## The Spatial Theory of Voting

### Introduction

I have always been fascinated by the idea of simulating social phenomena. In my youth I would spend hours playing President Elect, Balance of Power, SuperPower, and other games that purported to simulate political and economic phenomena. I have recently come across a mathematical model of voting, called the spatial theory of voting, which I shall discuss here.

### Ideological Spaces

#### The One-Dimensional “Political Spectrum”

We often speak in terms of the political spectrum, which we think of as a continuum along which we may identify a person’s ideology. On the right, we have right-wing ideology, and on the left we have left-wing ideology, and in the middle, we have centrist ideology. This continuum has certain properties. For example, we hear things like, “he’s to the left of Ted Kennedy,” or, “she’s more conservative than Ronald Reagan.” This implies that on this continuum we may rank ideologies as being “greater” or “lesser” than others, or more “rightward” or “leftward” than others—that is, given two ideologies $a$ and $b$, $a \neq b$, we can say that $a > b$ or $% $—without making any normative judgment, though people often use such rankings to make normative judgments.

#### Two-Dimensional Nolan Space

Careful thought, however, reveals that the arrangement of ideologies might be more complicated than placement on a one-dimensional continuum. This is the motivation of the Nolan chart, which places ideological positions in a two-dimensional space which I call Nolan space. One axis represents a person’s economic freedom “score,” while the other represents the person’s personal freedom “score.” Regions of the Nolan space correspond to general ideologies.

#### The General Case: $n$-Dimensional Space

There is no reason we cannot measure a person’s ideology with respect to more than two measures. We can therefore work with higher-dimensional spaces, though visualization may be more difficult for four or more dimensions.

A person’s ideology is therefore an $n$-vector $\mathbf{p}$ in the ideological space represented by $\mathbb{R}^n$. The Euclidean distance between two vectors $d(\mathbf{p}_1, \mathbf{p}_2) = \sqrt{(\mathbf{p}_1 - \mathbf{p}_2)^\top (\mathbf{p}_1 - \mathbf{p}_2)}$ is the “distance” between two ideologies and can be a measure of the similarity of two ideologies. A weighting matrix $\mathsf{W}$ can be added to produce a weighted distance

which allows us to specify the importance to which an elector gives each element of the elector’s ideological preference. When $\mathsf{W} = \mathsf{I}$, then the distance reduces to the un-weighted Euclidean distance.

For the purposes of this discussion, we will restrict ourselves without loss of generality to Nolan space.

### Elections

#### Electors and Candidates

An elector (or voter) is a rational agent that has a “bliss point” $\mathbf{p}$ in the ideological space that represents that elector’s ideological preferences. A candidate is a person who is up for election by the electors. A candidate has an ideology $\mathbf{c}$ in the ideological space. Let $C = \{ \mathbf{c}_1, \mathbf{c}_2, \dots, \mathbf{c}_i \}$ be the set of candidates, represented by their ideologies, in a particular election. An elector with ideological preferences $\mathbf{p}_i$ has a utility $u_{\mathbf{p}_i}(\mathbf{c}_j)$ of voting for a particular candidate with ideology $\mathbf{c}_j$. This utility function $u : \mathbb{R}^n \to \mathbb{R}$ is monotonically increasing as the Euclidean distance $d$ decreases. A rational, ideological elector will vote for that candidate whose ideological position will give that elector maximum utility:

Note that if $d = 0$, then $\mathbf{p}_i = \mathbf{c}_j$, and $u_{\mathbf{p}_i}(\mathbf{c}_j)$ is at a maximum. Thus there would be no other candidate than that represented by $\mathbf{c}_j$ that gives the elector more utility. This implies that, in elections in which candidates are also electors, rational candidates should always vote for themselves.

#### The Electoral Zone

With the foregoing utility function, we see that each candidate will command an electoral zone in which each elector whose ideological preference is in the zone will have maximum utility of voting for that candidate. These zones are, in fact, Voronoi tessellations of the ideological space. For first-past-the-post elections, that candidate whose electoral zone encompasses the most ideological positions of the electors is the winner of the election. Figure 1. A three-candidate election in Nolan space and each candidate's electoral zone. In this election, candidate A might be a center-right candidate, candidate B a center-left candidate, and candidate C a classical liberal candidate.

#### Distribution of Electors

The distribution of electors in the ideological space is probably not uniform, especially when the number of electors is large. In American political parlance, there are some “red states” and some “blue states,” and within states, there are “red” and “blue” counties and townships. This implies that we could treat the distribution of electors as a probability distribution. In a particular election, the electorate could be distributed multivariate-normally or some other way.

Economists Antonio Merlo and Aureo de Paula at the University of Pennsylvania, in their paper “Identification and Estimation of Preference Distributions When Voters are Ideological,” assert that by studying election results, one can estimate the ideological distribution of voters in an election. But suppose one already knows (or suspects) the distribution of the electorate. This spatial model would make it possible to predict—or at least simulate—electoral outcomes. This would be a matter of creating a Voronoi tessellation of the ideological space and “counting” the number of electors in each electoral zone, according to the distribution of electors.

### Campaigning

Viewed in the light of the spatial model, campaigning and electioneering can be seen as the efforts of candidates to maximize the number of voters in their electoral zones. This can be achieved either through convincing the electorate to move their ideological bliss points into the candidate’s electoral zone, or by moving the candidate’s ideological position so as to acquire a greater number of electors, which has echoes of the median voter theorem.

### Is This Worth Anything?

This model of voting is all nice and neat, but is it actually useful in describing reality? Let’s revisit some of its assumptions.

• Electors are ideological. Do electors always vote ideologically? Might they not vote for other reasons? Is it realistic to have a utility function that numerically describes an elector’s utility of choosing a particular candidate? Are electors’ utilities really cardinal and not just ordinal? Isn’t value subjective? Is it valid to compare two electors’ subjective utilities? Is it realistic to assign numerical values to ideologies and place them in a metric space?

• Electors have good information about their own ideological preferences and those of the candidates. Can each elector accurately “calculate” the elector’s distance in the ideological space to each candidate, or at least to the nearest candidates?

• All electors vote. This is clearly false in the real world. Could we modify this model to capture the fact that many electors do not vote? Would it matter if we could?

### Conclusion

This model may be predictively good in certain electoral situations, and it may be helpful in determining voter distributions, so long as the assumptions hold. But voting is primarily a matter of human action, which is notoriously difficult to predict accurately. But for less-serious, simulative purposes, this model might be useful for quasi-realistic prediction and simulation, such as for political simulation games.