Maps of candidates from coconsideration

G. Elliot Morris posted this embedding of the current Democratic presidential candidates in R^2 on Twitter:

where the edge weights (and thus the embeddings) derive from YouGov data, which for each pair of candidates (i,j) tell you which proportion of voters who report they’re considering candidate i also tell you they’re considering candidate j.

Of course, this matrix is non-symmetric, which makes me wonder exactly how he derived distances from it. I also think his picture looks a little weird; Sanders and Bloomberg are quite ideologically distinct, and their coconsiderers few in number, but they end up neighbors in his embedding.

Here was my thought about how one might try to produce an embedding using the matrix above. Model voter ideology as a standard Gaussian f in R^2 (I know, I know…) and suppose each candidate is a point y in R^2. You can model propensity to consider y as a standard Gaussian centered at y, so that the number of voters who are considering candidate y is proportional to the integral

$\int f(x) f(y-x) dx$

and the voters who are considering candidate z to

$\int f(x) f(y-x) f(z-x) dx$

So the proportions in Morris’s table can be estimated by the ratio of the second integral to the first, which, if I computed it right (be very unsure about the constants) is

$(2/3) \exp(-(1/12) |y-2z|^2$.

(The reason this is doable in closed form is that the product of Gaussian probability density functions is just exp(-Q) for some other quadratic form, and we know how to integrate those.) In other words, the candidate y most likely to be considered by voters considering z is one who’s just like z but half as extreme. I think this is probably an artifact of the Gaussian I’m using, which doesn’t, for instance, really capture a scenario where there are multiple distinct clusters of voters; it posits a kind of center where ideological density is highest. Anyway, you can still try to find 8 points in R^2 making the function above approximate Morris’s numbers as closely as possible. I didn’t do this in a smart optimization way, I just initialized with random numbers and let it walk around randomly to improve the error until it stopped improving. I ended up here:

which agrees with Morris that Gabbard is way out there, that among the non-Gabbard candidates, Steyer and Klobuchar are hanging out there as vertices of the convex hull, and that Warren is reasonably central. But I think this picture more appropriately separates Bloomberg from Sanders.

How would you turn the coconsideration numbers into an R^2 embedding?

6 thoughts on “Maps of candidates from coconsideration”

1. Your viz more accurately conveys the separation between Bernie and Bloomberg but is perhaps less useful for conveying overlap among all candidates. (Full disclosure: I support Warren.)
The Economist recently did a ranked-choice viz that shows Warren narrowly losing to Biden, although the underlying data is older and their assumptions about how multiple iterations play out is questionable.
https://www.economist.com/graphic-detail/2020/02/01/under-ranked-choice-voting-left-wing-purism-would-aid-joe-biden

2. There’s also been a decent amount of punditry and polling lately indicating that voters *don’t* vote in lanes, so separation by ideological position might be extraneous to voter preference.

3. JSE says:

What does it mean to say “voters don’t vote in lanes” — just that they are not taking candidate ideologically strongly into account?

4. Shecky R says:

yes, “voters don’t vote in lanes” I think means, or acknowledges, that voters verrry much vote “viscerally” from a sheer “gut feeling” or gut reaction to a candidate’s personality/voice/mannerisms/height etc. (having little to do with issues or policy stances — and even voters who say they care about issues, generally have only 1-2 issues that personally override all others). That’s just the sad state of the electorate.

5. Mark Meckes says:

That matrix is a nice example of a similarity matrix, as discussed for example here. Probably somewhere out there are some standard tricks for visualizing similarity matrices via distances in R^2, but searching is tricky due to the fact that all the words involved get used to mean lots of unrelated things.

6. Re “lanes”: It’s pretty well established that policy plans don’t have a big impact on how voters make their decisions but factors like how voters identify *themselves* do. So what Shecky R said, plus the fact that a large chunk of voters don’t think of themselves as “left” or “right,” so they don’t make decisions based on where candidates fall on the left-right spectrum. Lots of voters are undecided between Bloomberg and Sanders or Biden and Sanders.