Spatial Theory

POLSCI 240 / PSY 225: Political Psychology

March 17, 2025

How do people choose?

  • What are their decision rules?
  • What considerations do they take into account?
  • What are the implications for public policy?

Spatial theories of voting

New York Times

Spatial theories of voting

The core ideas of spatial theories of voting are:

  1. We can locate both voters and alternatives (candidates, parties) in a geometric space
  2. Voters choose alternatives using some rule that translates relative distance and orientation into utility
  3. Alternatives choose locations in space to maximize their utility (e.g., max votes)

Spatial theories of voting


Building blocks of a spatial theory

To build a spatial theory of voting, we need:

  1. A representation of the political “space” (e.g., left-right)
  2. An idea about how relative positioning affects voter evaluations (e.g., proximity)
  3. A function for translating relative positions into utility (e.g., quadratic loss)
  4. A decision rule (e.g., maximize expected utility)

Political space

Political space

Political space

Political space

Proximity voting

Proximity voting says: “the value of an alternative is decreasing in spatial distance”

But what is “distance” and how exactly does utility decrease as distance increases?

What does it mean to be “close”?

Measuring proximity

There are three important considerations to think about with respect to measuring proximity:

  1. What kind of distance metric should we use?
  • e.g., “city-block” or “Euclidean”?
  1. Are some dimensions more important than others?
  • e.g., care more about economic policy or social policy?
  1. Are preferences across dimensions separable or non-separable?
  • e.g., does proximity on one dimension depend on proximity on the other?

Distance metric

Dimension importance

If a voter cares about one dimension more than another, this is like multiplying the distances on the more important dimension by a factor greater than 1

  • It is like “stretching” and “squishing” the space along the dimension axes

Importance stretches along dimensions axes


Example

Example

Implications

The relative importance of dimensions can potentially be manipulated (e.g., agenda-setting)

  • A lot of political competition is about conceptualizing (defining the terms of) party differences

Implications

Voters make trade-offs across dimensions

  • Voters may be willing to sacrifice on one dimension to get better outcomes on another
  • Especially true when parties are far apart on a dimension they care about

Spatial voting and democratic norms

Graham and Svolik (2020)

Spatial voting and democratic norms

Graham and Svolik (2020)

Spatial voting and democratic norms

AP

Spatial voting and democratic norms

Graham and Svolik (2020)

Separable or non-separable?

We say that preferences are non-separable when the value of positions on one dimension depend on the other dimension

  • e.g., I may prefer higher taxes if we spend a lot on education, but lower taxes if we spend only a little on education

This is another kind of “dimension-stretching” but along the diagonals, instead of just up-down and left-right

  • When we want alternatives to “miss” in the same-direction, we call them complements
  • When we want them to “miss” in different directions, we call them substitutes

Non-separability stretches along diagonals


Example

Proximity model, mathematically

\[ \sqrt{(\boldsymbol{\text{V}} - \boldsymbol{\text{P}})' \mathbf{A} (\boldsymbol{\text{V}} - \boldsymbol{\text{P}})} \]

  • \(\boldsymbol{\text{V}}\) is the voter’s (vector-valued) position in space

  • \(\boldsymbol{\text{P}}\) is the party’s (vector-valued) position in space

  • \(\mathbf{A}\) is a matrix and is the metric for the space: it defines how distances are measured

    • Diagonal entries define the relative importance of each dimension
    • Off-diagonal elements define non-separability, with + meaning substitutes, and - meaning complements

\[ \mathbf{A} = \begin{pmatrix} 1 & 0.5 \\ 0.5 & 1 \end{pmatrix} \]

Example

# Voter's position
V = c(1, 1)

# Candidate's position
P = c(0.5, -0.25)

# Metric matrix
A = matrix(c(2, 0.5, 
             0.5, 1), 
           2, 2)

# Calculate distance
sqrt( t(V - P) %*% A %*% (V - P) )
        [,1]
[1,] 1.63936

A function to translate proximity into utility

Distance alone is not enough to make a decision

  • We need a rule that translates distance into value (utility)

An important consideration in choosing a function concerns the marginal returns to utility as a function of distance

  • Increasing marginal returns to distance (concave, e.g., quadratic loss)
  • Decreasing marginal returns to distance (convex, e.g., gaussian (normal) loss (at high distance values))

Loss functions

Final proximity model of voting (quadratic)

\[ U(\text{P}_1) = - \left( \sqrt{(\boldsymbol{\text{V}} - \boldsymbol{\text{P}_1})' \mathbf{A} (\boldsymbol{\text{V}} - \boldsymbol{\text{P}_1})} \right)^2 = - (\boldsymbol{\text{V}} - \boldsymbol{\text{P}_1})' \mathbf{A} (\boldsymbol{\text{V}} - \boldsymbol{\text{P}_1}) \]

\[ U(\text{P}_2) = - \left( \sqrt{(\boldsymbol{\text{V}} - \boldsymbol{\text{P}_2})' \mathbf{A} (\boldsymbol{\text{V}} - \boldsymbol{\text{P}_2})} \right)^2 = - (\boldsymbol{\text{V}} - \boldsymbol{\text{P}_2})' \mathbf{A} (\boldsymbol{\text{V}} - \boldsymbol{\text{P}_2}) \]

Decision rule: Choose party (\(\text{P}_1\) or \(\text{P}_2\)) that has higher utility

Evaluating the proximity model

The standard proximity spatial model captures important aspects of political psychology and party competition:

  • Highly intuitive! Vote for candidates “close” to you

  • Political discourse is often abstract and ideological (e.g., liberal, conservative, moderate, “far left”, “extreme right”)

  • Proximity model predicts a centripetal tendency toward the median voter (e.g., median primary voter, general election voter, etc.)

  • Helps to understand strategic aspects of party competition

    • emergence of smaller parties at extremes which exert a countervailing force on center parties
    • introduction (or emphasis) on “cross-cutting” ideological dimensions to shift lines of conflict in advantageous direction

Evaluating the proximity model

The standard proximity spatial model also seems to miss some important things about party competition and citizens’ thinking about politics

  • Parties don’t converge to the median voter completely, and polarization is increasing in some places

  • There is little to no role for the status quo in the standard proximity model

    • Political discourse and thought is often relative: too much spending, not enough attention to immigration, too liberal, etc.

  • What an elected official or party can accomplish in office is constrained by time and institutions: shouldn’t we discount candidates’ claims about their positions?

Proximity with discounting

The core idea is that voters move candidate/party positions closer to the status quo (SQ) and then vote on these “shadow” positions

  • How far they move candidates depends on a discount factor: a number between 0 and 1

    • Smaller values mean more movement toward the SQ
    • The amount of discounting may depend on factors of both the candidate/party (e.g., how competent are they?) and the context (e.g., divided government)

Discounting and polarization

The more citizens discount toward the status quo, the more incentive there is for candidates to push their rhetoric in extreme directions to ensure their shadow points line up with their targeted voters

Discounting, mathematically (1-dim)

Imagine we have one voter choosing between two candidates, the SQ is at 0.0, and a discount factor of 0.50

# voter
V = 0.25

# status quo
sq = 0.0

# candidate 1
C1 = 1.0

# candidate 2
C2 = -0.25

# discount factor
d = 0.50

# utility w/o discounting (C2 > C1)
-(V - C1)^2 # U(C1)
[1] -0.5625
-(V - C2)^2 # U(C2)
[1] -0.25
# utility w/ discounting (C1 > C2)
-(V - (sq + d*(C1 - sq)))^2 # U(C1)
[1] -0.0625
-(V - (sq + d*(C2 - sq)))^2 # U(C2)
[1] -0.140625

Directional models

Directional spatial voting models begin with similar observations: much of political discourse is about movement away from the status quo in one direction or another

  • Citizens heavily discount politicians’ claims about what they can accomplish

  • But they have a preference, of varying intensity, for moving policy either in a left- or right-wing direction

  • But they are worried about candidates’ credibility: will they actually be willing and able to move policy in the direction desired?

    • Candidates signal their commitment to a direction through rhetoric, and citizens’ take this into account
    • Their degree of commitment matters more when citizens want to move policy further in one direction or another

Directional models, intuitively


Directional models, intuitively


Directional models, mathematically (1-dim)

Utility is simply the product of the voter’s point relative to the status quo and the candidate’s point relative to the status quo

# SQ
sq = 5

# voter
v = 2

# candidate 1
c1 = 1

# candidate 2
c2 = 7

# utility for c1
(v - sq) * (c1 - sq)
[1] 12
# utility for c2
(v - sq) * (c2 - sq)
[1] -6

Directional models, 2 dimensions

Directional models, 2 dimensions

In the standard directional model, the utility of a candidate for a voter is the dot product (vector product) of their positions in space:

\[ \begin{align} \text{U}(\text{V}) &= \text{V} \cdot \text{C} \\ &= \cos \angle (\text{V}, \text{C}) ||\text{V}|| ||\text{C}|| \end{align} \]

where:

  • \(\cos \angle\) is the cosine of the angle between their vectors (ranges from -1 (180 degrees) to 1 (0 degrees))
  • \(||\text{V}||\) is the length of the vector for the voter (and similarly for \(||\text{C}||\))
  • So directional utility is the product of the two vector lengths multiplied by a number that is between 0 and 1 when vectors “point” in same direction, and between 0 and -1 when they point in opposite directions

Directional models, mathematically (2-dim)

Utility is simply the dot product of the voter’s point relative to the status quo and the candidate’s point relative to the status quo

# SQ
sq = c(0,0)

# voter
v = c(2,1)

# candidate 1
c1 = c(-1,-2)

# candidate 2
c2 = c(2,2)
# utility for c1 using dot product
(v - sq) %*% (c1 - sq)
     [,1]
[1,]   -4
cos(angle(v,c1))
[1] -0.8
norm(v, type="2")
[1] 2.236068
norm(c1, type="2")
[1] 2.236068
cos(angle(v,c1)) * norm(v, type="2") * norm(c1, type="2")
[1] -4
# utility for c2
(v - sq) %*% (c1 - sq)
     [,1]
[1,]   -4
cos(angle(v,c2))
[1] 0.9486833
norm(v, type="2")
[1] 2.236068
norm(c2, type="2")
[1] 2.828427
cos(angle(v,c2)) * norm(v, type="2") * norm(c2, type="2")
[1] 6

Advantages of directional

Advantages

  • Elegantly captures idea that status quo matters and political actors are constrained in what they can accomplish
  • Captures an important aspect of real-world discourse and campaigning: often about relative positions
  • Predicts polarization
  • Captures (to some degree) the observation that citizens often think of themselves as part of a “side” or team, rather than at a location in space

Thermostatic model of public opinion

One possible implication of directional models is that public opinion is thermostatic

  • Voters want to move policy left or right relative to status quo, so elect the directionally-appropriate party
  • But parties overshoot voters’ ideal point, so preferences swing back and forth over time

Thermostatic model of public opinion


Public mood

Stimson (2025)

Public mood

Stimson (2025)

Disadvantages of directional

Disadvantages

  • Proximity really does seem to matter! People say so all the time

    • Directional model predicts unbounded, increasing utility the more “extreme” a candidate gets
    • “Fixes” to this problem are not very compelling

  • Is this the best model for capturing “identity”-related considerations?

Discounting as compromise model

The discounting model can potentially capture the intuitions of directional models without losing the basic idea of proximity

  • Discounting remains a proximity model at heart: people want to elect candidates that produce policies “close” to their ideal points

  • But they are also skeptical that individual politicians or parties, in short periods of time, can change policy drastically

    • The status quo is hard to move
    • There are lots of decision makers and “veto points”

  • The more heavily voters discount candidates’ positions, the more the discounting model looks like a directional model