# Geometry, Dimensions, and Elections

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I found this to be an interesting way to ponder the theory of elections and group decision-making, so I’m writing to share. I have not done the research to become aware of what is previously known in this area, and I make no claim that any of the thoughts contained here are new.

It’s common in the United States to approximate political opinions using a spectrum from “left” to “right”, where the left end of the spectrum represents an emphasis on social justice, and the right an emphasis on free markets and traditional values. Libertarians, on the other hand, are famous for advocating their view that politics are better described by two orthogonal dimensions, as epitomized by David Nolan in his Nolan Chart. Leaving aside a bunch of details, the idea of the chart is that an individual’s political opinions can be approximately described by a point in a two-dimensional space.

There are many legitimate criticisms of the specific choice of dimensions in the Nolan Chart, but it does capture a first step toward the perspective that interests me here. Most generally, we can consider each individual’s political opinions as living in an infinite-dimensional space. However, such a space can then approximated by its projection down to however many dimensions are convenient for a particular purpose, with a corresponding loss of information as the number of dimensions gets smaller.

This has a fascinating interaction with Condorcet’s paradox. If you’re not familiar with the name, Condorcet’s paradox refers to a phenomenon described by the Marquis de Condorcet in the 18th century.

Condorcet’s Paradox: If each member of a group has consistently ordered individual preferences among three or more options, it is nevertheless still possible that the collective preferences of the group are cyclic. That is, a majority of the group may prefer option A to option B, a majority may prefer option B to option C, and yet a majority may also prefer option C to option A. Cycles are possible of any length greater than or equal to three.

For example, let’s think about an election for book club president, with three candidates: Alice, Bob, and Camille. We will write A>B>C to indicate that a member of the club prefers Alice as their first choice for president, followed by Bob, and finally Camille as their last choice. In all, there are six possible preference orders among the three candidates: A>B>C, A>C>B, B>A>C, B>C>A, C>A>B, and C>B>A. As Condorcet’s paradox predicts, there may be cycles in the overall preferences of the book club. For example, if 10% of members prefer A>B>C, 35% prefer A>C>B, 45% prefer B>C>A, and the remaining 10% prefer C>A>B, then you can verify that 55% of club members prefer Alice over Bob, 55% of voters prefer Bob over Camille, but 55% also prefer Camille over Alice!

This is quite inconvenient, because it means that in many elections, it’s possible for there to be no clear winner at all. But how does it relate to the dimensionality of political preferences?

Well, let’s assume for the sake of argument that political opinions are one-dimensional. I’ll describe the opinions as “left” or “right”, but the specific choice of dimension doesn’t matter. In such a model, the only question is how far left or right is optimal. Every voter would have a preference. Maybe it’s left-wing. Maybe it’s center-right. We won’t be concerned with which specific opinions the voter holds on an issue-by-issue basis, because in this world those are completely determined by just measuring how far left or right their opinions are. A voter’s preference among candidates is determined by how far each candidate is from that voter’s preferred political position.

Here are three candidates, as well as the ranges of voters who will express each possible preference. The dotted lines mark the midpoints between each of the three candidate pairs.

You may notice two of the six possible preference orders are missing. Voters never prefer A>C>B or C>A>B, because there is simply no position along the one-dimensional left-right axis that is closer to both A and C than it is to B. Because of this, there is also no possibility of a Condorcet cycle among these candidates. Indeed, if either A or C are preferred over B, it can only be because they are the first choice of a majority of voters, so they are preferred over any alternative.

We can go even further in this case: except for exact ties, the one unique candidate who will be preferred over all others by a majority of voters (possibly a different majority for each head-to-head contest, though!) will be the first choice of whichever voter has the median political preference among all the voters. However, I don’t see how to naturally generalize this observation to higher dimensions.

It is considering a second dimension that reveals the possibility of a Condorcet paradox among voters’ true preferences. That’s because the additional dimension lets candidates A and C have similarities that are not shared by B. With a second dimension, voter preferences are divided into areas, like this.

If you project this image onto the x axis, the candidates are the same as in the previous model. However, here we’ve added a second dimension, the y axis, in which candidate B differs significantly from A or C. There are now regions of voter preferences in which it is sensible to express candidate orderings A>C>B and C>A>B, restoring the possibility of a Condorcet paradox. Of course, we didn’t create Condorcet’s paradox by choosing to use a two-dimensional model. In a real-world scenario, voters would have expressed the preferences A>C>B and C>A>B anyway. A one-dimensional model would have to reject those voters as behaving irrationally, but a second dimension can explain them.

Similarly, suppose we add a fourth candidate into the two-dimensional model. We might see something like this:

There are 24 possible candidate orderings among 4 candidates, but only 18 of them appear here. Of the eighteen, 12 are open-ended regions around the outside of the diagram that include extreme positions, while the other 6 are bounded regions that sit strictly between the others as a kind of compromise or centrism. 6 more orderings, though, are missing from the diagram entirely! That is because, like before, the model has too few dimensions to recognize how a voter could adopt one of those preferences. In this case, the six missing preferences are D>A>B>C, D>A>C>B, A>D>C>B, A>D>B>C, D>C>A>B, and D>B>A>C. (Curiously, these are precisely the opposite preferences of the six bounded areas. The same thing occurs in the one-dimensional model, where the two unrepresentable orderings were the opposite preferences for the two bounded regions of the spectrum.)

You can see, then, that a two-dimensional model such as a Nolan Chart may be more expressive than a one-dimensional model, but still fails to capture some voter preferences (and this is entirely setting aside the question of whether the Nolan Chart in particular chooses the best pair of dimensions to consider). Beyond 2 dimensions, it’s more difficult to visualize, but the same things should occur. As more candidates are added, more dimensions will be needed to explain the various preferences voters may have.

There’s definitely some hand-waving involved in the above. The most obvious example is the notion of “distance” that is assumed to accurately determine a voter’s candidate preference. In my models, I used a Euclidean distance. In reality, each voter, in addition to having their own ideal candidate as a point in the space, may also have a different metric expressing how important each dimension is to them. These concerns can be dismissed as just another example of how “all models are wrong”, but this one would need some kind of validation to rely on it for real quantitative predictions. I don’t mean it that way; only as a framework for thinking about what can happen when you apply low-dimensional reasoning to what’s ultimately a high-dimensional concept.

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Software engineer, volunteer K-12 math and computer science teacher, author of the CodeWorld platform, amateur ring theorist, and Haskell enthusiast.