theory of issue voting".*American Political Science Review*.**83**(1): 93–121. doi:10.2307/1956436. JSTOR 1956436. - ↑
^{a}^{b}Alós-Ferrer, Carlos; Granić, Đura-Georg (2015-09-01). "Political space representations with approval data".*Electoral Studies*.**39**: 56–71. doi:10.1016/j.electstud.2015.04.003. hdl:1765/111247.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.

- ↑ Cho, Sungdai; Endersby, James W. (March 2003). "Issues, the spatial theory of voting, and British general elections: a comparison of proximity and directional models".
*Public Choice*.**114**(3): 275–293. doi:10.1023/A:1022616323373. JSTOR 30025956. - ↑ McCullough, B. Claire (Summer 1978). "Effects of variables using panel data: a review of techniques".
*Public Opinion Quarterly*.**42**(2): 199–220. doi:10.1086/268443. - ↑ Stoetzer, Lukas F.; Zittlau, Steffen (2015-07-01). "Multidimensional Spatial Voting with Non-separable Preferences".
*Political Analysis*.**23**(3): 415–428. doi:10.1093/pan/mpv013. ISSN 1047-1987.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. - ↑ 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"
- ↑ Tideman, T; Plassmann, Florenz (June 2008). "The Source of Election Results: An Empirical Analysis of Statistical Models of Voter Behavior".
Assume that voters care about the “attributes” of candidates. These attributes form a multi-dimensional “attribute space.”

Cite journal requires`|journal=`

(help) - ↑ Woon, Jonathan. "Introduction to spatial modeling" (PDF).
*University of Pittsburgh*. - ↑ Milner, Helen (2004). "Partisanship, Trade Policy, and Globalization: Is There a Left–Right Divide on Trade Policy" (PDF).
*International Studies Quarterly*.**48**: 95–120. doi:10.1111/j.0020-8833.2004.00293.x. - ↑ Knapp & Wright, p. 10.
- ↑ Adam Garfinkle, Telltale Hearts: The Origins and Impact of the Vietnam Antiwar Movement (1997). Palgrave Macmillan: p. 303.
- ↑ "Left (adjective)" and "Left (noun)" (2011),
*Merriam-Webster Dictionary*. - ↑ Roger Broad,
*Labour's European Dilemmas: From Bevin to Blair*(2001). Palgrave Macmillan: p. xxvi. - ↑ Yee, Ka-Ping (2006-12-08). "Voting Simulation Visualizations".
*zesty.ca*. Retrieved 2020-04-06. - ↑ Most people are sensitive to the fairness foundation
- ↑ W:A Conflict of Visions
- ↑ "Book sources",
*Wikipedia*, retrieved 2021-01-14 - ↑ Sowell, Thomas (1987).
*A conflict of visions*(1st ed. ed.). New York: W. Morrow. ISBN 978-0-688-06912-4.CS1 maint: extra text (link) - ↑ https://casnocha.com/2009/10/tragic-vs-utopian-view-of-human-nature.html
- ↑ Alós-Ferrer, Carlos; Granić, Đura-Georg (2015-09-01). "Political space representations with approval data".
*Electoral Studies*.**39**: 56–71. doi:10.1016/j.electstud.2015.04.003. hdl:1765/111247.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.