## What does it mean to overlap?

The ActiVote Political Matrix provides a visually appealing 2-dimensional view of where people are on the Political Spectrum. Generally speaking: the closer two people are in the Political Matrix, the more they will have in common. Thus, as a tool to determine who the most interesting candidates are to research and whether they deserve our vote, starting with those who are closest to you in the matrix is a good rule of thumb.

However, when you look closely you may see that “closeness in the Matrix” and “overlap” are not *perfectly* correlated. For example, let’s look at the following example matrix of a somewhat left-leaning centrist Independent (“YOU”) in relation to the National Figures in the Political Matrix in July 2023:

In this Political Matrix, lots of overlap numbers seem to make a lot of sense: the 5 Republicans and 1 Libertarian on the right hand side of the matrix have relatively low overlap (between 26% – 40%) as they are furthest away from this user’s green dot. Also: Bernie Sanders, Howie Hawkins (hidden behind Bernie) and Gavin Newsom, seem to be a bit more progressive than this left-leaning Independent, showing overlap percentages between 68% – 76% which are lower than some of the other Democrats. Then, Secretary Pete Buttigieg is closest and also has the highest overlap with 84%, while Kamala Harris, a bit further away has a slightly lower overlap with 81%. And Lisa Murkowski, a moderate Republican is about as far away from the green circle as Gavin Newsom, and scores the same overlap with 68%. So, for 12 out of 15 politicians the overlap percentages seem to make sense.

However, there are three outliers: Joe Biden and Andrew Yang are seemingly far away in the Matrix, but score just as high as Pete Buttigieg with 84%, while Joe Machin is further away than Lisa Murkowski, but scores a higher overlap with 78%. What’s up with these unexpected high overlap scores?

I will get to that in detail in the rest of this blog, but the short answer consists of two explanations:

(1) The Horizontal Axis is more important than the Vertical Axis (so, people who are closer horizontally are more likely to have a higher overlap than those who are close vertically, but not so much horizontally).

(2) It is *mathematically impossible *to place all people in the matrix in such a way that people who are closer *always* have higher overlap percentages than people who are farther away. The best we can do is get “pretty close”, as the example shows.

In practice, look at those people close to you, but also look at those that may seem a bit further away, but are close to you horizontally.

# Detailed Explanation (wonky)

Political opinions are not 1-dimensional or 2-dimensional: they include many dimensions, covering Economic Opportunity, Justice Reform, Healthcare, Individual Rights, Education, Environment, Democracy, Foreign Policy, National Security, Taxes, and much more. Actually, within ActiVote we have **18 **such** **policy categories, and for each policy category we have 20 or more questions that help determine where everyone stands on those policy categories.

Although opinions on some topics make it more likely that you can predict someone’s position on other positions, almost every possible unique combination of opinions is held by someone. That is why we measure for every ActiVote user, for each of these 18 categories where they fall on that policy category, ranging from “progressive” to “conservative”. That gives us for each of those 18 categories a value ranging from -1 (very progressive) to 1 (very conservative) or anything in between. If we now want to position everyone perfectly in the Political Matrix, we should just show their position as a dot in an “18-dimensional Cube” (which is almost impossible to imagine for us humans living in just 3 spatial dimensions). Also, the screen of our smartphones is only 2-dimensional, so we should somehow simplify that 18-dimensional picture to a 2-dimensional picture. Reducing dimensions in a representation is possible, but in the process we will lose some detail and it will therefore be imperfect.

Instead of jumping into reducing an 18-dimensional representation to a 2-dimensional representation, let’s look at a simplified challenge where we try to map a 2-dimensional picture (square) on a 1-dimensional (line) political spectrum, to see how reducing dimensions works in general, and why it can be imprecise.

In this example we assume that we have only two categories of policy positions: “Social” positions (vertical axis in our square) and “Economic” positions (horizontal axis in our square). In the picture we have shown 3 politicians (A, B and C) and where they fall on these topics. A and B agree on economic policy, but disagree on social policy, while A and C agree on social policy, but disagree economically. A and C are polar opposites on both topics. Suppose now that we could not use a 2-dimensional picture (square) to display this, but instead had to put them on a line (1-dimensional). Fortunately, there would seem to be a good solution:

We could create a single Social/Economic Axis and show that B and C are polar opposites by putting them on the extreme ends. The overlap that A has with each of them on 1 of the two categories puts them in the middle. Thus, everything seems perfect: B and C overlap 0%, while A overlaps 50% with both B and C and the picture reinforces that belief.

Now we would like to place “YOU” (the green dot in the middle of the Social/Economic square) somewhere on this line. Clearly, “YOU” overlaps with all three candidates equally: as a moderate in both areas, “YOU” overlaps 50% with A, 50% with B and 50% with C. But where should we place “YOU”? There is no good place we can find. In the middle between B and A would show a similar distance with both of them, but the overlap with C, much farther away on the line, would then surprisingly be the same. Placing “YOU” in the middle, right where A is, would correctly show that “YOU” are 50% away from B and C, but then it would be hugely surprising that the overlap with A is not 100% (same position), but is also just 50%. In this case there is no perfect solution: the imperfection resulting from trying to visualize a 2-dimensional picture in a simplified 1-dimensional representation.

Things get even worse if we would have a fourth politician on the 4th corner of the square, who opposes everything that A stands for. If we want to show A, B, C and D on a single line and then “YOU” equidistant to all four: we have an impossible problem on our hands again.

With tens of thousands of politicians in our Political Matrix and hundreds of thousands of ActiVote users, it is now easier to understand that it is equally impossible to plot everyone in such a way that for *everyone* the overlap on the 18 policy categories exactly fits what you would expect from the visual distance in the 2-dimensional Political Matrix.

Fortunately, data scientists have come up with several methods to find a reasonably good practical mapping anyway. One of those methods is called “Principal Component Analysis” (PCA) which is what ActiVote uses to create our Political Matrix. And while the results are necessarily imperfect, if we look at the example we started with, where 12 of 15 politicians looked great, and the other 3 were easy to explain, we think that the Political Matrix does a great job in showing where “YOU” stand relative to politicians and by zooming in on overlap percentages you can get extra information to help you determine who you may wish to support.

More importantly, PCA is *devoid of any political bias*. All calculations are done completely automatically and the best possible overall mapping in the Political Matrix is created. Therefore, whether someone ends up closer to Secretary Pete Buttigieg (as the example user in the matrix earlier in this blog), or ends up closer to anyone else, is not biased by any user, politician or party who would perhaps like their candidate to be close to many people. As a nonpartisan app, informing people without bias is of the utmost importance.

Want to find out where you fall in our Political Matrix? We encourage you to give it a try!